Nonlinear Waves: From Dissipative Solitons to Magnetic Solitons 9811967431, 9789811967436

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Emmanuel Kengne WuMing Liu

Nonlinear Waves From Dissipative Solitons to Magnetic Solitons

Nonlinear Waves

Emmanuel Kengne · WuMing Liu

Nonlinear Waves From Dissipative Solitons to Magnetic Solitons

Emmanuel Kengne Zhejiang Normal University Jinhua, China

WuMing Liu Institute of Physics Chinese Academy of Sciences Beijing, China

ISBN 978-981-19-6743-6 ISBN 978-981-19-6744-3 (eBook) https://doi.org/10.1007/978-981-19-6744-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

One of the authors, Emmanuel Kengne, dedicates this book to the memory of his parents, Papa François Mtopi and KUA Marie Djomou, who passed away, respectively, on October 1980 and September 2021. For if he had not believed that they would have wished him to give such help as he could toward making their life’s work of service to mankind, he should never have been led to co-author this book.

Preface

Mathematical modeling is a main key for understanding and investigating diverse phenomena in both linear and nonlinear systems. When nonlinear phenomena are governed by nonlinear partial differential equations (PDEs), it is important to obtain their exact analytical solutions if they exist, otherwise, to obtain the numerical solutions. Therefore, explorations of exact solutions of nonlinear PDEs are turned out to be a charming and challenging area of research for mathematicians, theoretical physicists, and research communities for many years because they are widely playing a significant role in the study of nonlinear physical phenomena in applied physics and mathematical studies with essential applications in several areas of engineering and natural science including fluid mechanics, chemistry, thermodynamic, physics, electromagnetism, bio-mathematics, mathematical physics, and vice versa. This is why it is very important to develop different analytical and numerical methods for finding either exact analytical or numerical solutions of nonlinear PDEs which describe nonlinear phenomena. This book focusses both on the mathematical modeling of nonlinear phenomena in nonlinear electric transmission networks and on the study of dynamics of both nonlinear modulated waves and matter waves in nonlinear systems. One of its main aims is, through exact analytical solutions of nonlinear PDEs, to present methodically several methods to engineer dissipative and magnetic nonlinear waves such as dissipative and magnetic solitons (MSs) propagating in nonlinear systems. In soliton theory, the development of dissipative, non-dissipative solitons as well as soliton signals might be a consequence of the nonlinear, dispersive, and diffusive results, whereas a soliton is also called solitary or soliton-like wave that maintains its shapes, velocity, and amplitude with the impact of others solitons. The book consists of two parts and thirteen chapters. Part I, titled “Engineering Nonlinear Modulated Waves in Nonlinear Transmission Networks”, consists of five chapters and investigates the generation and the transmission of nonlinear modulated waves, in general, and solitary/solitonlike waves in particular in lossless and in lossy electric nonlinear transmission networks (NLTNs). Part I of the book is composed of two parts, which deal with, respectively, one-dimensional (1D) and two-dimensional (2D) NLTNs. For both 1D and 2D NLTNs, we consider both lossless and dissipative

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NLTNs. The respective models are based on either 1D or 2D conservative and dissipative nonlinear Schrödinger (NLS) equations, derivative NLS equations cubic and cubic-quintic complex Ginzburg-Landau (CGL) equations, Kundu–Eckhaus equation, Chen-Liu and generalized Chen–Lee–Liu equations and cubic-quintic CGL equations with derivative terms. Chapter 2 deals with the NLS models for solitons propagation in 1D lossless nonlinear transmission networks, while Chap. 3 treats the generation of dissipative solitons in one-dimensional NLTNs. In Chaps. 4 and 5, we focus our attention, respectively, to the dynamics of rogue waves in 1D NLTNs and to the transmission of nonlinear modulated waves in multi-component nonlinear transmission networks. Part II of this book titled “Dynamics of Matter-Wave and Magnetic-Wave Solitons” is composed of eight chapters. In this part of the book, we develop basic theoretical results for the dynamics matter-wave and magnetic-wave solitons of nonlinear systems in general and of Bose–Einstein condensates (BECs) in particular, as well as of optical fields trapped in external potentials, combined with the time-modulated nonlinearity. The respective models are based on one-, two-, and three-component non-autonomous Gross–Pitaevskii equations or NLS equations with external potentials. Chapter 7 addresses the dynamics of one-component condensates with time modulation of the scattering length and trapping potential. Here, the approach outlined above leads to the construction and management of non-autonomous solitons of 1D cubic self-defocusing NLS equations with spatiotemporally modulated coefficients that may be transformed into the classical integrable NLS equation. In this setting, matter-wave soliton solutions are constructed in an analytical form, and it is shown that the instability of those solitons, if any, may be delayed or completely eliminated by varying the nonlinearity’s strength in time. In this Chap. 7, we also engineer non-autonomous matter-wave solitons in BECs with spatially modulated local nonlinearity and a time-dependent harmonic-oscillator (HO) potential. The modulational instability in that setting is considered too. Chapter 8 deals with the dynamics of matter rogue waves in BECs trapped in time-dependent external potential. Chapter 9 treats the soliton generation in multi-component Bose–Einstein condensates. In Chap. 10, we proceed to more general non-integrable models, which are treated by means of the semi-analytical variational approximation and direct numerical simulations. In this case, we address the dynamics of two- and three-dimensional condensates with the nonlinearity strength containing constant and harmonically varying parts, which can be implemented with the help of AC magnetic field tuned to FR. In particular, the spatially uniform temporal modulation of the nonlinearity may readily play the role of an effective trap that confines the condensate and sometimes enforces its collapse. Chapter 11 deals with the investigation of soliton states in a model based on a set of three coupled GP equations modeling the dynamics of the spinor BEC. Both integrable and non-integrable versions of the system are considered, and exact soliton solutions are demonstrated. The stability of the solutions was checked, in most cases, by direct simulations, and, in some cases, it was investigated in a more rigorous form, based on linearized Bogoliubov–de Gennes (BdG) equations for small perturbations. The motion of bright and dark matter-wave solitons in 1D BEC in the presence of spin-orbit coupling (SOC) is also treated in

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Chap. 11. Here, we demonstrate that the spin dynamics of the SOC solitons are governed by a nonlinear Bloch equation and affects the orbital motion of the solitons, leading to SOC effects in the dynamics of macroscopic quantum objects. The models addressed in Chap. 11, as well as the applied methods and the obtained results, are quite similar to those produced by the engineering. In particular, the macroscopic SOC phenomenology is explained by the fact that an effective timeperiodic force produced by rotation of the soliton’s (pseudo-) spin plays the role of temporal management which affects the motion of the same soliton. Chapters 12 and 13 deal with the dynamics of magnetization in ferromagnet with or without spin-transfer torque. This research book, which accurately treats the mathematical modeling of nonlinear phenomena and presents methodically several methods for engineering the propagation of nonlinear waves in nonlinear systems, is suitable for physicists, mathematicians, engineers, as well as for graduate and postgraduate students from schools of mathematics, physics, network, and information engineering. Jinhua, China Beijing, China

Emmanuel Kengne WuMing Liu

Acknowledgements

Nobody has been more important to Prof. Emmanuel Kengne of the Zhejiang Normal University (China), one of the authors, in the pursuit of the multiple projects leading to this book than the members of his family. He wishes to thank his loving and supportive wife, Eleonore Nkuojip Sado, and his three wonderful children, Kengneson Delma Djomo, Kengneson Weierstrass Owan Wambo, and Kengneson Cris-Carelle Djike all at Gatineau-Quebec-Canada, whose love and guidance are with him in whatever he pursues. They are the ultimate role models. This book was supported by the Chinese Academy of Sciences PIFI under the grant No. 2023VMA0019, National Key R&D Program of China under grants No. 2021YFA1400900, 2021YFA0718300, 2021YFA1402100, and NSFC under grants Nos. 11835011, 61835013, 12234012, Space Application System of China Manned Space Program.

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Part I

Engineering Nonlinear Modulated Waves in Nonlinear Transmission Networks

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Nonlinear Schrödinger Models for Solitons Propagation in 1D Lossless Nonlinear Transmission Networks (NLTNs) . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Standard Nonlinear Schrödinger Equations for Modulated Waves Propagation in a Lossless Electrical Transmission the Network: Effects of the Dispersive Element C S . . . . . . . . . . . . 2.2.1 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Bright and Dark Single-Solitary Waves Propagating in the 1D Lossless Network of Fig. 2.1 . . . . 2.3 Modulational Instability and Transmission of Chirped Femtosecond Signals Through the Lossless Electrical Network of Fig. 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Modulational Instability and Evolution of Chirped Femtosecond Solitary Signals Embedded on a Non-vanishing CW Background . . . . . . . . . . . . . . . . 2.3.3 Sister Femtosecond Nonlinear Modulated Waves with Two Nonlinear Chirp Terms . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Transmission of Dissipative Solitonlike Signals Through One-Dimensional Transmission Networks . . . . . . . . . . . . . . . . . . . . . . . 3.1 Chirped Lambert W-Kink Waves Propagation in a Lossy Electrical Transmission Network . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Amplitude Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 5 7 7

8 9 13

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23 30 45 47 47 47 49

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3.1.3 3.1.4

Baseband Modulational Instability Analysis . . . . . . . . . . 54 Evolution of Chirped Lambert W-Kink Pulses in the Network of Fig. 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2 Spatiotemporal Modulation of Damped Solitonlike Wave Signals in a Nonlinear RLC Transmission Network . . . . . . . . . . . . 67 3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2.2 Linear Dispersion Relation and Spatial Decreasing Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2.3 Amplitude Equation and Modulated Damped . . . . . . . . . 72 3.2.4 Linear Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.2.5 Spatiotemporal Modulated Signals Propagating Through the Network of Fig. 3.8 . . . . . . . . . . . . . . . . . . . . 80 3.2.6 Transmission of Spatiotemporal Modulated Damped Envelope Signals Through a Lossy Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.3 Modulated Wavetrains in a Dissipative Bi-Inductance Transmission Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.3.1 Introduction and Circuit Equations . . . . . . . . . . . . . . . . . . 93 3.3.2 Generalized Cubic-Quintic Complex Ginzburg-Landau Equation for a Lossy Network . . . . . . 94 3.3.3 Linear Analysis and Modulational Instability . . . . . . . . . 96 3.3.4 Coherent Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4

Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Emission of Rogue Wave Signals Through the Modified Noguchi Electric Transmission Network . . . . . . . . . . . . . . . . . . . . . 4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Modulated Waves and Linear Analysis . . . . . . . . . . . . . . . 4.1.3 Construction of Rogue Waves in the Lossless Network Under the Condition P Q > 0 . . . . . . . . . . . . . . 4.1.4 Emission of Rogue Wave Signals Through the Network of Fig. 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Conclusion and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Generation of Network Modulated Rogue Waves Under the Action of the Quintic Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 The Kundu–Eckhaus Model for Non-autonomous Modulated Rogue Waves in a Lossless Electric Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Generation of First-Order Non-autonomous Modulated Rogue Waves (Alias Peregrine Solitons) for a Lossless Electric Network . . . . . . . . . . . . . 4.2.3 Conclusion and Discussions . . . . . . . . . . . . . . . . . . . . . . . .

109 109 109 110 113 118 124 124

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Chirped Super Rogue Waves Propagating Along a Lossless Nonlinear Electrical Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Generalized Nonlinear Schrödinger Equation for a Lossless Electric Network . . . . . . . . . . . . . . . . . . . . . 4.3.2 Phase Engineering Chirped Super Rogue Waves for a Lossless Electric Network . . . . . . . . . . . . . . . . . . . . . 4.3.3 Computational/Numerical Simulations . . . . . . . . . . . . . . . 4.3.4 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5

Emission of Nonlinear Modulated Waves in Multi-coupled Nonlinear Transmission Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Transmission of Solitonlike Wave Signals in a Two-Dimensional Lossless Dispersive Nonlinear Transmission Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Description of the and Network Equations . . . . . . . . . . . . 5.1.2 Amplitude Equation for the Dynamics of Modulated Waves in the Two-Dimensional Lossless Electric Network . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Computational Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Coherent Structures for a Multi-coupled Nonlinear Transmission Network with Dissipative Elements . . . . . . . . . . . . . 5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Amplitude Equation for a Two-Dimensional Lossy Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Modulational Instability in a Two-Dimensional Lossy Transmission Network . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Coherent Structures for a Two-Dimensional Lossy Nonlinear Transmission Network . . . . . . . . . . . . . . . . . . . 5.3 Soliton Signals in an Alternate Right-Handed and Left-Handed Multi-coupled Lossy Nonlinear Transmission Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Description of the Model and Basic Equations . . . . . . . . 5.3.2 Complex Ginzburg-Landau Equation for the Dynamics of Modulated Waves in a Multi-coupled Lossy Network . . . . . . . . . . . . . . . . . . 5.3.3 Linear Stability Analysis: Modulational Instability of Stokes Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Dissipative Effects on Electrical Modulated Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

138 139 145 153 156 159 163

163 163

165 167 169 170 172 173 177

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183 189 191 209 210

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Part II

Dynamics of Matter-Wave and Magnetic-Wave Solitons

6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

7

Dynamics of One-Dimensional Condensates with Time Modulation of the Scattering Length and Trapping Potential . . . . . . 7.1 Non-autonomous Solitons in Bose-Einstein Condensates with a Spatially Modulated Scattering length . . . . . . . . . . . . . . . . . 7.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Cubic Inhomogeneous Nonlinear Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Matter-Wave Solitons in an Inhomogeneous Nonlinear Schrödinger Equation with the Spatiotemporal HO Potential . . . . . . . . . . . . . . . . 7.1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Soliton Management (SM) in One-Dimensional Bose-Einstein Condensates with Two- and Three-Body Inter-atomic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Modulational Instability in Bose-Einstein Condensates Trapped in a Spatiotemporal-Dependent Dissipative Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Soliton Management in Bose-Einstein Condensate Systems with Two- and Three-Body Interactions Trapped in a Spatiotemporal-Dependent Dissipative Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Conclusion and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Chirped Solitons and Chirped Double-Kink Solitons in Bose-Einstein Condensates with Time-Varying Atomic Scattering Length in a Complicated External Potential . . . . . . . . . 7.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Derivation of the Higher-Order Nonlinear Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Dynamics of Chirped Femtosecond Solitons and Chirped Double-Kink Solitons in Bose-Einstein Condensates with Time-Dependent Atomic Scattering Length in a Complex Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Conclusion and Discussions . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8

9

Rogue Matter Waves in Bose-Einstein Condensates Trapped in Time-Varying External Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Non-autonomous Rogue Matter Waves in Bose-Einstein Condensates With Dissipative External Potential . . . . . . . . . . . . . . 8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Model and Analytical Exact Firstand Second-Order Rational Solutions of the Gross-Pitaevskii Equation . . . . . . . . . . . . . . . . . . . . 8.1.3 Management of Dissipative Rogue Matter Waves in Bose-Einstein Condensates with Complicated Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Conclusion and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Chirped Rogue Matter Waves in Bose-Einstein Condensates with a Variable Scattering Length in an Expulsive Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Phase Engineering and Chirped Wave Solutions . . . . . . . 8.2.3 Evolution of Chirped Rogue Waves Under the Action of the Time-Dependent Atomic Scattering Length and Parabolic Potential . . . . . . . . . . . . 8.2.4 Conclusion and Discussions . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamics of Matter-Wave Solitons in Multi-component Bose-Einstein Condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Soliton Management in a Binary Bose-Einstein Condensate . . . . 9.1.1 The Model and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Conclusion and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Soliton Stability in Binary Bose-Einstein Condensate Under Temporal Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 The Physical Model and the Lax Pair . . . . . . . . . . . . . . . . 9.2.2 Analytical and Numerical Results for Two-Component Bright Solitons in the Integrable System . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Dynamics of Higher-Dimensional Condensates with Time Modulated Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Dynamics of Two- and Three-Dimensional Bose-Einstein Condensates with Time Modulated Nonlinearities . . . . . . . . . . . . . 10.1.1 The Model and Variational Approximation (VA) . . . . . . 10.1.2 Two-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.3 The Three-Dimensional Gross-Pitaevskii Model . . . . . . . 10.1.4 Conclusion and Discussions . . . . . . . . . . . . . . . . . . . . . . . .

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10.2 Stable Vortex Modes in Two-Dimensional Bose-Einstein Condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Model Description and Main Transformation . . . . . . . . . 10.2.2 Exact Vortex-Soliton Solutions for the Attractive Nonlinearity (g0 < 0) when E = 0 . . . . . . . . . . . . . . . . . . 10.2.3 Exact Analytical Vortex-Soliton Solutions for the Repulsive Nonlinearity (g0 > 0) . . . . . . . . . . . . . . 10.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Engineering Matter-Wave Solitons in Spinor Bose-Einstein Condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Formulation of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Exact Analytical One-, Two-, and Three-Component Soliton Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Single-Component FM Solitons . . . . . . . . . . . . . . . . . . . . . 11.2.2 Single-Component Polar Solitons . . . . . . . . . . . . . . . . . . . 11.2.3 Two-Component Polar Solitons . . . . . . . . . . . . . . . . . . . . . 11.2.4 Three-Component Polar Solitons . . . . . . . . . . . . . . . . . . . . 11.2.5 Multistability of Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.6 Finite-Background Solitons . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The Darboux Transform and Nonlinear Development of Modulational Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Engineering Magnetic Solitons in Nonlinear Systems . . . . . . . . . . . . . 12.1 Dynamics of Dissipative Magnetic Matter-Wave Solitons in a Spinor Polariton Bose-Einstein Condensate . . . . . . . . . . . . . . . 12.1.1 Introduction and Model Equations . . . . . . . . . . . . . . . . . . . 12.1.2 Exact Analytical Solutions and Dissipative Magnetic Polariton Solitons in a Spinor Polariton Bose-Einstein Condensate . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.3 Conclusion and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Nonlinear Magnetization Dynamics of a Classical Ferromagnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Macroscopic Description and Equation of Motion . . . . . 12.2.3 Soliton Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.4 Asymptotic Behavior of Multisoliton Solutions . . . . . . . 12.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Nonlinear Magnetization Dynamics in the Presence of Spin-Polarized Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Long Ferromagnetic Nanowire with a Uniform Cross Section (High-Q Model, Q > 1) . . . . . . . . . . . . . . .

363 364 365 367 370 371 373 373 377 377 378 380 382 384 385 387 393 393 395 395 396

398 403 403 403 404 412 428 430 431 431 432

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12.3.3 Short Ferromagnetic Nanowire with a Uniform Cross Section (Low-Q Model, Q < 1) . . . . . . . . . . . . . . . 438 12.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 13 Current Driven Dynamics of Magnetization in Ferromagnet with Spin Transfer Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Current Driven Dynamics of Domain Wall in Ferrimagnets . . . . . 13.2.1 Domain-Wall Resonance Induced by Spin-Polarized Current . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Screw-Pitch Effect and Velocity Oscillation of Domain Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.3 Non-autonomous Helical Motion of Magnetization in Ferromagnetic Nanowire . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Current Driven Dynamics of Soliton . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Dark Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Bright Soliton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.3 Current Driven Interaction of Spin Wave and Soliton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.4 Dynamics of Magnetic Rogue Wave . . . . . . . . . . . . . . . . . 13.4 Ferromagnetic Resonance in Magnetic Trilayers . . . . . . . . . . . . . . 13.4.1 Ferromagnetic Resonance in a Perpendicular-Analyzer Magnetic Trilayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Stability Analysis of Perpendicular Magnetic Trilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

447 447 448 448 453 458 468 469 473 474 484 490

491 498 505 506

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

About the Authors

Emmanuel Kengne obtained both his Master’s degrees in mathematics and physics and his Ph.D. in Physicomathematical Sciences from the School of Mathematics and Mechanical Engineering at Kharkov State University (now Kharkov National University), Ukraine, on, respectively, July 1991 and March 1994. He also obtained a Master of Education with concentration in Teaching, Learning and Evaluation at the University of Ottawa (Canada) on April 2012. Emmanuel Kengne is an applied mathematician and theoretical physicist, full professor at the School of Physics and Electronic Information Engineering of Zhejiang Normal University (China), adjunct professor at the Department of Computer Science and Engineering of the University of Quebec at Outaouais (Canada), and adjunct researcher at the Institute of Physics, Chinese Academy of Sciences (China). A multidisciplinary researcher, his research plan mainly encompasses four main themes that lead to interdisciplinary collaborations among different fields such as applied mathematics and pure mathematics, nonlinear physics, mathematical physics, electrical and telecommunication engineering, biomedical engineering, computational sciences, nonlinear theory and its applications, and biological sciences. Emmanuel Kengne has made major contributions to a vast number of fields, including the theory of well-posedness boundary value problems for partial differential equations, wave propagation on nonlinear transmission networks, optical and heat solitons, nonlinear dynamical lattices, Ginzburg-Landau models, Boson–Fermion models, bio-thermal physics, light propagation, thermal therapy for tumors, as well as many other physicomathematical fields. WuMing Liu obtained his Ph.D. degree from the Institute of Metal Research, Chinese Academy of Sciences, Shenyang, China, in June 1994. He became an associate professor at the Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, China, in 1996, and is a full professor at the Institute of Physics at the same Academy since 2002. He has served as an editorial board member for several international journals, including Scientific Reports and Journals of Physics.

xxi

Abbreviations and Acronyms

1D 2D 3D AC AM AMI BC BdG equation BEC C.C. CGL equation CLL equation CQ CW DC DM DM DNLS DT DW EODE FP FR GCQ GCQD-NLS GI equation GL equation GLM equation GP equation GVD HBM

One dimensional Two dimensional Three dimensional Alternating Current Ansatz method Azimuthal modulational instability Boundary condition Bogoliubov–de Gennes equation Bose–Einstein condensate Complex Conjugation Complex Ginzburg-Landau equation Chen–Lee–Liu equation Qubic-quintic Continuous wave Direct Current Direct method Dispersion management Derivative nonlinear Schrödinger equation Darboux transform Domain wall Elliptic ordinary differential equation Fixed point Feshbach resonance Generalized cubic-quintic Generalized cubic-quintic derivative nonlinear Schrödinger Gerdjikov–Ivanov equation Ginzburg-Landau equation Gel’fand-Levitan-Marchenko equation Gross–Pitaevskii equation Group velocity dispersion Hirota bilinear method xxiii

xxiv

HO HO IST KdV KE equation KN equation LH lhs LL equation LLG equation LLGS equation MI MRW MS NLS equation NLTN NM NMDs NPDE NSW ODE PDE PIT PT RH rhs RKMK method RW SM SOC SPM SRW SW TS wave VA VS XPM

Abbreviations and Acronyms

Harmonic oscillator Higher order Inverse scattering transformation Korteweg–de Vries Kundu–Eckhaus equation Kaup–Newell equation Left handed Left-hand side Landau–Lifshitz equation Landau–Lifshitz–Gilbert equation Landau–Lifshitz–Gilbert–Slonczewski equation Modulational instability Magnetic rogue wave Magnetic soliton Nonlinear Schrödinger equation Nonlinear transmission network Nonlinearity management Nonlinear magnetization dynamics Nonlinear partial differential equation Nonlinear spin wave Ordinary differential equation Partial differential equation Phase engineering transformation Parity time Right handed Right-hand side Runge–Kutta–Munthe–Kaas method Rogue wave Soliton management Spin-orbit coupling Self-phase modulation Super rogue wave Spin wave Tollmien–Schlichting wave Variational approximation Vortex soliton Cross-phase modulation

Part I

Engineering Nonlinear Modulated Waves in Nonlinear Transmission Networks

Chapter 1

Introduction

Abstract This Chapter introduces the first part of the Book. It presents nonlinear transmission networks in general and, lossless/lossy nonlinear electric transmission networks, in particular. Also, we present here scientific motivations for studying nonlinear excitations in both lossless and dissipative nonlinear transmission networks.

While various nonlinear partial differential equations have been explored in detail, used to model the real systems in different scientific domains, and many solutions of these equations have been found, there are only a few systems where solitons are easily and directly observed in controlled laboratory experiments. Nonlinear transmission networks (NLTNs)—most often, electric transmission lines—are relevant examples of such experimentally available setups [1, 2]. Being discrete systems, they approximate continuum media quite well, as shown in this part of our book. Thus, NLTNs provide a reliable platform for studying the dynamics of excitations in nonlinear media [2]. An electric NLTN is built as a transmission line with periodically inserted varactors, in which the capacitance nonlinearity arises from a variable depletion-layer width, that depends both on the dc bias voltage and ac voltage of the propagating waves. Usually, diodes with reverse-biased capacitance are employed as nonlinear capacitors [3]. NLTNs constructed according to this scheme are very convenient tools to study the wave propagation in one-dimensional (1D) nonlinear dispersive media [4]. A growing interest to the work with NLTNs has been drawn since the pioneering works by Hirota and Suzuki and by Nagashima and Amagishi on a single electrical network simulating the Toda lattice [1, 5–10]. The multiplicative process in NLTNs is understood as a direct consequence of the propagation of modulated waves in this medium. Qualitatively, the origin of nonlinear waves, such as solitons, in NLTNs is explained by the balance between the effect of dispersion, due to the periodic location of capacitors, and nonlinearity due to the voltage dependence of the capacitance [11]. Discrete electrical transmission networks have been used in the design of thinfilm and diffusion resistors, capacitors, and conductors, as well as for evaluation of © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 E. Kengne and W. Liu, Nonlinear Waves, https://doi.org/10.1007/978-981-19-6744-3_1

3

4

1 Introduction

undesirable interaction between different components of integrated circuits. These networks can be successfully used to simulate characteristics of some active microcircuit elements, such as the field-effect transistors [12, 13]. In particular, dissipative transmission networks with a nonlinear load are very important for the design of high-speed electronic circuits [14]. To properly consider increasing frequencies and bit rates, one has to take transmission network effects into account. In particular, these are the signal delay, refection, attenuation, and crosstalk [15]. In this context, nonlinear transmission networks have proven to be of great practical use for the focusing and shaping of extremely wide-band signals (with frequencies from dc to 100 GHz), i.e., controllably changing basic features of incoming signals, such as the frequency content, pulse width, and amplitude, which is usually a hard problem [16]. Further, topoelectric (i.e., topological electric [17], also called “topoelectric" [18]) two-dimensional (2D) circuit networks can be employed to emulate various topological states of matter, including surface and corner modes in topological insulators with a spatially periodic [18–20] and quasiperiodic [21] structure. Interest has also been recently drawn to the creation of non-Hermitian topologically structured electric networks incorporating parity-time (PT ) symmetric or chiral anti-PT symmetric elements [22]. In biomedical engineering, the RLC networks are used for modeling blood flow in biological tissues [23, 24]. In the linear regime, NLTNs can be used as phase shifters in phased antenna arrays, where the time delay can be controlled by means of a dc bias applied to Schottky diodes acting as variable reactance [25]. Under large-signal conditions, NLTNs can serve as impulse compressors or frequency multipliers. Efforts have been made to study, both analytically and numerically, nonlinear excitations in NLTNs. In particular, it has been shown experimentally that modulated waves can exist in such systems [16]. A decade ago, the backward-wave propagation and formation of discrete bright and dark solitons in a nonlinear electrical lattice were addressed [26]. Still earlier, the existence of stable discrete solitons in an electric transmission network emulating an ac-driven lossy Toda lattice was experimentally demonstrated by Kuusela [27]. Those results confirm the stable motion of the solitons at resonant velocities, as predicted in [28] and [29]. The aim of this part of the book is to survey recent and most essential advances in the application of nonlinear PDEs for the generation of nonlinear modulated waves in real NLTNs. We present mathematical models of nonlinear phenomena occurring along NLTNs and produce a study of dynamics of nonlinear modulated waves propagating in these networks, aiming to make it sufficiently complete. For this purpose, essential elements of the derivation of nonlinear PDEs are included, aiming to make the presentation accessible to the general readership. To put the new findings in the context and clarify core application of nonlinear PDEs to NLTNs, we are particularly concerned with topics such as the modulational instability, the generation of envelope and/or hole solitons (dissipative or non dissipative), the evolution and decay of chirped solitons, and dynamics of rogue waves (RWs). It is important to note that research on modulation instability and dynamical behavior of the localized

References

5

wave solutions (as for examples solitons, breathers, and rogue waves) for the nonlinear systems is a fascinating subject in the field of contemporary nonlinear science, which has attracted widespread attention from many experts and scholars. In nonlinear dispersive media, MI is a nonlinear process, in which a continuous plane wave generates amplitude and frequency self-modulation through a nonlinear dispersive medium, resulting in exponential growth of small perturbations superimposed on a plane wave. This part of the book is composed of two parts, which deal with, respectively, single-(1D) and multi-component (2D) NLTNs. In each part, we consider both lossless and the dissipative NLTNs. The respective models are based on either 1D or 2D conservative and dissipative nonlinear Schrödinger (NLS) equations, derivative NLS (DNLS) equation, cubic and cubic-quintic complex Ginzburg-Landau (CGL) equations, Kundu–Eckhaus (KE) equation [30–34], Chen-Liu [35] and generalized Chen-Lee-Liu equation [36], and a cubic-quintic (CQ) CGL equation with derivative terms. The first part of this book, which addresses the dynamics of modulated waves in single (1D) and two-dimensional (alias multi-component) NLTNs, is composed of four Chapters. This chapter deals with the NLS models for solitons propagation in one-dimensional lossless nonlinear transmission networks, while Chap. 2 treats the generation of dissipative solitons in one-dimensional NLTNs. Chapters 3 and 4 deal with respectively the dynamics of rogue waves in 1D NLTNs and the transmission of modulated waves in multi-component nonlinear transmission networks.

References 1. R. Hirota, K. Suzuki, Theoretical and experimental studies of lattice solitons in nonlinear lumped networks. Proc. IEEE 61, 1483 (1973) 2. K. Lonngren, Solitons in Action. eds. by K. Lonngren, A. Scott (New York: Academic, 1978) 3. R. Marquié, J.M. Bilbault, M. Remoissenet, Nonlinear Schrödinger models and modulational instability in real electrical lattices. Phys. D 87, 371–374 (1995) 4. A.C. Newell, J.V. Moloney, Nonlinear Optics (Addison-Wesley, 1991) 5. R. Hirota, K. Suzuki, Studies on lattice solitons by using electrical networks. J. Phys. Soc. Japan 28, 1366 (1970) 6. H. Nagashima, Y. Amagishi, Experiment on the Toda lattice using nonlinear transmission lines. J. Phys. Soc. Japan 45, 680 (1978) 7. M. Toda, Wave propagation in an harmonic lattices. J. Phys. Soc. Japan 23, 501 (1967) 8. P. Marquié, J.M. Bilbault, M. Remoissenet, Generation of envelope and hole solitons in an experimental transmission line. Phys. Rev. E 49, 828 (1994) 9. E. Kengne, A. Lakhssassi, W.M. Liu, Nonlinear Schamel-Korteweg de Vries equation for a modified Noguchi nonlinear electric transmission network: analytical circuit modeling. Chaos, Solitons Fractals 140, 110229 (2020) 10. E. Kengne, B.A. Malomed, S.T. Chui, W.M. Liu, Solitary signals in electrical nonlinear transmission line. J. Math. Phys. 48 11. A.C. Scoot, F.Y.F. Chu, W. Mclaughlin, The soliton: a new concept in applied science. Proc. IEEE 61, 1443 (1973) 12. R.S.C. Cobbold, Theory and Application of Field-Effect Transistors (Wiley, New York, 1970) 13. V.P. Popov, T.A. Bickart, RC transmission line with nonlinear. resistance: large-signal response computation. IEEE Trans. Circuits Syst. CAS 21, 666 (1974)

6

1 Introduction

14. A.R. Djordjevic, T.K. Sarkar, R.F. Harrington, Analysis of Lossy transmission lines with arbitrary nonlinear terminal networks. IEEE Trans. Microwave Theory Tech. MTT 34, 660 (1986) 15. Q. Gu, Y.E. Yang, J.A. Kong, Transient analysis of frequency-dependent transmission line systems terminated with nonlinear loads. J. Electromagn. Waves Appl. 3, 183 (1989) 16. E. Afshari, A. Hajimiri, Nonlinear transmission lines for pulse shaping in silicon. IEEE J. Solid-State Circuits 40, 744 (2005) 17. C.H. Lee, S. Imhof, C. Berger, F. Bayer, J. Brehm, L.W. Molenkamp, T. Kiessling, R. Thomale, Topoelectrical circuits. Commun. Phys. 1, 39 (2018) 18. J.C. Bao, D.Y. Zou, W.X. Zhang, W.J. He, H.J. Sun, X.D. Zhang, Topoelectrical circuit octupole insulator with topologically protected corner states. Phys. Rev. B 100, 201406 (2019) 19. S. Imhof, C. Berger, F. Bayer, J. Brehm, L.W. Molenkamp, T. Kiessling, F. Schindler, C.H. Lee, M. Greiter, T. Neupert, R. Thomale, Topolectrical-circuit realization of topological corner modes. Nat. Phys. 14, 925–929 (2018) 20. X. Ni, Z.C. Xiao, A.B. Khanikaev, A. Alu, Robust multiplexing with topoelectrical higher-order Chern insulators. Phys. Rev. Appl. 13, 064031 (2020) 21. B. Lv, R. Chen, R. Li, C. Guan, B. Zhou, G. Dong, C. Zhao, Y. Li, Y. Wang, H. Tao, J. Shi, D.H. Xu, Realization of quasicrystalline quadrupole topological insulators in electrical circuits. Commun. Phys. 4, 108 (2021) 22. A. Stegmaier, S. Imhof, T. Helbig, T. Hofmann, C.H. Lee, M. Kremer, A. Fritzsche, T. Feichtner, S. Klembt, S. Höfling, I. Boettcher, I.C. Fulga, L. Ma, O.G. Schmidt, M. Greiter, T. Kiessling, A. Szameit, R. Thomale, Topological defect engineering and PT symmetry in non-Hermitian electrical circuits, Phys. Rev. Lett. 126 23. R. Gowrishankar, D.A. Stewart, G.T. Martin, J.C. Weaver, Transport lattice models of heat transport in skin with spatially heterogeneous, temperature-dependent perfusion. Biomed. Eng. Online 3, 42 (2004) 24. I. Kokalari, T. Karaja, M. Guerrisi, Review on lumped parameter method for modeling the blood flow in systemic arteries. J. Biomed. Sci. Eng. 6, 92–99 (2013) 25. R. Liu, X. An, H. Zheng, M. Wang, Z. Gao, E. Li, Neutralization line decoupling tri-band multiple-input multiple-output antenna design. IEEE Access 8, 27018–27026 (2020) 26. L.Q. English, S.G. Wheeler, Y. Shen, G.P. Veldes, N. Whitaker, P.G. Kevrekidis, D.J. Frantzeskakis, Backward-wave propagation and discrete solitons in a left-handed electrical lattice. Phys. Lett. A 375, 1242 (2011) 27. T. Kuusela, Soliton experiments in a damped ac-driven nonlinear electrical transmission line. Phys. Lett. A 167, 54–59 (1992) 28. B.A. Malomed, Propagating solitons in ac-driven chains. Phys. Rev. A 45, 4097–4101 (1992) 29. T. Kuusela, J. Hietarinta, B.A. Malomed, Numerical study of solitons in the damped ac-driven Toda lattice. J. Phys. A 26, L21–L26 (1993) 30. A. Kundu, Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger-type equations. J. Math. Phys. 25, 3433 (1984) 31. D. Qiu, J. He, Y. Zhang, K. Porsezian, The Darboux transformation of the Kundu-Eckhaus equation. Proc. R. Soc. A 471, 20150236 (2015) 32. F. Yuan, D. Qiu, W. Liu, K. Porsezian, J. He, On the evolution of a rogue wave along the orthogonal direction of the (x, t)-plane. Commun. Nonlinear Sci. Numer. Simul. 44, 445–457 (2017) 33. S. Xu, J. He, L. Wang, The Darboux transformation of the derivative nonlinear Schrödinger equation. J. Phys. A 44, 305203 (2011) 34. Y. Zhang, L. Guo, S. Xu, Z. Wu, J. He, The hierarchy of higher order solutions of the derivative nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 19, 1706–1722 (2014) 35. H.-H. Chen, C.-H. Liu, Solitons in nonuniform media. Phys. Rev. Lett. 37, 693–697 (1976) 36. H.H. Chen, Y.C. Lee, C.S. Liu, Integrability of nonlinear Hamiltonian systems by inverse scattering method. Phys Scr. 20, 490 (1979)

Chapter 2

Nonlinear Schrödinger Models for Solitons Propagation in 1D Lossless Nonlinear Transmission Networks (NLTNs)

Abstract In this Chapter, we consider the lossless 1D discrete dispersive NLTN sketched in Fig. 2.1. When lattice effects are considered, the reductive perturbation method in the semi-discrete limit can be used to show that the dynamics of modulated waves can be modeled by PDEs of Schrödinger type, which describes the MI and propagation of modulated waves on a continuous-wave (CW) background (also referred to as the constant-amplitude state). Analytical solutions of these PDEs are used to investigate the transmission of modulated nonlinear waves in the network under consideration. Here, we also investigate the effects of the linear dispersive element C S on the dynamics of modulated nonlinear waves. We show that the linear dispersive parameter C S can be used to manipulate the motion of bright, dark, and kink solitons in the network.

2.1 Introduction Nonlinear Schrödinger type equations are models emerging from a wide variety of fields and playing a significant role in many fields of science such as fluids, nonlinear optics, the theory of deep water waves, plasma physics, nonlinear transmission networks, nuclear physics, Bose–Einstein condensates, condensed matter physics, and so on. The present chapter deals with the study of a modified NLTN of the Noguchi type (which was introduced in [1]) depicted in Fig. 2.1. Such a model is composed of N identical cells, each of which contains a linear inductor L1 shunted by a linear capacitor C S in the series branch, and in the transverse branch, a nonlinear capacitor C shunted by another linear inductor L2 . The nonlinear capacitor is voltage-dependent and is realized as a reverse-biased diode. This physical model differs from the Hirota-Suzuki model [2] by the presence of the linear dispersive capacitance C S . Built by Noguchi to experimentally study the propagation of the first-order Korteweg-de Vries (KdV) solitons [1], the linear dispersive capacitance C S was absent in its first version. The physical model shown in Fig. 2.1 has been used by Ichikawa et al. [3] to predict the realization of KdV-like solitons in it; it has been generalized by Yoshinaga et al. [4] to experimentally study properties of solitons of the KdV type. Most recently, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 E. Kengne and W. Liu, Nonlinear Waves, https://doi.org/10.1007/978-981-19-6744-3_2

7

8

2 Nonlinear Schrödinger Models for Solitons Propagation …

Fig. 2.1 Schematic representation of an elementary cell of the lossless dispersive nonlinear transmission network with the linear inductor in the series branch, and the nonlinear capacitor in the parallel branch. A linear capacitor (dispersive element) is connected in parallel to the linear inductor. The line is composed of N identical cells. Reprint from Ref. [8], Copyright 2022, with permission from American Physical Society

Pelap et al. [5] addressed the existence of solitary waves in this network system. Some years ago, Marklund and Shukla [6] investigated by means of Kengne and Liu model [7] the MI of partially coherent signals in the transmission lines shown in Fig. 2.1. The main purpose of this Chapter is to present various PDEs of Schrödinger type that are used to model and to investigate the dynamics of nonlinear modulated waves propagating in the network of Fig. 2.1. A particular attention will be paid to the effects of linear dispersive capacitance C S on the soliton propagation.

2.2 Standard Nonlinear Schrödinger Equations for Modulated Waves Propagation in a Lossless Electrical Transmission the Network: Effects of the Dispersive Element C S The present Section deals with the study of dynamics of solitons in the modified Noguchi lossless electrical transmission network displayed in Fig. 2.1. Employing the reductive perturbation method in the semi-discrete limit, we establish that the dynamics of modulated waves are described by one-dimensional NLS equations. Based on these equations, the analytical results predict either two or four frequency regions with different behavior concerning the MI. Applying the Darboux transform to the found network equations, we produce exact bright and dark solitons embedded in the CW background. Our investigation reveal that the physical system under consideration may support the propagation of bright, dark, and kink soliton signals. Also, we show that when a wave packet travels along the lattice, it experiences velocity fluctuations which become more and more pronounced with the increase of lattice

2.2 Standard Nonlinear Schrödinger Equations…

9

effects. Through analytical exact solutions of the network equations, we establish that the dispersion parameter C S can be used to control the motion of bright, dark, and kink solitary waves propagating in the network. The methodology presented in this Section offers an efficient tool for systematically investigating the dynamics of solitons in 1D nonlinear transmission networks.

2.2.1 Model Equations In order to study the effects of the dispersive element C S on nonlinear modulated waves propagating in the network system of Fig. 2.1, we adopt the voltagecapacitance dependence as   Q n (Vn ) = C0 Vn − αVn2 + βVn3 , C0 being the characteristic capacitance, and α and β are positive nonlinear coefficients. Applying the Kirchhoff’s laws to our network system yields the following system of equations    1 1 d2  CS d2 2 3 V + − αV + βV V − + (Vn−1 − 2Vn + Vn+1 ) = 0, (2.1) n n n n dt 2 C0 L 2 C0 L 1 C0 dt 2

n = 1, 2, ..., N . In the linear limit Vn ∼ exp [i (kn − ωt)], we derive the following linear dispersion relation, similar to that of a typical passband filter:       1 4 2 k 2 k − = 0. + sin ω C0 + 4C S sin 2 L2 L1 2 2

(2.2)

Throughout this Section, we focus ourselves to the network system of Fig. 2.1 with linear dispersion element C S that satisfies the condition C S < (L 2 /L 1 ) C0 so that the group velocity υg = dω/dk remains positive in the Brilloum zone 0 ≤ k ≤ π . During the numerical simulations, the following network parameters will be used [9]: L 1 = 28 μH, L 2 = 14 μH, C0 = 540 pF, V0 = 1.5 V, α = 0.16 V−1 , β = 0.0197 V−2 , (2.3)

with C S ∈ [0, C0 /2] pF. For the network parameters 2.3, the linear dispersion curves of a typical passband filter are represented in Fig. 2.2 for different values of coefficient C S . These dispersion curves show (a) the propagating frequency f = ω/2π and (b) dispersion coefficient Pl = d 2 ω/dk 2 as functions of the wavenumber k. It is clearly seen from the plots of Fig. 2.2a that the frequency decreases when C S increases. Figure 2.2b reveals that for any dispersion element C S ∈ [0, C0 /2], equation Pl (k) = 0 admits one and only one solution k z ∈ [0, π ] that corresponds to the frequency f z = f (k z ), and, k z → 1/2 as C S → C0 /2 − 0.

10

2 Nonlinear Schrödinger Models for Solitons Propagation …

Fig. 2.2 Linear dispersion curves showing frequency f = ω/2π (a) and dispersion coefficient Pl = d 2 ω/dk 2 (b) versus wavenumber k for different values of dispersive parameter Cs . The network parameters used in these plots are given by Eq. (2.3). Reprint from Ref. [8], Copyright 2022, with permission from American Physical Society

In order to focus ourselves on waves with a slowly varying envelope in time and space with regard to a given carrier wave with angular frequency ω = 2π f and wavenumber k and use the reductive perturbation method, we introduce two slow envelope variables as [10] ξ = ε(n − υg t),

τ = ε2 t,

(2.4)

with a small parameter ε  1. Here, υg = dω/dk, ω and k being related by the linear dispersion relation (2.2). Then, we seek solutions for system (2.1) in the general form Vn (t) = εV01 (ξ, τ ) exp(iθ ) + ε2 [V02 (ξ, τ ) + V12 (ξ, τ ) exp(2iθ )] + c.c.,

(2.5)

where θ = kn − ωt is the rapidly varying phase and c.c. stands for the complex conjugation. In the general form (2.5), the term proportional to ε2 is included in the dc part to take into account the asymmetry of the charge-voltage relation. Substituting ansatz (2.5) into the nonlinear discrete system (2.1) and neglecting terms of higher order in ε, one arrives, after keeping up to the second-order derivatives of Vn (t) (this is done to secure the balance of the dispersion and nonlinearity), to the following results V02 (ξ, τ ) = − α02 |V01 |2 , V12 (ξ, τ ) =

αω2 12

(V01 ) , 2

V01 + P ∂∂ξV201 + Q |V01 |2 V01 = 0. i ∂∂τ 2

(2.6a) (2.6b) (2.6c)

2.2 Standard Nonlinear Schrödinger Equations…

11

Here,   4L 1 C S ω2 − 1 sin2 k 1 02 = , 12 = ω − + , (2.7a) 1 − L 1 C0 υg2 4L 2 C0 L 1 C0   υg2 4C S 1 k CS CS 1+ + P =− sin2 ω cos k − 2 υg sin k, − 2ω C0 2 2L 1 C0 ω 2C0 C0 2L 1 C0 α 2 ωυg2

2

(2.7b) 3β α 2 ω3 Q = . ω + 02 − 2 12

(2.7c)

As we can see from Eqs. (2.6a), (2.6b), (2.7a), and (2.7c), the two last terms in the expression for nonlinear coefficient Q of the NLS equation (2.6c) come from the ε2 terms in the general expression (2.5) for Vn (t). From expression (2.7b), it is seen that the ε2 terms does not affect the dispersion coefficient P. This means that, assuming a solution in the form of (2.5) without ε2 terms, one arrives at another NLS equation similar to Eq. (2.6c), except for the nonlinear coefficient which is Q = 3βω/2.

(2.8)

The above methodology shows that one may obtain two NLS equations with the same dispersion coefficient P, but different nonlinear coefficients Q, allowing the investigation of the dynamics of solitons in the network of Fig. 2.1. If we apply now the MI criterion for the NLS equation [11], we can conclude that a uniform wavetrain propagating along the network of Fig. 2.1 is unstable under modulation for P Q > 0, and remains stable for P Q < 0. In the situation when the nonlinearity parameter Q of the NLS equation is given by Eq. (2.8), product P Q and P have the same sign as soon as ω is always positive; this behavior can be seen from Fig. 2.3b. Now, considering that the nonlinear coefficient Q is given by Eq. (2.7c), one of the three scenarios shown in Fig. 2.3 occurs, depending on the value of dispersive parameter C S . Indeed, max Q(k) = Q max (C S ) is an increasing continuous function of C S , with Q max (0) < [0,π]

0 < Q max (C0 /2). Therefore, there exists a unique value, C S = C S0 ∈ [0, C0 /2] that satisfies the relationship Q max (C S0 ) = 0. For every C S < C S0 , one has Q (k) < 0, while for every C S > C S0 , equation Q(k) = 0 has two solutions, kq1 and kq2 , in interval [0, π ]. Different shapes of the region of the MI by the wave solution of the NLS equation (2.6c) with the nonlinear coefficient (2.7c) are displayed in Fig. 2.3. In Fig. 2.3d–f, f k shows the value of the frequency for the wavenumber k = k z that satisfies the relationship P(k z ) = 0, that is, f z = f (k z ); in these same figures, f q1 and f q2 are frequencies associated with Q(k) = 0. Figure 2.3d shows two MI regions, the one with f ∈ [ f min , f z ] corresponding to P Q < 0, hence the modulationally stable CW and hole (dark) solitons, and region f ∈ [ f z , f max ] corresponding to P Q > 0, that is, MI and bright envelope solitons. In Fig. 2.3e, three regions of the modulational instability can be observed, the one f ∈ [ f min , f z ] corresponding to P Q < 0 (modulational stability and dark solitons), and two regions f ∈ [ f z , f q1 ] ∪ [ f q1 , f max ] corresponding to P Q > 0 (MI and envelope solitons). From Fig.

12

2 Nonlinear Schrödinger Models for Solitons Propagation …

Fig. 2.3 Linear dispersion curves (bottom) and plots of the product P Q (top) versus wavenumber k, showing different scenarios of the MI of solutions of the NLS equation (2.6c) with nonlinear coefficient ( 2.7c). a, d Product P Q and frequency f = ω/2π for C S = C0 /8 < C S0 ; b, e Product P Q and frequency f = ω/2π for C S = C S0 ≈ C0 /5.87; c, f: Product P Q and frequency f = ω/(2π ) for C S = C0 /5 > C S0 . To generate these plots, parameters (2.3) are used. Reprint from Ref. [8], Copyright 2022, with permission from American Physical Society

2.3f, we can clearly see four regions of MI: two regions of f ∈ [ f min , f z ] and f ∈ [ f q1 , f q2 ], corresponding to P Q < 0 (modulational stability and hole solitons), and two regions f ∈ [ f z , f q1 ] and f ∈ [ f q2 , f max ] corresponding to P Q > 0 (once again, this implies the presence of the MI and bright solitons). In Fig. 2.4, we depict the product P Q versus the wavenumber k for different values of dispersion parameter C S . Plots of Fig. 2.4a correspond to the NLS equation (2.6c) with nonlinear coefficient (2.8), while Fig. 2.4b and c are obtained with the nonlinear coefficient Q given by Eq. (2.7c) for C S < C S0 and C S > C S0 , respectively. It is clearly seen from Figs. 2.2a and 2.4a, that the region of negative P Q (modulational stability) for NLS equation (2.6c) with nonlinear coefficient (2.8) enhances when the dispersion parameter C S increases. From Figs. 2.2a and 2.4b, we can see that the region of negative P Q (i.e., the region of the modulational stability) of NLS equation (2.6c) with nonlinear coefficient (2.7c) shrinks when the dispersion parameter C S increases, being capped by C S < C S0 . As one can seen from Figs. 2.2a and 2.4c, using nonlinear coefficient (2.7c), the modulational stability region (P Q < 0) shrinks for low frequencies and enhances for higher frequencies, when C S > C S0 increases. In  other words, the size of interval f min , f z decreases ( f z is getting smaller) while that of interval [ f q1 , f q2 ] increases ( f q1 decreases and f q2 increases) when C S > C S0 increases.

2.2 Standard Nonlinear Schrödinger Equations…

13

Fig. 2.4 Plots of product P Q showing the regions of positive and negative values for different values of the dispersion parameter C S . a Plots of product P Q with nonlinear coefficient Q given by Eq. (2.8) for different C S . b The plot of product P Q generated with nonlinear coefficient (2.7c) for different parameters C S < C S0 ≈ C0 /5.87. c The plot of product P Q generated with nonlinear coefficient (2.7c) for different parameters C S > C S0 ≈ C0 /5.87. These plots are generated with parameters (2.3). Reprint from Ref. [8], Copyright 2022, with permission from American Physical Society

2.2.2 Bright and Dark Single-Solitary Waves Propagating in the 1D Lossless Network of Fig. 2.1 With the use of the NLS equations (2.6c) with the nonlinear coefficient Q given by Eq. (2.7c) and by Eq. (2.8), we here aim to study analytically the propagation of bright and dark solitary waves through the network of Fig. 2.1. For this purpose, we first find analytical soliton like solutions of the NLS equation (2.6c).

2.2.2.1

Analytical Solitonlike Solutions of the NLS Equation

Here, we focus ourselves on solutions on top of a CW background, V01c (ξ, τ ) =

Acb exp

(iφcb ) , if P Q > 0

Q exp (iφcd ) , if P Q < 0, −Acd − 2P

with  φcb = kcb

2  A2 Q 2 + 2P 2 kcd Q 2 Q 2 ξ+ 2 Acb − kcb τ + δ0 ; τ, φcd = kcd ξ − cd 2P 2 2P

14

2 Nonlinear Schrödinger Models for Solitons Propagation …

here, Acb , Acd , kcb , kcd , and δ0 are arbitrary real constants. If we apply the Darboux transformation [12] for the NLS equation (2.6c) to the CW background V01c (ξ, τ ), we obtain the following antidark and dark soliton solutions d1 cosh θ + cos ϕ d2 sinh θ + d3 sin ϕ + i Asb V01 (ξ, τ ) = Acb + Asb cosh θ + d1 cos ϕ cosh θ + d1 cos ϕ × exp (iφcb ) , (2.9a)  Q V10 (ξ, τ ) = − − (2.9b) [Acd + i Asd tanh ζ ] exp (icd ) , 2P 

respectively. It is important to notice that “antidark” are solitons of the bright type sitting on top of a flat background. Different parameters appearing in solutions (2.9a) and (2.9b) are given as 

Q [(kcb + ksb ) M R − Asb M I ] θ = MR ξ− Qτ − θ0 , 2P 2  Q [(kcb + ksb ) M I + Asb M R ] ξ− Qτ − ϕ0 , ϕ = MI 2P 2   2 Asd Acd Q 2 − 4P 2 kcd Q ξ+ τ + ζ0 , ζ = Asd 2P 2P   2 P2 Q 2 A2sd + A2cd + 2kcd τ + δ0 , cd = kcd ξ − 2P D − 2 A2cb Acb (M R − Asb ) Acb (ksb − kcb + M I ) , d2 = , d3 = , d1 = D D D (M R − Asb )2 + (ksb − kcb + M I )2 D = A2cb + , 4

M=

(−Asb + iksb − ikcb )2 − 4 A2cb = M R + i M I .

Here, Asb , Asb , θ0 , ϕ0 , ksb , ζ0 , and ksd are arbitrary real parameters. The antidark soliton solution (2.9a) and the dark-soliton solution (2.9b) carry over into the CW background V01c (ξ, τ ) when Asb = 0 and when Asd = 0, respectively (the modulational stability of the background is considered below). On the other hand, for the vanishing CW background, i.e., when Acb = 0, and when Asd = 0, they respectively, turn into the following bright soliton solution and the kink solution      2 Q ξ − ksb − A2sb Qτ − ϕ Asb exp i ksb 2P 0 2    V01 (ξ, τ ) = (2.10a) , Q cosh Asb 2P ξ − ksb Qτ − ξ0      A2 Q 2 Q Q Asd V10 (ξ, τ ) = −i Asd − , (2.10b) tanh ξ − ξ0 exp i − sd τ − τ0 2P 2P 2P

2.2 Standard Nonlinear Schrödinger Equations…

15

ξ0 and τ0 being arbitrary real constants. This means that solutions (2.9a) and (2.9b) produce solitary waves embedded into the CW background. Substituting expressions (2.9a) and (2.9b) into Eqs. (2.6a) and (2.6b) leads to analytical expressions for V02 (ξ, τ ) and V12 (ξ, τ ). If we then insert the expressions of V10 (ξ, τ ), V02 (ξ, τ ) and V12 (ξ, τ ) in the general form (2.5) for voltage Vn (t) and going back to the original coordinate n and time t by means of Eq. (2.4), we will obtain the analytical expressions for Vn (t). These results for Vn (t) can be used to investigate analytically the dynamics of solitons in the network system of Fig. 2.1. Next, we separately discuss the dynamics of solitons in the MI region (envelope solitons given by Eq. (2.9a)) and in the region of the modulational stability (hole solitons defined by Eq. (2.9b)).

2.2.2.2

Dynamics of Solitons in the Regions of Positive P Q (Modulational Instability)

Using the zeros of functions P = P(k) and Q = Q(k), the regions of the MI for the NLS equation (2.6c) are displayed in Fig. 2.3 for a given set of the network parameters, with nonlinear coefficient (2.7c) or (2.8). In the regions of MI, that is, the regions of positive P Q, the solitons are envelope modes propagating on top of the CW background, defined by expression (2.9a). Without the loss of generality, we consider here the NLS equation (2.6c) with dispersion coefficient (2.7b). First of all, we notice that the situation with 4 A2cb − A2sb = 0 leads to a constant solution. In what follows, we focus on three special cases, viz., the bright soliton with a vanishing CW background (Acb = 0), the case of ksb = kcb and A2sb − 4 A2cb < 0, and, finally, the one with ksb = kcb and A2sb − 4 A2cb > 0. Case Acb = 0 By taking Acb = 0, solution (2.9a) of NLS equation (2.6c) for positive P Q (i.e., solution (2.10a)) can be written as follows: V10 (ξ, τ ) =





cosh Asb 

where (ξ, τ ) = ksb

Asb Q ξ 2P

 exp (i) ,

(2.11)

− ksb Qτ − ξ0

  2 ksb − A2sb Q Q ξ− τ − ϕ0 , 2P 2

ϕ0 and ξ0 being two arbitrary real constants. Expression (2.11) is an envelope soliton solution of the NLS equation (2.6c) when P Q > 0. Analyzing solution (2.11), it √ appears that the coordinate of the center of the envelope soliton is η(τ ) = 2P/Q (ksb Qτ + ξ0 ) and the width of the envelope soliton is inversely proportional to (Q/2P)1/2 , meaning that one can use Eq. (2.11) to describe the compression of envelope solitons when (Q/2P)1/2 increases with the variation of frequency f = ω/2π . It is obvious that for P(k)Q(k) > 0, one has (Q/2P)1/2 → +∞ as k → k z ; therefore, for a given set of parameters, the width of the envelope soliton

16

2 Nonlinear Schrödinger Models for Solitons Propagation …

Fig. 2.5 The wave propagation in the region of low frequencies (top) and higher frequencies (bottom), showing the extension and compression of envelope solitons, respectively. These plots are obtained for parameters (2.3) with C S = C0 /5 and ε = 10−3 , Asb = 10, ksb = −0.6, ξ0 = −10, ϕ0 = 0. The frequencies are a f = 2.018 MHz, b f = 2.047 MHz, c f = 2.077 MHz, d f = 2.247 MHz, e f = 2.265 MHz, f f = 2.281 MHz. Reprint from Ref. [8], Copyright 2022, with permission from American Physical Society

shrinks as f (k) is approaching f z = f (k z ) for P Q > 0. Also, it is seen that the center of the soliton moves at a constant velocity, dη/dτ = ksb Q (2P/Q)1/2 . To demonstrate the dynamics of envelope soliton propagating in the network of Fig. 2.1, we use parameters (2.3) with C S = C0 /5. In this case, the region of positive P Q (that is, the region of the existence of envelope solitons) are (in MHz) intervals of low frequencies, f ∈ [1.999, 2.1125], and of higher frequencies, f ∈ [2.231, 2.3631]. Function (Q/2P)1/2 decreases for f ∈ [1.999, 2.1125] and increases for f ∈ [2.231, 2.3631]. Using these parameters, we depict in Fig. 2.5 the evolution of voltage Vn (t) at cell n = 1 for different frequencies. Figure 2.5a–c and d–f show the pulses propagating, respectively, with low and high frequencies. As we can see from Fig. 2.5a–c, the width of the envelope solitons increases with the frequency; Fig. 2.5d–f reveal that the width decreases as the frequency increases. These findings are explained by the fact that quantity (Q/2P)1/2 (which determines the compression or expansion of the solitons) is a decreasing/increasing function of the propagating frequency f in the region of low/high frequencies. Case ksb = kcb and A2sb − 4 A2cb < 0 In the present case, we can rewrite solution (2.9a) in the following form:

⎞ −Asb cosh θ + 2 Acb cos ϕ + i 4 A2cb − A2sb sinh θ ⎠ V01 (ξ, τ ) = ⎝ Acb + Asb 2 Acb cosh θ − Asb cos ϕ ⎛

  × exp i kcb



 Q Q 2 2 2 Acb − kcb τ ξ+ 2P 2

 ,

(2.12)

2.2 Standard Nonlinear Schrödinger Equations…

17

√ with θ = (1/2)Asb 4 A2cb − A2sb Qτ − θ0 and ϕ = 4 A2cb − A2sb Q/2Pξ − kcb Qτ ) − ϕ0 ; here, θ0 and ϕ0 are arbitrary real parameters.From the expressions for θ and ϕ and from transformation (2.4), we can see that solution (2.5) associated with solution (2.12) is periodic with the period 2π N = ε



2P   Q 4 A2cb − A2sb

in the spatial coordinate n and

aperiodic in time t, if the wavenumber of the CW background is chosen as kcb =  k 4 A2cb − A2sb , where  k is an integer. Any wavenumber

kcb =  k 4 A2cb − A2sb will lead to an aperiodic solution in both time and space. We display in Fig. 2.6 the evolution of the MI corresponding to solution (2.12) for the network parameters (2.3) with C S = C0 /5. Figure 2.6a–c and d–f show the transmission of the soliton-like signals through the network at frequencies f = 2.018 MHz and f = 2.256 MHz, respectively. Therefore, the envelope solitons in Fig. 2.6a–c propagate at the frequency taken from the MI region [ f min , f z ], while the pulses of Fig. 2.6g, h are associated with the frequency taken from the MI region, [ f q2 , f max ]. Figure 2.6g, h show solitons propagating at different frequencies taken from the same MI region; here, it is clearly seen that the CW background becomes unstable. From Fig. 2.6, we can conclude that the choice of the propagating frequency in the MI region affects the soliton propagation. Figure 2.6 also reveals that, with the best choice of solution parameters, the solution is periodic in time, with the period increasing with the underlying frequency. This can be clearly seen in Fig. 2.6c, f, as well as in Fig. 2.6g, h. Moreover, Fig. 2.6g, h reveal that the envelope soliton features different behavior at frequencies taken from the low- and high-frequency regions of the MI. We conclude from Fig. 2.6g and h that the temporal period of the wave is related to the propagating frequency f of the envelope soliton: In the low-frequency MI region, the period increases with f , while in the high-frequency region the period decreases. Case ksb = kcb and A2sb − 4 A2cb > 0 In the special case when ksb = kcb and A2sb − 4 A2cb > 0, one can compute coefficients of solution (2.9a) and obtain the solution in the form  ⎞ ⎛

2 2 2 2 Asb − 4 Acb Asb − 4 Acb cos ϕ − i Asb sin ϕ ⎟ ⎜ ⎟ + −A V01 (ξ, τ ) = ⎜ cb ⎠ ⎝ Asb cosh θ − 2 Acb cos ϕ   × exp i kcb



 Q 2 Q 2 ξ+ 2 Acb − kcb τ 2P 2

 ,

(2.13)

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2 Nonlinear Schrödinger Models for Solitons Propagation …

Fig. 2.6 The evolution of the MI associated with Eq. (2.12) for parameters (2.3) with dispersion coefficient C S = C0 /5 and the frequency taken from the lower region, f = 2.018 MHz (top), and from the higher region, f = 2.273 MHz (middle). a, d: The spatiotemporal evolution of the voltage signal; b, e: The spatial profile of the signal at t = 50; c, f The temporal evolution of the voltage at cell n = 350; g: The evolution of the voltage (at cell n = 350) with frequencies taken from the MI region [ f min , f z ] ( f = 2.018 MHz ⇐⇒ k = 0.90 and f = 2.047 MHz ⇐⇒ k = 1.05); h The evolution of the voltage (at cell n = 350) with frequencies taken from the MI region [ f g2 , f max ] ( f = 2.256 MHz ⇐⇒ k = 1.90 and f = 2.273 MHz ⇐⇒ k = 2.0). The

curves are generated with parameters ε = 10−3 , Asb = 10−4 , Acb = Asb /1.8, kcb = ksb = 2 4 A2cb − A2sb , ϕ0 = 0,

and θ0 = −10−8 . Reprint from Ref. [8], Copyright 2022, with permission from American Physical Society

with θ =

A2sb

−

4 A2cb



Q ξ 2P

− ksb Qτ

− θ0 and ϕ = − 21 Q Asb A2sb − 4 A2cb τ −

ϕ0 , where θ0 and ϕ0 are two arbitrary real constants. Solution (2.13) is aperiodic in both space and time. Going back to the original variables n and t, one concludes that the motion of the envelope soliton is aperiodic in both n and t, as shown in Fig. 2.7. It is seen from Fig. 2.7c and f that the envelope solitons feature different behavior when propagating at frequencies taken from the low- and high-frequency MI regions: For the solitons propagating in the low-frequency region, waves with a smaller frequency reach their peak earlier than ones with a higher frequency. The opposite situation is observed for the solitons in high-frequency MI region.

2.2 Standard Nonlinear Schrödinger Equations…

19

Fig. 2.7 The evolution of envelope solitons on top of the CW background given by Eq. (2.13) −3 −4, A = A /3, k = for cb sb cb

the same network as in Fig. 2.6 with parameters ε = 10 , Asb = 10 2 2 2 Asb − 4 Acb , ϕ0 = 0, θ0 = −4. a, d Spatial profile of bright solitons at time t = 50 s propagating at frequencies f = 2.018 MHz (a) and f = 2.256 MHz (d); b, e The evolution of the envelope soliton at cell n = 350 with frequencies f = 2.018 MHz (b) and f = 2.256 MHz (e); c Voltages at n = 350 as function of time t for different frequencies taken from the MI region of low frequencies, [ f min , f z ] (c), and from the MI region of high frequencies, [ f g2 , f max ] (f). Reprint from Ref. [8], Copyright 2022, with permission from American Physical Society

2.2.2.3

Dynamics of Solitons in the Region of the Negative PQ (Modulational Stability Region)

Employing now solution (2.9b), we address here the dynamics of solitons propagating in the network at frequencies taken from the regions of the modulational stability. As we have already mentioned, the analytical solution (2.9b) describes the propagation of The analytical dark (hole) solitons on the non-vanishing CW background V01c (ξ, τ ). √ expression (2.9b) shows that the soliton amplitude is proportional to −Q/(2P). Firstly, it is important to notice that for vanishing CW background, i.e., when Acd = 0, solution (2.9b) is associated with the propagation of a kink soliton through the network, as it is shown in Fig. 2.8, which is produced by the NLS equation (2.6c) with the nonlinearity coefficient (2.7c). Figure 2.8 shows the temporal profile of stable kink solitary waves at the 350-th cell at low and high frequencies (left and right panels, respectively). Plots of the top, middle, and bottom panels correspond to the network with the dispersion parameter C S = C0 /5, C S = C0 /4.5 and C S = C0 /4.1, respectively. The left and right panels of Fig. 2.8 show that, independently of the value of C S , the soliton’s speed decreases as the propagating frequency increases. Secondly, in the non-vanishing CW background (i.e., at Acb = 0), solution (2.9b) leads to the propagation of dark solitons embedded in the CW background. Thus, expression (2.9b), may lead to the dark soliton solution of Eq. (2.1) which corresponds to waves propagating on a CW background. An example of the propagation of the dark soliton is shown in Fig. 2.9. The top and the bottom panels of Fig. 2.9 show respectively dark solitons propagating at different frequencies, taken from

20

2 Nonlinear Schrödinger Models for Solitons Propagation …

Fig. 2.8 The propagation of stable kink solitons through the network for the system’s parameters (2.3) with ε = 10−3 and solution parameters ksd = π/5, Asd = −10−2 , ζ0 = 2, and δ0 = 0. The top panel: C S = C0 /5; the middle panel: C S = C0 /4.5; the bottom panel: C S = C0 /4.1. Reprint from Ref. [8], Copyright 2022, with permission from American Physical Society

the low-frequency region [ f min , f z ] of the modulational stability, and dark solitons propagating in the high-frequency region of the modulational stability, i.e., [ f q1 , f q2 ]. Different plots of Fig. 2.9 are generated with the use of the solution parameters kcd = 0.1, Acd = −10−4 , ksd = 0.35, Asd = 3 × 10−4 , δ0 = 0, and ζ0 = −0.532.

2.3 Modulational Instability and Transmission of Chirped Femtosecond Signals Through the Lossless Electrical Network of Fig. 2.1 In this Section, we employ a cubic-quintic nonlinear Schrödinger equation with selfsteepening and self-frequency shift terms to study the MI phenomenon and to show

2.3 Modulational Instability and Transmission…

21

Fig. 2.9 The signal voltage (measured in volt) at cell n = 350 as a function of time t (in ms). The temporal profile of dark solitons propagating at different frequencies (in MHz) for parameters (2.3) with C S = C0 /5 and ε = 10−3 . a f = 1.837, b f = 1.854, c f = 1.882, d f = 2.119, e f = 2.146, f f = 2.171. Reprint from Ref. [8], Copyright 2022, with permission from American Physical Society

that the competing cubic–quintic nonlinearity induces propagating chirped femtosecond pulses, signals of first-order rogue wave type, and two sister modulated nonlinear wave signals along the modified Noguchi electric network of Fig. 2.1. We show the relationship between the nonlinear chirp and the intensity of the corresponding signal. We introduce the concept of two sister signals and show how the corresponding chirps are related. Our findings reveal that chirping’s features and behavior depend on both the self-steepening and self-frequency shift, while its amplitude can be managed by varying the parameters of the group velocity dispersion and cubic nonlinearity.

2.3.1 Introduction Femtosecond pulses have been extensively investigated for the past few decades, due to their wide applications in many different physical systems such as ultrahigh-bitrate optical communication systems [13], ultrafast physical processes [14], infrared time-resolved spectroscopy [15], and optical sampling systems [16]. Nonlinear partial differential equations with higher-order nonlinear and dispersive terms of Schrödinger types have been used to explain a variety of effects in the propagation of pulses in such systems. These equations in general and the NLS models in particular are generally not completely integrable and cannot be solved exactly by using traditional methods [17]. Therefore, these models do not have soliton solutions in general. However, they can admit solitary-wave solutions. The generation and propagation of solitons in nonlinear systems result, as we well-know, generally from the phenomenon of the MI [18, 19]. The phenomenon of the modulational instability

22

2 Nonlinear Schrödinger Models for Solitons Propagation …

appears in many nonlinear dispersive systems and indicates that due to the interplay between nonlinearity and the dispersive effects, a small perturbation on the envelope of a plane wave may induce an exponential growth of the wave amplitude, resulting in the carrier-wave breakup into a train of localized waves [20]. The main purpose of this Section is the analytical transmission of chirped femtosecond soliton signals and periodic pulse trains through the modified Noguchi nonlinear electric transmission network system of Fig. 2.1. Some years ago, Liu and Kengne [10] presented a model for wave propagation on the discrete electrical transmission network of Fig. 2.1 based on the generalized cubic-quintic derivative nonlinear Schrödinger (GCQD-NLS) equation, derived in the small amplitude and long wavelength limit using the standard reductive perturbation technique and complex expansion on the governing nonlinear equations. This GCQD-NLS equation, also referred to as the cubic-quintic nonlinear Schrödinger equation with derivative terms, reads ∂ψ ∗ ∂ψ ∂ 2ψ ∂ψ + P 2 + Q 3 |ψ|2 ψ + Q 5 |ψ|4 ψ + iχ1 ψ 2 + iχ2 |ψ|2 = 0. ∂t ∂x ∂x ∂x (2.14) Here, ψ(x, t) is the complex envelope of the electric field, t and x are respectively the temporal and the spatial coordinates, and P, Q 3 , Q 5 , χ1 , and χ2 coefficients expressed in terms of the network parameters. Throughout this Section, we will mostly use the following equivalent form of Eq. (2.14) i

i

∂ψ ∂  2 ∂  2  ∂2ψ |ψ| ψ + i Q 14 ψ |ψ| = 0, + P 2 + Q 3 |ψ|2 ψ + Q 5 |ψ|4 ψ + i Q 13 ∂t ∂x ∂x ∂x

(2.15) where Q 13 = χ2 − χ1 and Q 14 = 2χ1 − χ2 . In the context of NLS models, P is the group velocity dispersion (GVD) parameter, Q 3 and Q 5 represent respectively the cubic or Kerr nonlinearity and the quintic or non-Kerr nonlinearity, while Q 13 and Q 14 are respectively the self-steepening and the self-frequency shift coefficients which arise from stimulated Raman scattering. In cases of self-focusing (Q 3 > 0), self-defocusing (Q 3 < 0), and zero Kerr nonlinearities, cubic nonlinearity Q 3 can be normalized to 1, −1, and 0. The nonlinearity that arises because of the fifth-order susceptibility can be obtained in many optical materials such as semiconductor-doped glasses, semiconductors, polydiacetylene toluene sulfonate, AlGaAs, chalcogenide glasses, and some transparent organic materials [21]. The cubic term |ψ|2 ψ in the context of fiber optics is often referred to as self-phase modulation and its coefficient Q 3 can be scaled out to the GVD term, which will be termed normal (anomalous) dispersion depending on whether Q 3 < 0 (Q 3 > 0) [21]. Arising from an intensitydependent group velocity, the self-steepening of an optical fiber pulse produces a temporal pulse distortion and an asymmetry in the pulse spectrum [22, 23]. The last term of Eq. (2.15) related to Q 14 incorporates the intrapulse Raman scattering and originates from the delayed Raman response [note that for model (2.15), the intrapulse Raman scattering coefficient is Q 13 + Q 14 = χ1 ], which causes a self-frequency shift. The self-frequency shift is the main driving mechanism of several wavelength

2.3 Modulational Instability and Transmission…

23

conversion and supercontinuum generation processes [24]. The self-frequency shift enhances the effects introduced by self-steepening and also shifts the frequency of the pulse. In the mathematical point of view, the chirping, δω(x, t), of a wave (wave solution of the NLS equation (2.15)) of amplitude ρ(x, t) and phase φ(x, t) is the negation of the spatial derivative of phase φ, that is, δω(x, t) = −∂φ(x, t)/∂ x. In the following, we employ Eq. (2.15) which combines the cubic, quintic and cubic derivative nonlinearities in the description of the propagation of chirped femtosecond pulses along the modified Noguchi electric network of Fig. 2.1. Here, we intend to develop via Eq. (2.15) an analytical framework to show that the lossless network of Fig. 2.1 supports the propagation of (1) chirped solitonlike signals and chirped first-order rogue wave signals embedded on non-vanishing CW background and (2) special solitonlike and periodic nonlinear modulated wave pulses named “two sister modulated nonlinear waves”, with chirping that varies simultaneously as directly and inversely proportional to the intensity of the wave with nonzero boundary conditions. Further, we show that the amplitude of chirping can be managed by varying the self-steeping and self-frequency shift terms.

2.3.2 Modulational Instability and Evolution of Chirped Femtosecond Solitary Signals Embedded on a Non-vanishing CW Background Here, we focus ourselves on the MI and the transmission through the network of Fig. 2.1 of femtosecond solitonlike pulses, propagating on a non-vanishing CW background with nonlinear chirp. In order to build such pulses and investigate their MI, we start by performing the Madelung transformation [25, 26] ψ(x, t) = φ(x, t) exp [iθ (x, t)] ,

(2.16)

where φ(x, t) is the new unknown complex amplitude and θ (x, t) is a real function, relating with φ(x, t) as follows ∂θ(x, t) (2.17a) = α10 |φ(x, t)|2 , ∂x  ∗ ∂φ(x, t) ∂θ(x, t) ∂φ (x, t) + α13 |φ(x, t)|4 ; (2.17b) = iα12 φ ∗ (x, t) − φ(x, t) ∂t ∂x ∂x

in Eq. (2.17b), φ ∗ (x, t) is a the complex conjugate of φ(x, t) and α10 , α12 , and α13 are three real parameters given in terms of the network parameters as α10 = −

16P Q 5 + 3Q 213 − 4Q 14 (Q 13 + Q 14 ) 3Q 13 + 2Q 14 Q 13 + 2Q 14 , α12 = − , α13 = . 4P 4 16P

(2.17c) Replacing Eq. (2.16) into Eq. (2.15) and employing the relationships (2.17a), (2.17b) and (2.17c), we obtain the following cubic NLS equation for φ :

24

2 Nonlinear Schrödinger Models for Solitons Propagation …

i

∂ 2φ ∂φ + P 2 + Q 3 |φ|2 φ = 0. ∂t ∂x



If we set φ=

(2.18a)

 2P ± u= Q3

κ0

2P u and T = 2κ0 Pt, Q3

with κ0 = sign[P Q 3 ] , Eq. (3.63) under the condition P Q 3 = 0 will take the form i

κ0 ∂ 2 u ∂u + + |u|2 u = 0. ∂T 2 ∂x2

(2.19a)

Equation (2.18a) (or its equivalent form (2.19a)) is the standard NLS equation. As we well-know, Eq. (2.19a) admits envelope (hole) solitons if P Q 3 > 0, i.e., κ0 = +1, (P Q 3 < 0, i.e, κ0 = −1) [27]. Its soliton solutions obtained in the vanishing boundary conditions are known. In order to obtain our results in terms of the network parameters, our preference of the equation to be used here is Eq. (2.18a) (remember that the coefficients of Eq. (2.15) are expressed, according to Ref. [10], in terms of the network parameters). To study the phenomenon of the MI, we need to find exact solutions of Eq. (2.18a) on the non-vanishing CW background   2 − Q 3 A2cw t, φcw (x, t) = Acw exp [icw ] , cw = kcw x − Pkcw

(2.20)

where Acw and kcw are two arbitrary real constants. For seeking such exact solutions, we distinguish two cases, P Q 3 > 0 and P Q 3 < 0. (i) Case of Envelope Solitons: P Q 3 > 0 In the case of positive P Q 3 , we use solution (2.20) as the seed solution and seek for the solution solutions of the NLS equation (2.18a) in the form [28] ⎛

⎞ d cosh [ϑ] + cos [ϕ] + A A sol ⎜ cw cosh [ϑ] + d cos [ϕ] ⎟ ⎟ exp [icw ] . φ(x, t) = ⎜ ⎝ b sinh [ϑ] + c sin [ϕ] ⎠ +i Asol cosh [ϑ] + d cos [ϕ]

(2.21)

Here, cw is given by Eq. (2.20), d, b, and c are real parameters to be determined, and ϑ = ϑ(x, t) and ϕ(x, t) are two real functions to be determined. Substituting Eq. (2.21) into Eq. (2.18a) and removing exponential terms lead to an algebraicdifferential system having as solutions the following ϑ(x, t) =

Q 3 [(kcw + ksol ) M R − Asol M I ] Q3 M R x− t − ϑ0 , 2P 2

ϕ(x, t) =

Q 3 [(kcw + ksol ) M I + Asol M R ] Q3 MI x− t − ϕ0 , 2P 2

2.3 Modulational Instability and Transmission…

25

Fig. 2.10 Spatiotemporal plots showing the propagation of bright solitary pulses on cw background, according to the exact soliton solution given by Eq. (2.24b) and the corresponding frequency chirp defined in Eq. (2.25a) for P = 0.5, Q 3 = 1, Q 13 = −0.5, Q 14 = 1/8, with Asol = 1, ksol = 0.2, and ϑ0 = ϕ0 = 0. a, d Acw = 0, kcw = 0; b, e Acw = 2, kcw = 0.2; c, f Acw = 0.25, kcw = 0.2. Reprint from Ref. [19], Copyright 2022, with permission from Elsevier

b=

D − 2 A2cw Acw (M R − Asol ) Acw (ksol − kcw + M I ) ,c = ,d = , D D D

4 A2cw + (M R − Asol )2 + (ksol − kcw + M I )2 (2.22) , 4  2 A2sol − 4 A2cw − (ksol − kcw )2 + (ksol − kcw )2 + 4 A2cw − A2sol + 4 A2sol (kcw − ksol )2 2 , MR = 2 D =

M I2 =

(ksol − kcw )2 + 4 A2cw − A2sol +



2 (ksol − kcw )2 + 4 A2cw − A2sol + 4 A2sol (kcw − ksol )2 2

;

here, ϑ0 and ϕ0 are two arbitrary real constants. (ii) Case of Hole Solitons: P Q 3 < 0 When P Q 3 is negative, the NLS equation (2.18a) admit the exact solution φ(x, t) = (Acw + i Asol tanh [ξ(x, t)]) exp [idr ] ,      Q3 Q3 Q3 x + 2P Asol − ξ(x, t) = ± − k1 + Acw − t, 2P 2P 2P   2 dr = k1 x − Pkcw − Q 3 A2cw t, where k1 is an arbitrary real constant.

(2.23a) (2.23b) (2.23c)

26

2 Nonlinear Schrödinger Models for Solitons Propagation …

Replacing now Eqs. (2.21) and (2.23a) in Eq. (2.16) yields respectively the following envelope and hole soliton solutions of Eq. (2.15) on the non-vanishing CW background: ⎛

⎞ d cosh [ϑ] + cos [ϕ] + A A sol ⎜ cw cosh [ϑ] + d cos [ϕ] ⎟ ⎟ ψ(x, t) = ⎜ ⎝ b sinh [ϑ] + c sin [ϕ] ⎠ +i Asol cosh [ϑ] + d cos [ϕ] × exp [icw + iθ (x, t)] , ψ(x, t) = (Acw + i Asol tanh [ξ(x, t)]) exp [idr + iθ (x, t)] ,

(2.24a)

(2.24b)

cw and dr being given respectively by Eq. (2.20) and Eq. (2.23c). The exact soliton solutions (2.24b) and (2.24b) describe respectively envelope and hole soliton signals embedded on a CW background. Using now Eqs. (2.17a), (2.17c), and (2.20), we compute the chirping corresponding to Eqs. (2.24b) and (2.24b) and find δω(x, t) = −

3Q 13 + 2Q 14 Q 3 kcw ∂ + (cw + θ ) = − ∂x 2P 4P ⎡

⎤ d cosh [ϑ] + cos [ϕ] 2 A + A cw sol ⎢ cosh [ϑ] + d cos [ϕ] ⎥ ⎢ ⎥ ×⎢  2 ⎥ , ⎣ b sinh [ϑ] + c sin [ϕ] ⎦ +A2sol cosh [ϑ] + d cos [ϕ] (2.25a) ∂ δω(x, t) = − (cw + θ ) ∂x  3Q 13 + 2Q 14  2 Acw + A2sol tanh2 [ξ(x, t)] . = −k1 + 4P

(2.25b)

Here, b, c, d, ϕ and ϑ and ξ are given respectively by Eqs. (2.22) and (2.23b), and Acw , kcw , Acw , and Asol are arbitrary real constants. It is well seen from Eqs. (2.25a) and (2.25b) that the nonlinear part of the frequency chirp strongly depends of the GVD parameter P and both the self-steepening and the self-frequency shift coefficients Q 13 and Q 14 through the nonlinear chirp parameter μ = −α10 , α10 being given by Eq. (2.17c); on the other hand, the pulse intensity |ψ(x, t)|2 is independent of Q 13 and Q 14 . This means that we can control the chirping amplitude by varying the selfsteeping and self-frequency shift terms of Eq. (2.15). It is important to notice that solutions (2.24b) and (2.24b) and the corresponding chirping (2.25a) and (2.25b) are different from those obtained in Ref. [29] and can be used to investigate the dynamics of chirped femtosecond bright and dark solitons in the network of Fig. 2.1. 3 kcw or −k1 The nonlinear chirp parameter μ and the constant chirp parameters − Q2P are similar to those obtained in Ref. [29].

2.3 Modulational Instability and Transmission…

27

Now, we turn to the study of the dynamics of chirped bright/dark solitons embedded on a CW background and the corresponding chirping. For this aim, we distinguish three special cases, namely, the case Acw = 0 corresponding to soliton signals propagating on the vanishing CW background, the case kcw = ksol and 4 A2cw − A2sol > 0, and the case kcw = ksol and 4 A2cw − A2sol < 0 corresponding respectively to M R2 = 0, M I2 = 4 A2cw − A2sol , and M R2 = A2sol − 4 A2cw , M I2 = 0. When kcw = ksol and 4 A2cw − A2sol > 0, we have ϑ(x, t) = ±

Q 3 Asol 4 A2cw − A2sol 2

t − ϑ0 ,

Q 3 4 A2cw − A2sol  1 and ϕ(x, t) = ± x − 2ksol t − ϕ0 2 P so that the amplitude of the bright soliton is periodic in the x− coordinate with period Tspat =

4π P

Q 3 4 A2cw − A2sol

and aperiodic in the t−coordinate. In the special case when kcw = ksol and 4 A2cw − A2sol < 0, our computations give

Q 3 A2sol − 4 A2cw  1 ϑ(x, t) = x − 2ksol t − ϑ0 , 2 P

Q 3 Asol A2sol − 4 A2cw t − ϕ0 ; ϕ(x, t) = ∓ 2 this means that the wave amplitude is periodic in the t−coordinate with period Ttemp =

4π

Q 3 Asol A2sol − 4 A2cw

,

and aperiodic in the x−coordinate. In Figs. 2.10 and 2.11, we show the dynamics of respectively bright soliton signals and dark soliton signals with the corresponding chirping propagating along the network of Fig. 2.1. Generated with the nonlinear chirp parameter μ = −0.625, Fig. 2.10a and d, b and e, and c and f show the propagation of the bright soliton signals and the corresponding chirping when Acw = 0, kcw = ksol and 4 A2cw − A2sol > 0, and kcw = ksol and 4 A2cw − A2sol < 0, respectively. As we can see from these figures, the chirping associated with the bright pulse in each of the cases Acw = 0 and kcw = ksol

28

2 Nonlinear Schrödinger Models for Solitons Propagation …

Fig. 2.11 Transmission of the dark solitary pulse through the network according to the exact solution given by Eq. (2.24b) and the corresponding chirping (2.25b) for P = 0.5, Q 3 = −1, Q 14 = 0.5, with Asol = 0.5 and k1 = −2. a, d Acw = 0, Q 13 = −1.8; b, e Acw = 0.2, Q 13 = 1.8; c, f: Acw = 0.2, Q 13 = −1.8. Reprint from Ref. [19], Copyright 2022, with permission from Elsevier

and 4 A2cw − A2sol < 0 is a dark soliton (see the first and last columns of Fig. 2.10), while that of bright pulse in the case kcw = ksol and 4 A2cw − A2sol > 0 is a bright solitary wave, as we can see from the plots of the middle column of Fig. 2.10. As we can clearly see from plots of the middle (right) columns, the pulse amplitude and the corresponding chirping are periodic (aperiodic) in the x− coordinate and aperiodic (periodic) in the t− coordinate when kcw = ksol and 4 A2cw − A2sol > 0 (kcw = ksol and 4 A2cw − A2sol < 0). In Fig. 2.11a and d, we show the evolution of the dark solitary signal propagating on a vanishing CW background and the corresponding chirping when the nonlinear chirping parameter μ is negative. From the plots of the middle and left panels of Fig. 2.11, we can observe the propagation of dark pulses embedded on a non-vanishing CW background for respectively positive and negative nonlinear chirping parameter μ. From the plots of the left and right panels of Fig. 2.11, we can conclude that the chirping associated with dark solitary signals corresponding to a negative nonlinear chirping parameter μ are bright pulses, while that with positive nonlinear chirping parameter μ corresponding to the dark solitary pulse is of dark soliton type, as we can well see from Fig. 2.11b and e. We thus conclude from the plots of Figs. 2.10 and 2.11 that the chirping profile of a bright (Fig. 2.10) or a dark (Fig. 2.11) solitary signal can be either of bright type or of dark soliton type. Remark: It is important to notice that when kcw = ksol , Acw 0 with kcw = ksol [17, 30]. In order to understand such a modulational instability process, we compute, under the conditions kcw = ksol and 4 A2cw − A2sol > 0, the initial value of solution (2.24b) for ϑ0 = 0 as follows

2.3 Modulational Instability and Transmission…

29

Fig. 2.12 The dynamics of (left) bright solitary signals given by # and (middle and # Eq. (2.24b) right) the corresponding chirping defined by Eq. (2.25a) for small #4 A2cw − A2sol # with the nonlinear chirping parameter μ of different sign, ϑ0 = ϕ0 = 0, kcw = ksol = 0.2, P = 0.5, and Q 3 = 1. a, b Acw = 2, Asol = 3.9, Q 13 = 0.5, and Q 14 = −1/8 (meaning that μ = 0.625); c Acw = 2, Asol = 3.9, Q 13 = 0.5, and Q 14 = −1/8 (meaning that μ = −0.625); d, e Acw = 0.95, Asol = 2, Q 13 = 0.5, and Q 14 = −1/8 (meaning that μ = 0.625); f Acw = 0.95, Asol = 2, Q 13 = −0.5, and Q 14 = 1/8 (meaning that μ = −0.625). Plots of the top (bottom) panels show first-order rogue wave signals which are periodically loaded along the x− (t−) direction. Reprint from Ref. [19], Copyright 2022, with permission from Elsevier

# # # # A2 − 4 A2cw |ψ(x, 0)|2 = ## Acw + sol 2 Acw # 1− #

Asol 2 Acw

#2 # # # 1  ## . (2.26)  √ Q 4 A2 −A2 cos 3 2Pcw sol x + ϕ0 ## 

It follows from conditions 4 A2cw −A2sol

> 0 and 1 −

Asol 2 Acw

cos

Q3

√

4 A2cw −A2sol x 2P

 + ϕ0 >

0 for all x that the initial value of solution (2.24b) for ϑ0 = 0 given by Eq. (2.26) is defined for all real x. Therefore, |ψ(x, 0)|2 diverges as A2sol → 4 A2cw − 0. This means that a small periodic perturbation in the CW background may lead to the modulational instability. It is interesting to study the behavior of the chirped bright soliton # given by#Eq. (2.24b) and the corresponding chirping (2.25a) when kcw = ksol and #4 A2cw − A2sol # → 0. In such a situation, Tspat → +∞ and Ttemp → ∞. This means that the bright soliton given by Eq. (2.24b) and the corresponding frequency chirp (2.25a) under the parametric condition 0 < 4 A2cw − A2sol 0, then u(z) is periodic; if  = 0, g2 ≥ 0 and g3 ≤ 0, then u(z) is solitary wavelike [33]. Conditions under which Eq. (2.29) admits real and bounded solutions u(z) can be obtained by considering the phase diagrams of F(u) [33]. In the following, we focus ourselves to special cases of Eq. (2.29) leading to solitonlike and periodic solutions, that leads respectively to chirped solitary pulses and chirped periodic pulses propagating along the network of Fig. 2.1. Definition: We call two wave solutions u 1 (z) and u 2 (z) of Eq. (2.29) two sister wave solutions if there exist two real constants B0 = 0 and C0 = 0 so that u 1 (z) = A0 + B0 y(z), u 2 (z) = A0 + C0 w(z), C0 = B0 ,

(2.30a)

where A0 = 0 is any real solution of equation   ε + 2 A0 2δ + 3γ A0 + 2β A20 = 0,

(2.30b)

and y(z) and w(z) are solutions of the nonlinear ordinary differential equations

32

2 Nonlinear Schrödinger Models for Solitons Propagation …

 

dy dz

dw dz

2 = 4β B0 y 3 + 6 (γ + 2β A0 ) y 2 + 4 2

δ + 3γ A0 + 3β A20 y, B0

= 4βC0 w 3 + 6 (γ + 2β A0 ) w 2 + 4

(2.30c)

δ + 3γ A0 + 3β A20 w, (2.30d) C0

respectively. In what follows, any two wave solutions of Eq. (2.15) obtained with the use of two sister wave solutions of Eq. (2.29) will be referred to as two sister wave solutions of the NLS equation (2.15). The corresponding electric signals will be referred to as two sister electric signals. Sister Solitary Wave Solutions of the Higher-Order NLS Equation (2.15) In order that Eqs. (2.30c) and (2.30d) admit simultaneously solitary solutions, parameters β, γ , δ, and A0 must satisfy the condition    2  16   = 36β 2 δ + 3γ A0 + 3β A20 (γ + 2β A0 )2 − β δ + 3γ A0 + 3β A20 = 0, 9 g2 ≥ 0 and g3 ≤ 0; here [34]   g2 = 3 (γ + 2β A0 )2 − 16β δ + 3γ A0 + 3β A20 ,  

 g3 = (γ + 2β A0 ) 8β δ + 3γ A0 + 3β A20 − (γ + 2β A0 )2 . In what follows, we limit ourselves to the situation when δ + 3γ A0 + 3β A20 = 0. If we insert ε = −4Q 21 /P 2 into Eq. (2.30b), we obtain an equation for finding the free parameters Q 1 and A0 = 0. The strategy here is to consider A0 = 0 as an arbitrary real parameter and solve Eq. (2.30b) in Q 1 and then, replace δ by its expression obtained from the condition  = 0. For example, in the situation when δ + 3γ A0 + 3β A20 = 0, we obtain, after using Eq. (2.29), the following result: Q 21 = −

2 A0  2 υ + 4Pυ + 2P Q 3 A0 , A0 P Q 3 < 0. 32P Q 3

(2.31)

Integrating Eqs. (2.30c) and (2.30d) and taking into account the condition (2.31), we obtain the following sister solitary wave solutions of Eq. (2.29)  3A0 + 2 A0 − ρ 2 (z) = u(z) =

  3 υ 2 +4Pυ+6P Q 3 A0 sinh2 8P Q 3 B0

3 + 2 sinh2





υ 2 +4Pυ+6P Q 3 A0 z 4P 2

υ 2 +4Pυ+6P Q 3 A0 z 4P 2



 ,

(2.32a)

2.3 Modulational Instability and Transmission…  3A0 + 2 A0 − ρ 2 (z) = u(z) =

33

  3 υ 2 +4Pυ+6P Q 3 A0 sinh2 8P Q 3 C0

3 + 2 sinh2





υ 2 +4Pυ+6P Q 3 A0 z 4P 2

υ 2 +4Pυ+6P Q 3 A0 z 4P 2



 ;

(2.32b) here, A0 , B0 , C0 , and υ must be chosen from conditions B0 = C0 A0 > 0, P Q 3 < 0, υ 2 + 4Pυ + 6P Q 3 A0 > 0, 1 B0

>

16P Q 3 A0 , 1 3(υ 2 +4Pυ+6P Q 3 A0 ) C0

>

16P Q 3 A0 . 3(υ 2 +4Pυ+6P Q 3 A0 )

(2.33)

It is important to note that each pair of B0 and C0 satisfying conditions (2.33) leads to two sister solitonlike solutions (2.32a)-(2.32b) of Eq. (2.29). As we can see from Eqs. (2.32a) and (2.32b), two sister solitonlike solutions (2.32a)-(2.32b) will take the same value, A0 at z = 0. Now, if we use Eqs. (2.32b), (2.32a), (2.27), and (2.16), we obtain, under conditions (2.33), the following sister solitonlike solutions of Eq. (2.15) $    %   2 % υ 2 +4Pυ+6P Q 3 A0 % 3A0 + 2 A0 − 3 υ +4Pυ+6P Q 3 A0 sinh2 (x − υt) 8P Q 3 D0 % 4P 2 %   ψ(x, t) = ±% % & υ 2 +4Pυ+6P Q 3 A0 3 + 2 sinh2 (x − υt) 2 4P

× exp [i {λ(z) − υt + θ (x, t)}] .

(2.34)

In Eq. (2.34), one should take D0 = B0 for Eq. (2.32a) and D0 = C0 for Eq. (2.32b), θ (x, t) being the real function defined by Eqs. (2.17a) and (2.17b), and λ(z) is the solution of Eq. (2.28c) with ρ 2 (z) given by Eq. Eq. (2.32a) for D0 = B0 and Eq. (2.32b) for D0 = C0 . Using now Eq. (2.28d), we find the corresponding sister chirping as follows: υ 3Q 13 + 2Q 14 + 2P 4P      3 υ 2 +4Pυ+6P Q 3 A0 υ 2 +4Pυ+6P Q 3 A0 3A0 + 2 A0 − − υt) sinh2 (x 2 8P Q 3 D0 4P   × 2 +4Pυ+6P Q A υ 3 0 − υt) 3 + 2 sinh2 (x 4P 2

δω(x, t) = −

−

Q1 P

3 + 2 sinh2



υ 2 +4Pυ+6P Q 3 A0 4P 2

 (x − υt)

 .   3 υ 2 +4Pυ+6P Q 3 A0 ) υ 2 +4Pυ+6P Q 3 A0 sinh2 − υt) 3A0 + 2 A0 − ( (x 2 8P Q 3 D0 4P

(2.35)

34

2 Nonlinear Schrödinger Models for Solitons Propagation …

Fig. 2.13 a and c: The transmission of two sister bright-dark solitary pulses along the network of 3P 3P Fig. 2.1, generated with the use of the wave solution given by Eq. (2.34) for B0 = 2Q , C0 = − 4Q , 3 3 respectively. b and d Frequency chirp given by Eq. (2.35) corresponding to two sister bright-dark solitary signals of plots (a) and (c), respectively. Different parameters used in generating these plots are given in the text. Reprint from Ref. [19], Copyright 2022, with permission from Elsevier

It is important to notice that depending on the choice of D0 = B0 or/and D0 = C0 and the coefficients of the NLS equation (2.15), the sister solitonlike solutions (2.34) with the corresponding chirp have diverse behaviors, as we can see from the examples below where we use    3A0 Q 3 υ = υ± = −2P 1 ± 2 − , 2P so that υ 2 + 4Pυ + 6P Q 3 A0 = 4P 2 . Transmission of two sister bright-dark solitonlike wave signals along the network To experience the transmission of two sister bright-dark solitonlike wave signals along the network under consideration, we consider the following set of parameters for our first example: A0 = 1, B0 =

 √  3P 3P , C0 = − , 3Q 13 + 2Q 14 = 4P, Q 3 = −3 + 2 2 P. 2Q 3 4Q 3

2.3 Modulational Instability and Transmission…

35

Fig. 2.14 Spatiotemporal evolution of two sister dark-dark solitary pulses obtained with the use of 3P P the analytical solution defined by Eq. (2.34) for a B0 = − 5Q and b C0 = − 2Q . The frequency 3 3 chirp δω(x, t) given by Eq. (2.35) are shown in plots (b) and (d) and correspond respectively 3P P to B0 = − 5Q and C0 = − 2Q . Reprint from Ref. [19], Copyright 2022, with permission from 3 3 Elsevier

From Eq. (2.31), we then compute Q 1 = −2P. For P = − 21 and υ = υ+ . For these parameters, we depict in Fig. 2.13 the spatiotemporal evolution of two sister brightdark solitonlike wave pulses and the corresponding chirping along the network of Fig. 2.1. Figure 2.13a, c show the evolution of bright and dark solitonlike wave pulses, respectively; in Fig. 2.13b and d, we show respectively the evolution of a dark solitonlike signal and a dark-bright-dark solitonlike signal. Thus, the frequency chirp Fig. 2.13b of the bright solitonlike wave (a) is of a dark solitonlike wave type, while the dark solitonlike wave signal plotted in Fig. 2.13c leads to a dark-bright-dark solitonlike wave (dark two-soliton). Evolution of two sister dark-dark solitonlike pulses through the network To demonstrate the evolution of two sister dark-dark solitonlike pulses through the network of Fig. 2.1, we consider the same set of parameters as in the previous example, but with 3P P and C0 = − . B0 = − 5Q 3 2Q 3 For those parameters, we display in Fig. 2.14 the evolution of two sister dark-dark solitonlike pulses and the corresponding chirping. As we can see from different plots

36

2 Nonlinear Schrödinger Models for Solitons Propagation …

Fig. 2.15 Transmission of two sister bright-bright solitary signals through the lossless network of 3P Fig. 2.1 generated with the help of exact the solution given by Eq. (2.34) for a B0 = − 5Q and c 3

P C0 = − 2Q with P = − 21 , and the corresponding frequency chirp given by Eq. (2.35). b and d are 3

3P P and C0 = − 2Q , respectively. Other parameters are given in the text. obtained with B0 = − 5Q 3 3 Reprint from Ref. [19], Copyright 2022, with permission from Elsevier

of Fig. 2.14, the frequency chirp corresponding to the two sister dark-dark solitonlike signals of Fig. 2.14a and c are dark-bright-dark pulses, as it is well seen from Fig. 2.14b and d. Propagation of Two Sister Bright-Bright Solitonlike Signals Along the Network Two sister bright-bright solitonlike signals can be generated with the following set of parameters A0 = 2, B0 =

 √  P 3P , C0 = , 3Q 13 + 2Q 14 = 4P, Q 3 = −P, υ = υ+ = −2 1 + 5 P. Q3 2Q 3

From Eq. (2.31), we then obtain Q 1 = −3P. For P = − 21 , we depict in Fig. 2.15 the evolution of the two sister bright-bright solitonlike pulses (Fig. 2.15a and c); the corresponding chirping are depicted in Fig. 2.15b and d, respectively. As we can see from Fig. 2.15b and d, the corresponding chirping for the two sister bright-bright solitonlike signals shown in Figs. 2.15a and c are dark-bright-dark solitonlike waves.

2.3 Modulational Instability and Transmission…

37

Fig. 2.16 a and c The evolution of two sister solitary wave pulses along the network of Fig. 2.1 for 3P respectively B0 = − QP3 and C0 = − 4Q ; b and d Plots of the chirping corresponding to the two 3 sister dark-dark solitary signals plotted on (a) and (c), respectively. Other parameters are given in the text. Reprint from Ref. [19], Copyright 2022, with permission from Elsevier

Engineering Two Sister Solitonlike Pulses With Bright-Dark-Bright Chirping We now turn to engineer two sister solitonlike pulses with bright-dark-bright chirping. For this end, we consider the following set of parameters:  √  1 P 3P , B0 = − , C0 = − , Q 3 = −2 15 + 4 14 P, 2 Q3 4Q 3

 √ υ = −P 2 − 98 + 24 14 , 3Q 13 + 2Q 14 = −12P, Q 1 = 2P = 1.

A0 =

For this set of parameters, we display in Fig. 2.16a and c two sister dark-dark solitonlike signals. The corresponding chirping are displayed in Fig. 2.16b for B0 = − QP3 3P and 2.16d for C0 = − 4Q . As one can seen from Fig. 2.16b and d, the chirping 3 of the two sister dark-dark solitonlike waves plotted in Fig. 2.16a and c show a bright-dark-bright solitonlike (bright two-soliton solitonlike) behavior. Two Sister Periodic Wave Solutions of the Higher-Order NLS Equation (2.15) To find periodic solutions of Eqs. (2.30c) and (2.30d), we consider only the case when  = 0 [34]. Inserting Eq. (2.29) into Eq. (2.30b) and solving the resulting equation in δ yield

38

2 Nonlinear Schrödinger Models for Solitons Propagation …

Fig. 2.17 Top: Amplitude profile of the two sister dn cnoidal signals obtained with the use of analytical solutions (2.43a) (solid line) and (2.43b) (dashed line) for two different values of QP3 : a

P P Q 3 = −0.1, and b Q 3 = −0.245. Bottom: Effects of the GVD parameter P and cubic nonlinearity Q 3 on the femtosecond cnoidal pulses propagating through the electric network of Fig. 2.1 and modeled by the higher-order NLS Eq. (2.15). Plots (c) and (d) are generated with the help of respectively the dn periodic solution (2.43a) and the nd periodic solution (2.43a), for QP3 = −0.1

(solid line) and QP3 = −0.245 (dashed line). Other parameters are given in the text. Reprint from Ref. [19], Copyright 2022, with permission from Elsevier

δ=

  2P Q 3 A20 + υ 2 + 4Pυ A0 Q 21 + . P 2 A0 4P 2

(2.36a)

First of all, we notice that if B0 and C0 satisfy the relationship   2 υ + 4Pυ + 4P Q 3 A0 A20 − 4Q 21 δ + 3γ A0 + 3β A20 = B0 C0 = β 2P Q 3 A0

(2.36b)

and y(z) is a periodic wave solution of Eq. (2.30c), then w(z) = 1/y(z) will be a periodic wave solution of Eq. (2.30d); therefore, u 1 (z) = A0 + B0 y(z),   2 υ + 4Pυ + 4P Q 3 A0 A20 − 4Q 21 1 u 2 (z) = A0 + 2P Q 3 A0 B0 y(z)

2.3 Modulational Instability and Transmission…

39

Fig. 2.18 Chirping profile (2.28d) for the two sister dn-nd periodic pulses (top) in Fig. 1.26a and +2Q 14 +2Q 14 . Top panels: a 3Q 134P = −5, (bottom) in Fig. 1.26b for three different values of 3Q 134P 3Q 13 +2Q 14 3Q 13 +2Q 14 3Q 13 +2Q 14 +2Q 14 b = −3.5, and c = 5; Bottom panels: d = −8, e 3Q 134P = 4P 4P 4P +2Q 14 −12, and f 3Q 134P = −16. Plots in solid line correspond to solution (2.43a), while those in dashed line are obtained with the use of solution (2.43b). Other parameters are given in the text. Reprint from Ref. [19], Copyright 2022, with permission from Elsevier

will be two sister periodic wave solutions of Eq. (2.29). Henceforth, we consider A0 = 0 as an arbitrary real parameter so that δ could be expressed in terms of A0 as showed in Eq. (2.36a). If we use Eq. (2.36a) and introduce α1 , α2 , and α3 as follow, α1 = −

  4Q 21 − υ 2 + 4Pυ + 4P Q 3 A0 A20 2Q 3 υ 2 + 4Pυ + 6P Q 3 A0 , α = D0 , α2 = − 3 P P2 P 2 D0 A 0

(2.37a) for D0 ∈ {B0 , C0 }, equations. (2.30c) and (2.30d) will take the common form 

dy dz

2 = α1 y 3 + α2 y 2 + α3 y.

(2.38)

In the following subsections, we describe two classes of two sister periodic wave solutions of Eq. (2.38) with the help of which one can build two sister periodic solutions of the higher-order NLS equation (2.15) that model the propagation of two sister periodic wave pulses through the network of Fig. 2.1. Two sister Cnoidal wave solutions Solutions y(z) of this class is sought as a quadratic function of sn [μz, m] , namely, y(z) = A + Bsn2 [μz, m] ,

(2.39a)

where sn [μz, m] is the Jacobi elliptic sine function with the elliptic modulus m ∈ [0, 1], μ being an arbitrary real parameter. Asking that function (2.39a) satisfies Eq.

40

2 Nonlinear Schrödinger Models for Solitons Propagation …

(2.38) leads, under the conditions α22 − 4α1 α3 > 0, α2 ±

α22 − 4α1 α3 > 0,

(2.39b)

to the periodic Cnoidal wave solution ⎤ 

2 −α2 ± α2 − 4α1 α3 ⎥ ⎢ α2 + 4μ ⎢− ⎥ ⎢ ⎥ 2α3 ⎢ 4α1 α3 ⎥; y(z) = −   ⎢ ⎥  α2 + 4μ2 ⎢ ⎥ α1 α3 1 2 ⎣  ⎦ +sn μz, −  μ 2 α2 + 4μ2 ⎡



2



(2.39c)

here, μ is any real root of the algebraic equation 

32μ4 + 4 α2 ± 3 α22 − 4α1 α3 μ2 − 4α1 α3 − α22 ± 3α2 α22 − 4α1 α3 = 0 (2.40a) satisfying the condition −1≤

α1 α3   < 0. 2 α2 + 4μ2 μ2

(2.40b)

Two Sister dn-nd Wave Solutions When α1 , α2 , and α3 satisfy condition α22 − 4α1 α3 > 0 and one of the conditions

− α2 ±

α22 − 4α1 α3 > 0 and



  α2 + 4μ2 −α2 ± α22 − 4α1 α3 4α1 α3

> 1,

(2.40c) solution (2.39c) will satisfy the condition y(z) = 0 for all z, so that w(z) = 1/y(z) will also be a Cnoidal wave solution of Eq. (2.30d) in the special situation when B0 C0 is defined by equation (2.36b). Expressing sn2 in terms of dn2 , we derive from Eq. (2.39c), under the condition α22 > 4α1 α3 > 0 and α2 > 0,

(2.40d)

the following dn periodic solution of Eq. (2.30c)

y(z) = −

α2 + α22 − 4α1 α3 2α1

$ ⎡ $

% % % % α + α 2 − 4α α % 2 α22 − 4α1 α3 & ⎢ 2 1 3 1 2

dn2 ⎢ z, % & ⎣2 2 α + α 2 − 4α α 2

2

⎤ ⎥ ⎥. ⎦

1 3

(2.41a)

2.3 Modulational Instability and Transmission…

41

Fig. 2.19 Effects of the GVD parameter P and the cubic nonlinearity parameter Q 3 on the frequency chirp for the two sister dn-nd periodic signals given by Eqs. (2.43a) and (2.43b) for the same set +2Q 14 of parameters as in Fig. 2.18 for 3Q 134P = −16 and three values of QP3 , QP3 = −0.22 (solid line), QP3 = −0.23 (dashed line), and QP3 = −0.24 (dot-dashed line). a Chirping profile for the dn periodic wave pulse (2.43a); b Chirping profile for the nd periodic wave signal (2.43b). Reprint from Ref. [19], Copyright 2022, with permission from Elsevier

It is important to note that Eq. (2.41a) and Eq. (2.39c) are two distinct solutions of Eq. (2.30c); indeed, they are obtained under different conditions of α1 , α2 , and α3 . Because y(z) = 0 for all real z and B0 and C0 satisfy the Eq. (2.36b), solution (2.41a) gives birth to the solution $ ⎡ $ ⎤

% % % % α + α 2 − 4α α 2 − 4α α 2 α % & ⎢ ⎥ 2 1 3 1 3 2α1 1 2 2 ⎥,

w(z) = − nd2 ⎢ z, % & ⎣ ⎦ 2 2 α2 + α22 − 4α1 α3 α2 + α22 − 4α1 α3

(2.41b) of Eq. (2.30d), where nd(x, m) = 1/dn (x, m). Two Sister Periodic Wave Solutions of Eq. (2.29) By using the above two sister periodic wave solutions (2.39c) and (2.41a) of Eq. (2.38), we can, with the use of Eq. (2.30a), write down the following two sister periodic wave solutions of Eq. (2.29), ρ 2 (z) = u 1 (z) = A0 + B0 y B0 (z), ρ 2 (z) = u 2 (z) = A0 + C0 yC0 (z),

(2.42a)

where y B0 (z) and yC0 (z) are obtained from Eqs. (2.39c) and (2.41a) by replacing respectively D0 by B0 and C0 in α1 and α3 , as one can see from Eq. (2.37a). For example, in the special case when B0 and C0 satisfy Eq. (2.36b), Eqs. (2.41a) and (2.41b) lead to the following two sister dn-nd periodic wave solutions of Eq. (2.29)

ρ 2 (z) = u 1 (z) = A0 +



2 P α2 + α2 − 4α1 α3 4Q 3

dn2

42

2 Nonlinear Schrödinger Models for Solitons Propagation …

$ ⎡ $

% % % α + α 2 − 4α α % 2 α 2 − 4α α % & ⎢1 2 1 3 1 3 2 2 %

×⎢ z, & ⎣2 2 α + α 2 − 4α α 2

⎤ ⎥ ⎥, ⎦

(2.43a)

1 3

2

 

 2 υ 2 + 4Pυ + 4P Q 3 A0 A20 − 4Q 21  ρ (z) = u 2 (z) = A0 +

P 2 A0 α2 + α22 − 4α1 α3 2

$ ⎡ $

% % % α + α 2 − 4α α % 2 α 2 − 4α α % & ⎢ 2 1 3 1 3 2 2 1

z, % ×nd2 ⎢ & ⎣2 2 α + α 2 − 4α α 2

2

⎤ ⎥ ⎥ . (2.43b) ⎦

1 3

Here, α1 , α2 , and α3 in which D0 = B0 satisfy conditions (2.40d). Henceforth, solutions (2.43a) and (2.43b) will be referred to as dn and nd type solutions of Eq. (2.29). From the equality u(z) = ρ 2 (z), various free parameters A0 = 0, B0 = 0, Q 1 = 0, and υ must be chosen so that u 1 (z) and u 2 (z) should be positive. Figures 2.17 and 2.18 display the amplitude profile of typical two sister dn wave (2.43a) and nd wave (2.43b) and the corresponding chirping for the following numerical values of parameters    1 10 199 199 Q3 , α1 α3 = , υ = P −2 + A0 − 199 , −15 000 α2 = 2500 625 50 P   P 1 P Q3 A0 + α2 A20 , A20 = 1, m = 0.99. − α1 α3 A0 − 2 Q1 = 2 2 Q3 P For the above numerical values of parameters, we show in Fig. 2.17 the typical periodic solutions generated with the use of Eqs. (2.43a) and (2.43b); different plots of this figure behave like periodic dark solitonlike waves. As we can clearly see from plots of Fig. 2.17, the amplitude of the nd wave given by Eq. (2.43b) is higher than that of the dn wave given by Eq. (2.43a). Also, Fig. and d show that the # # 2.17c #P# amplitude and the width of the wave increase when # Q 3 # increases. We can then conclude that parameter P of the GVD and parameter Q 3 of the cubic nonlinearity can be used to control two sister nonlinear modulated waves of the higher NLS equation (2.15). Figure 2.18 show how parameters Q 13 and Q 14 of self-steeping and self-frequency shift can be used to manage the motion of the chirping. Depending on the choice of these two parameters, the chirping corresponding to the two sister dn-nd periodic wave solutions (2.43a) and (2.43b) can behave like (i) periodic bright solitons as it is seen from Fig. 2.18a, (ii) periodic bright two-solitons as one can see from dashed plot of Fig. 2.18b, (iii) periodic dark solitons as shown in Fig. 2.18c, or (iv) periodic bright-dark-bright solitons, as we can see from dashed plots of Fig. 2.18d–f. Comparing plots of the top panels of Fig. 2.18 obtained with P/Q 3 = −1

2.3 Modulational Instability and Transmission…

43

Fig. 2.20 Top: Spatiotemporal evolution of the amplitude |ψ(x, t)|2 showing the evolution of two sister pulses of type dn (a) and b periodic waves according to the wave solutions given by Eqs. (2.44a) and (2.44b), respectively, for P = 21 , Q 3 = −5, 3Q 13 + 

Q3 199 199 1 2Q 14 = −15P, α2 = 10 2500 , α1 α3 = 625 , υ = P −2 + 50 −15 000 P A0 − 199 , Q 1 =    Q3 P 1 P 2 2 − 2 Q 3 α1 α3 A0 − 2 P A0 + α2 A0 , and A0 = 1. Bottom: The spatiotemporal chirping # # +2Q 14 2 υ + 3Q 134P ρ (z) − QP1 ρ 21(z) # for (c) the dn periodic modulated wave δω(x, t) = − 2P z=x−υt

plotted in (a) with ρ 2 (z) given by Eq. (2.43a) and (d) the nd periodic modulated wave plotted in (b) with ρ 2 (z) given by Eq. (2.43b). Reprint from Ref. [19], Copyright 2022, with permission from Elsevier

with those of the bottom panels generated with P/Q 3 = −0.245, we conclude that the behavior of the chirping also depends on parameter P of the GVD and parameter Q 3 of the cubic nonlinearity. As we can clearly see from plots of Fig. 2.18, the amplitude and the width of the chirping increase as |P/Q 3 | increases. For a better understanding, we display in Fig. 2.19 the chirping profiles for both the dn and nd periodic wave solutions (2.43a) and (2.43b) for different values of P/Q 3 . In order to obtain two sister periodic wave solutions of Eq. (2.15), we go back to ψ(x, t) by substituting Eq. (2.42a) into Eq. (2.27) and then, inserting the resulting equation into Eq. (2.16). For example, two sister dn and nd periodic wave solutions of the higher-order NLS equation (2.15) generated from Eqs. (2.43a) and (2.43b) are found to be ψ(x, t) $  $ ⎡ $

%

% % % % % α + α 2 − 4α α P α2 + α22 − 4α1 α3 % % 2 α22 − 4α1 α3 & 2 ⎢ 1 3 % 1 2

dn2 ⎢ =% (x − υt) , % & ⎣2 & A0 + 4Q 3 2 α + α 2 − 4α α 2

× exp [i {λ(z) − υt + θ(x, t)}] ,

2

⎤ ⎥ ⎥ ⎦

1 3

(2.44a)

44

2 Nonlinear Schrödinger Models for Solitons Propagation …

ψ(x, t) ⎧ ⎡ $

% ⎪

   % ⎪ 2 ⎨ ⎢ 1 & α2 + α2 − 4α1 α3 2 υ 2 + 4Pυ + 4P Q 3 A0 A20 − 4Q 21 2 ⎢  = A0 + nd ⎣ (x − υt) ,

⎪ 2 2 ⎪ ⎩ P 2 A0 α2 + α22 − 4α1 α3 $ % % % × % &

⎤⎫ 2

⎪ ⎬ 2 α22 − 4α1 α3 ⎥⎪ ⎥

exp [i {λ(z) − υt + θ(x, t)}] ⎦ ⎪ α2 + α22 − 4α1 α3 ⎪ ⎭ 1

(2.44b)

where α1 , α2 , and α3 are given by Eq. (2.37a) with which D0 = B0 and satisfy conditions (2.40d), θ (x, t) is any real function satisfying conditions (2.17a) and (2.17b), λ(z) is any solution of Eq. (2.28c), with z = x − υt. The free parameters A0 , Q 1 = 0, υ, δ, and B0 must be suitably chosen from Eq. (2.36a), that is, they must be chosen so that the conditions (2.40d) are fulfilled and the main square root in Eq. (2.44a) is defined. Chirping δω(x, t) associated with solutions (2.44a) and (2.44b) are obtained by inserting respectively Eqs. (2.43a) and (2.43b) with z = x − υt into Eq. (2.28d). In Fig. 2.20a and b, we depict the spatiotemporal evolution of the two sister dn and nd periodic nonlinear modulated waves obtained with the help of respectively solutions (2.44a) and (2.44b) of the higher-order NLS equation (2.15). The evolution of the corresponding chirping is shown in Fig. 2.20c and d, respectively. From Fig. 2.20a and b, it is clearly seen that the two sister dn-nd modulated wave corresponding to solutions (2.44a) and (2.44b) behave like periodic dark solitons. The chirping shown in Fig. 2.20c has the behavior of periodic bright solitons. It is clearly seen from Fig. 2.20d that the chirping for the nd modulated wave plotted in Fig. 2.20b behaves like periodic bright two-solitons. In conclusion, we have showed that the propagation of ultrashort pulses along the lossless network of Fig. 2.1 are modeled by a higher-order nonlinear Schrödinger equation with derivative terms. With the help of this NLS equation, the phenomenon of the modulational instability in the network under consideration has been investigated and our results reveal that the competing higher-order nonlinearity induces propagating on CW background solitonlike bright (dark) solitary wave and rogue wave signals in the lossless network of Fig. 2.1. The concept of two sister nonlinear modulated waves was introduced and their properties were explored; through bright solitonlike waves and periodic waves, the propagation of such nonlinear modulated waves in the network under consideration is analyzed. We have shown that the nonlinear chirp associated with each of electric solitary pulses embedded on a CW background is directly proportional to the intensity of the pulse, while chirping associated with each two sister modulated nonlinear waves is formed of two nonlinear parts, one of which is directly proportional to the intensity of the wave, while another part is inversely proportional to the intensity of the wave. Also, we have obtained that the amplitude of the chirping can be managed by varying the parameters of the GVD and cubic nonlinearity.

References

45

Our investigations also revealed the amplification [the compression] of two sister chirped femtosecond modulated signals can be enhanced by either increasing |P| or decreasing |Q 3 | [decreasing |P| or increasing |Q 3 |], and the wave structure is not destroyed during the process. It is also established that these two parameters P and Q 3 have important effects of the feature of the chirping. They also have some effect on the unlimited transmission of the compressed amplified chirping in an inhomogeneous media. Through our results, one can conclude that the self-steeping and self-frequency parameters Q 13 and Q 14 have a universal influence on chirping dynamics.

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Chapter 3

Transmission of Dissipative Solitonlike Signals Through One-Dimensional Transmission Networks

Abstract This chapter presents analytical results for the dynamics of nonlinear modulated waves in dissipative electrical nonlinear transmission networks. We employ the reductive perturbation method in the semi-discrete limit to derive a dissipative NLS equation that models the propagation of nonlinear modulated waves in the given network. Based on both the direct method and the method of Weierstrass elliptic functions, we present classes of bright, kink, dark, or chirped Lambert W-kink solitonlike solutions of the amplitude equations. These solitonlike solutions are then used to investigate analytically the transmission of dissipative solitonlike signals through the networks under consideration. Effects of the dissipation are also investigated. Naturally, the wave’s amplitude decreases and its width increases under the action of the dissipation. We show how the dissipative elements of the networks can be used to control the motion of solitonlike signals during their propagation along the dissipative network system under consideration.

3.1 Chirped Lambert W-Kink Waves Propagation in a Lossy Electrical Transmission Network 3.1.1 Introduction By means of both dissipative CQ NLS equations and complex CQ CGL equations, the dynamics of nonlinear modulated waves in lossy electrical nonlinear transmission networks have been studied in details [1–3] and it has been proven that both the cubic CGL equations as well as the cubic-quintic ones support a class of localized solutions such as stationary solitons, sources, sinks, Lambert W-kink solitons, moving solitons and fronts with fixed velocity [4, 5]. Also, these models admit erupting soliton solutions, which periodically exhibit explosive instability [6, 7]. Such erupting solitons were found analytically, numerically, and experimentally in ultrafast fiber lasers with normal dispersion and in passively mode-locked lasers [8, 9]. The Lambert function W(x), also called the product logarithm, defined as one inverse to x = W exp(W) [10–12] finds many applications in physics, applied mathematics, and computer science [13, 14]. In the present Section, we mainly aim to © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 E. Kengne and W. Liu, Nonlinear Waves, https://doi.org/10.1007/978-981-19-6744-3_3

47

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3 Transmission of Dissipative Solitonlike Signals …

demonstrate through a generalized CQ CGL equation [15] how the Lambert function W function can be used to investigate the propagation of chirped pulses along a lossy nonlinear electric transmission network. The model under consideration is the discrete nonlinear transmission network with dissipative elements made of N identical blocks, illustrated in Fig. 3.1 [1]. Each single block contains a linear inductor with inductance L1 in the series branch and a linear inductor with inductance L2 in parallel with the bias-dependent nonlinear capacitor C(V ) in the shunt branches. To take into consideration the dissipative effects of the network, two linear conductances G 1 and G 2 are connected in parallel with, respectively, L1 and L2 . In this model, the nonlinear capacitance C(V ), responsible for nonlinearity of the system, consists of a reversed-biased diode with differential capacitance which is a function of voltage Vn across the n-th capacitor, biased by a constant voltage Vb . For low voltage, it is of the form [1] C(Vb + Vn ) =

  dQn  C0 1 − 2αVn + 3βVn2 ; dVn

(3.1a)

here, C0 = C(Vb ) is the capacitance of the nonlinear diode at the dc bias-voltage Vb , and α and β are nonlinear parameters of the stored electrical charge Qn . Here, α is assumed to be positive [16]. If we apply the Kirchhoff’s laws to the network in Fig. 3.1, we arrive to the following difference differential system for the propagation of modulated waves in the model    2  d d d 2 2 V + ω + 2u  σ + u + 2u σ (2Vn − Vn−1 − Vn+1 ) 0 2 n 0 1 0 0 dt 2 dt dt  d2  = 2 αVn2 − βVn3 , (3.2) dt with n = 1, 2, . . . , N − 1; here, u02 = (C0 L1 )−1 and ω02 = (L2 C0 )−1 are the dimensionless capacitance and squared characteristic frequency of the network, respecσ2 are dimensionless conductances tively, while σ1 and   σ1 =

 L1 G1,  σ2 = 4C0

L1 G2. 4C0

(3.3)

Throughout this Section, numerical results will be displayed for the network parameters L1 = 220 µH, L2 = 470µH, C0 = 370 pF, α = 0.21 V−1 , β = 0.0197 V−2 . (3.4) For the numerical simulation, the total number N of cells in the network must be chosen so to eliminate wave reflection at the edge of the network.

3.1 Chirped Lambert W-Kink Waves Propagation in a Lossy Electrical …

49

3.1.2 Amplitude Equation In order to derive the governing equation (amplitude equation) that models the propagation of slowly modulated waves in the lossy network of Fig. 3.1, we employ the reductive perturbation method in the semidiscrete limit. In other words, we introduce the continuum approximation for the envelope and keep a discrete description for the carrier wave, whose wavenumber kp ranges over the Brilloum zone (0 ≤ kp ≤ π ), with the corresponding frequency ωp = 2π fp . For this aim, two slow time scales are introduced as T1 = εt and T2 = ε2 t, in addition to the original one, T0 ≡ t. Also, one slow spatial coordinate X1 is introduced as X1 = εx, in addition to original one, x ≡ n. When studying the dissipative effects on the dynamics of modulated waves propagating through the network system of Fig. 3.1, Yemélé et al. [1] have found that the most important one comes from conductance G 1 in the series branch of the network. Conductance G 2 in the shunt branch may then be disregarded when addressing the linear propagation, while effects originating from G 2 and the nonlinearity appear in the same perturbation equation; therefore we set  σ2 ≡ ε2 σ2 . Next, the solution Vn (t) of the difference-differential system (3.2) is sought in the general form [2] Vn (t) =

        εu exp i θ + ε u20 + u22 exp 2i θ + ε3/2 u30 + u33 exp 3i θ      2 θ + u44 exp 4i θ (3.5) +ε u40 + u42 exp 2i       7/2  5/2 +ε u50 + u53 exp 3i , θ + u55 exp 5i θ + c.c. + O ε

√

where  θ = (k + iχ ) n − ωt =  kn − ωt is the rapidly varying complex phase, c.c. stands for the complex conjugation, while u and ulm are complex amplitudes of variables X1 , T1 , T2 (l = 2, 3, 4, 5, m = 0, 2, 3, 4, 5); here, ω is the real angular frequency, while k = kp and χ are respectively the wavenumber and the spatial linear parameter. The spatial linear parameter χ is introduced to take into account dissipative effects on the linear oscillations originating from conductance G 1 . Expanding Vn±1 (t) into Taylor series and inserting Eq. (3.5) in (3.2) yields a power series in ε1/2 and exp (iθ ), where θ = kn − ωt. Equating to zero coefficients of like powers of ε1/2 , exp (iθ ) yields a series of equations for u and ulm .

Fig. 3.1 The schematic representation of the dissipative nonlinear discrete electrical transmission network with a linear inductor placed in the series branch, and a nonlinear capacitor in the parallel branch. Dissipative elements are connected in parallel with two linear inductors. This network is built of N identical blocks. Reprint from Ref. [17], Copyright 2022, with permission from Springer

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3 Transmission of Dissipative Solitonlike Signals …

  At order ε1/2 , exp (iθ ) , we obtain that ω, k, and χ must satisfy the complex equation       k − ω02 − 2u02 1 − cosh i k = 0. ω2 + 4iωσ1 u0 1 − cosh i

(3.6)

Expanding Eq. (3.6) leads respectively to the following linear dispersion relation and the spatial linear parameter   1 ω2 8σ12 − 1 + 2u02 + ω02   cos k = μ(ω) 2 u02 + 4ω2 σ12

 χ (ω) = ln μ(ω) + μ2 (ω) − 1 ,

and

(3.7a)

(3.7b)

where

   1 μ(ω) =  2 

⎡



⎢ 4σ12 ω2 ω2 − ω02 ⎢ ⎢1 + ⎣ 4u02 E12

2

+ u02 E22

⎤

⎛ ⎞2  2  2 2 2 2 2 2  4σ1 ω ω − ω0 + u0 E2 E22 ⎥ ⎥ ⎟ ⎜ + ⎝1 + ⎠ − 2 ⎥, 2 2 4u0 E1 E1 ⎦

(3.7c) with

  E1 = u02 + 4ω2 σ12 , E2 = ω02 + 2u02 − ω2 1 − 8σ12 .

(3.7d)

To investigate the effects of the dissipative parameter σ1 on both the propagating frequency fp = ω/2π and the linear dissipation parameter χ , we depict in Fig. 3.2a, b the linear dispersion relation (3.7a) and the variation of the spatial linear dissipation parameter χ given by Eq. (3.7b) for different values of σ1 . It is seen from Fig. 3.2a that, at the first order with respect to σ1 (e.g., σ1 ∼ 10−2 ), the linear dispersion curve of the network (3.7a) can be approximated by the well-known dispersion relation of the typical bandpass filter, ω2 − ω02 − 4u02 sin2 (k/2) = 0,

(3.8)

obtained from Eq. (3.6) by setting σ1 = 0. Figure 3.2 shows that the spatial linear dissipation parameter χ is an increasing function of the angular frequency ω in the operating range of the network, meaning that the amplitude of the propagating wave decays more rapidly when the frequency of input wave increases.  At order ε3/2 , eiθ , one gets ∂u ∂u + υg = iN22 e−2χn |u|2 u, ∂T1 ∂X1

(3.9)

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51

Fig. 3.2 a The linear dispersion curve of the network defined by Eq. (3.7a) for different values of the dimensionless conductance σ1 : σ1 = 0 (the solid line), σ1 = 0.01 (the dashed-dotted line), and σ1 = 0.03 (the dashed line). b The variation of the spatial linear dissipation parameter χ as a function of frequency ω, according to Eq. (3.7b) for three values of σ1 : σ1 = 0.01 (the solid line), σ1 = 0.015 (the dashed-dotted line), and σ1 = 0.02 (the dashed line). Network parameters used to generate different plots are given in the text. Reprint from Ref. [17], Copyright 2022, with permission from Springer

with   k u0 (2σ1 ω + iu0 ) sinh i 1 ω2 (3β − 2α (2α + N12 )) dω    , υg =    , N22 = =− 2 ω + 2iσ1 u0 1 − cosh i k k d k ω + 2iσ1 u0 1 − cosh i (3.10) where N12 given by N12 =

4αω2 . 4ω2 − ω02 − 2u0 (u0 − 4iσ1 ω) {1 − cosh [2 (χ − ik)]}

(3.11)

Denoting by  kc the critical wavenumber satisfying condition N22 ( kc ) = 0 and setting    k = k − kc we obtain that       kc +  kc N22  k − N22  dN22 lim =  k→0  k d k k=k

c

so that

      dN22     N22 k = N22 kc + k ≈ k . d k k=kc

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3 Transmission of Dissipative Solitonlike Signals …

 with Q  =  Assuming that  k = O (ε) and letting N22 ≈ εQ k





dN22 , we follow d k  kc k=

  Kakutani and Michihiro  3/2  [18] and focus ourselves on a vicinity of kc , setting N22 = εQ , Eq. (3.9) becomes so that at order O ε ∂u ∂u + υg = 0. ∂T1 ∂X1

(3.12)

Equation (3.12) meansthat u is a function of two variables ξ = X1 − υg T1 and T2 , that is, u(X1 , T1 , T2 ) = ψ ξ = X1 − υg T1 , T2 . We then conclude that in the reference frame moving with group velocity υg = υgr + iυgi , the amplitude u of the signal remains constant with respect to ξ = X1 − υg T1 . Thus, the rhs of Eq.  (3.9) will be shifted to the corresponding nonsecular condition at order ε5/2 , eiθ . Equations of order ε5/2 eiθ lead to the following amplitude equation, a generalized CQ CGL equation for the dynamics of modulated waves in the network system of Fig. 3.1 ∂ψ ∂ψ ∗ ∂ 2ψ ∂ψ + iQ4 ψ 2 ; + P 2 + iRψ = Q1 |ψ|2 ψ + Q2 |ψ|4 ψ + iQ3 |ψ|2 ∂T2 ∂ξ ∂ξ ∂ξ (3.13a) here, i

      k = Q1r + iQ1i , P=P  k = Pr + iPi , R = R  k = Rr + iRi , Q1 = Q1        k = Q2r + iQ2i , Q3 = Q3  k = Q3r + iQ3i , and Q4 = Q4  k = Q4r + iQ4i Q2 = Q2 

are complex functions of complex variable  k = k + iχ , given by    k υg υg + 4σ1 u0 sinh i     , P=−  k 2 ω + 2iσ1 u0 1 − cosh i σ2 u0 ω    , R= k ω + 2iσ1 u0 1 − cosh i     dN22 Q1 = e−2χn  k − kc , d k k=kc

(3.14a) (3.14b) (3.14c)

  ∗ ω2 e−4χn 2αN12 N33 − 16α NN4341 − 3β 4α 2 + N33 + 4αN12 + 2 |N12 |2      , Q2 = k 2 ω + 2iσ1 u0 1 − cosh i (3.14d) ωυg N41 [3β − 2α (2α + N12 )] − 2iαω2 N42      , k N41 ω + 2iσ1 u0 1 − cosh i ωυg∗ [3β − 2α (2α + N12 )]    . Q4 = e−2χn k ω + 2iσ1 u0 1 − cosh i Q3 = 2e−2χn

(3.14e) (3.14f)

3.1 Chirped Lambert W-Kink Waves Propagation in a Lossy Electrical …

53

Complex amplitudes ujk used in expressions (3.5) are given in terms of ψ by u20 = 2αe−2χ n |ψ|2 , u22 = N12 ψ 2 , u30 = u40 = u50 = 0, u33 = N33 ψ 3 , u44 = N44 ψ 4 , u55 = N55 ψ 5 , (3.15) N43 ∂ψ N N42 ∂ψ N |ψ|2 ψ 2 , u53 = 6 53 ψ 2 u42 = 4 ψ − 8e−2χ n − e−2χ n 52 |ψ|2 ψ 3 , N41 ∂X1 N41 N51 ∂X1 N51

(3.16) where N12 and N22 are given, respectively by Eqs. (3.10) and (3.11), and other Njk are N33 = N41 =

9ω2 (β − 2αN12 ) , 2 2 ω0 − 9ω + 2u0 (u0 − 6iωσ1 ) (1 − cosh [3 (χ − ik)])    k , ω02 − 4ω2 + 2u0 (u0 − 4iωσ1 ) 1 − cosh 2i

        k + 2υg σ1 u0 N12 1 − cosh 2i N42 = u02 N12 − 4iωσ1 u0 N12 sinh 2i k + iωα − iωN12 , N43 N44

(3.17)

     k + ω2 2α 2 N12 + αN33 − 3αβ − 3βN12 , = N22 iω (α − N12 ) + σ1 u0 N12 1 − cosh 2i    2 16ω2 3βN12 − α N12 + 2N33    , = 2 ω0 − 16ω2 + 2u0 (u0 − 8iωσ1 ) 1 − cosh 4i k 

   N51 = ω02 − 9ω2 + 2u0 (u0 − 6σ1 iω) 1 − cosh 3i k ,   2 − 54βω2 N N52 = 36α 2 ω2 N33 + 18αω2 N44 − 27βω2 4αN12 + N12 33 − 144αω2

    N43 + 6N22 3iω (2αN12 − N33 − β) + 2σ1 u0 N33 1 − cosh 3i k , N41

    k N53 = υg 3iω (2αN12 − N33 − β) + 2σ1 u0 N33 1 − cosh 3i   N42 − 12αω2 − u0 (u0 − 6iωσ1 ) N33 sinh 3i k . N41

(3.18)

Because k is a function of the angular frequency ω as seen from Eq. (3.7a), one concludes that coefficients (3.14a)–(3.14f) of Eq. (3.13a) are functions of the propagating frequency f = fp = ω/2π and spatial linear parameter χ . It is important to remember that coefficient Q1 of the cubic nonlinearity is produced by mismatch  k which is assumed to be ∼ O (ε)), where N22 ( kc ) = 0, with N22 given k − kc (≡  by Eq. (3.10). The CQ-CGL equation (3.13a) is a generic amplitude equation that governs the nonlinear evolution of the patterns near the criticality. A number of works have been done for chirped pulses in special cases of the CQ-CGL equation (3.13a) with terms such as the self-steepening, the self-frequency shift, and the third-order dispersion. They are relevant to the design of solitary-wavebased communications links, fiber-optical amplifiers, and optical pulse compressors

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3 Transmission of Dissipative Solitonlike Signals …

[19, 20]. In the following, we consider the general case of Eq. (3.13a) and focus our attention on its chirped Lambert W-kink solutions.

3.1.3 Baseband Modulational Instability Analysis By mean of the CQ-CGL equation (3.13a), we establish in this subsection the conditions under which a uniform wavetrain in the lossy network of Fig. 3.1 is unstable against small perturbations with infinitesimally small wavenumbers. The MI of this type is referred to as the baseband MI [21]. First of all, we note that Eq. (3.13a) admits Stokes’ waves of the form ψ(ξ, T2 ) = ρ0 exp (iK0 ξ − i0 T2 ) ,

(3.19)

where ρ0 is a constant real amplitude, and K0 and 0 are adjustable real wavenumber and frequency, given by   0 = Pr K02 + (Q4r − Q3r ) ρ02 K0 + Q1r + Q2r ρ02 ρ02 + Ri ,

(3.20a)

with algebraic equation  Q2i ρ04 + (Q4i − Q3i ) K0 + Q1i ρ02 + Pi K02 − Rr = 0

(3.20b)

for ρ02 . Next, we focus on the situation when the “global existence condition (GEC)” 4Pi Q2i − (Q3i − Q4i )2 > 0

(3.21)

is satisfied. This avoids the situation when ρ02 → +∞ as K0 → +∞. In the limit K0 → 0, Eq. (3.20b) leads to ρ02 →

−Q1i ±

2 Q1i + 4Rr Q2i

2Q2i

,

which imposes the following restrictions called “local existence conditions (LECs)” 2 Q1i

 + 4Rr Q2i > 0andQ2i −Q1i ±

 2 Q1i

+ 4Rr Q2i

> 0.

(3.22)

Using the network parameters (3.4), we show in Fig. 3.3d the domain (in term of frequency ω) in which GEC (3.21) and LECs (3.22) are satisfied simultaneously; in this figure, we have used the sign “+” in (3.22).

3.1 Chirped Lambert W-Kink Waves Propagation in a Lossy Electrical …

55

Fig. 3.3 (Color online) Domain of the existence of the Stokes wave solution (3.19) when Q2i = 0. a–c show the plots of, respectively, curve GEC = 4Pi Q2i − (Q3i − Q4i )2 associated with the 2 + 4R Q associated with the first of the “global existence condition” (3.21), curve LEC1 = Q1i  r 2i  “local existence conditions” (3.22), and curve LEC2 = Q2i −Q1i +

2 + 4R Q Q1i r 2i associated

with the second of the “local existence conditions” (3.22) versus the propagating angular frequency ω. Panel (d) simultaneously shows curves GEC and two LECs; here, the sign “+” refers to the domain where both GEC (3.21) and two LECs (3.22) are satisfied simultaneously, while sign “−” refers to the frequency range for which at least one of GEC (3.21) and two LECs (3.22) is not satisfied. Different plots are produced for network parameters (3.4) with σ1 = 0.461 × 10−2 , σ2 = 0.01, kc = 0.1, and χc = 10−4 ; for the LECs (3.22), sign “+” is used. Reprint from Ref. [17], Copyright 2022, with permission from Springer

Under the above GEC and LECs, one has K0 min ≤ K0 ≤ K0 max , where K0 min and K0 max are given as

K0 min = K0 max =

 2  + 4Rr Q2i − Rr Q2i (Q3i − Q4i )2 Q1i (Q4i − Q3i ) − 2 Pi Q2i Q1i 4Pi Q2i − (Q3i − Q4i )2  2  + 4Rr Q2i − Rr Q2i (Q3i − Q4i )2 Q1i (Q4i − Q3i ) + 2 Pi Q2i Q1i 4Pi Q2i − (Q3i − Q4i )2

< 0, > 0.

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3 Transmission of Dissipative Solitonlike Signals …

It is seen from the dispersion relation (3.20a) that plane wave (3.19) is nonlinear, hence the superposition principle cannot be applied to it. To investigate the MI of the carrier wave under the GEC (3.21), we consider a small perturbation of the form ψ(ξ, T2 ) = (ρ0 + δψ(ξ, T2 )) exp (iK0 ξ − i0 T2 ) ,

(3.23)

where δψ(ξ, T2 ) is a small perturbation, solution of the equation i

∗   ∂ 2 δψ ∂δψ 2 ∂δψ 2 ∂δψ − iQ +P + i 2PK − Q ρ ρ 0 3 4 0 0 ∂T2 ∂ξ 2 ∂ξ ∂ξ 2   2 ∗ + (Q3 − Q4 ) K0 − Q1 − 2Q2 ρ0 ρ0 δψ + δψ = 0.

(3.24)

Next, we seek the perturbation δψ(ξ, T2 ) in the form   δψ(ξ, T2 ) = b1 exp (iKξ + T2 ) + b∗2 exp −iKξ + ∗ T2 ,

(3.25)

K and  being the wavenumber and complex frequency of the modulation waves, respectively. Following Chen et al. [21], we limit our consideration to perturbations with infinitesimally small K. Substituting Eq. (3.25) in Eq. (3.24) yields the following dispersion law for the perturbations:   = ± = Q1i + (Q4i − Q3i ) K0 + 2Q2i ρ02 ρ02 + Pi K 2 ±  ⎡ ⎤ √ 2 + Y2   −X + X ⎦, +i ⎣ Q3r ρ02 − 2Pr K0 K ± 2

 X+

√

X2 + Y2 2 (3.26)

where X and Y are real quantities given in work [17]. Following the discussion carried out in Ref. [17], one obtains, under GEC (3.21) and LECs (3.22), the following baseband-MI condition for the Stokes waves (3.19): Q1i + (Q4i − Q3i ) K0 + 2Q2i ρ02 > 0,

(3.27)

where ρ02 is any solution of Eq. (3.20b) with Q2i = 0. Under the MI criterion (3.27), the wavenumber K of the perturbation in the case when Pi < 0 satisfies the conditions 0≤K < 2

2 + 4Rr Q2i ! Q1i Q1i − 2Pi Q2i

" 2 Q1i

+ 4Rr Q2i .

(3.28)

Relation (3.28) means that Q1i > 0 and Rr Q2i < 0. The propagating frequency f = fp = ω/2π , as well as dimensionless conductances σ1 and σ2 significantly modify the instability domain. In fact, different values of either fp or σ1 and σ2 correspond to

3.1 Chirped Lambert W-Kink Waves Propagation in a Lossy Electrical …

57

different instability diagrams. Each of these three parameters enhances the instability of the system. This behavior is shown in Fig. 3.4 through the MI growth rate:

Re − (K, K0 , ρ0 ) = Q1i + (Q4i − Q3i ) K0 + 2Q2i ρ02 ρ02 + Pi K 2 − 



 X+

√

X2 + Y2 , 2

(3.29) when K0 = 0. Figure 3.4a–c show that the MI growth rate increases when each of fp , σ1 , and σ2 increases. From the plots of Fig. 3.4, we can also conclude that new ranges of wavenumber K may become available to the MI, causing expansion of the MI domain. Figure 3.4a confirms the behavior shown in Fig. 3.2a, that is, the propagating frequency f = fp increases when the dimensionless conductance σ1 increases.

Fig. 3.4 The instability growth rate as per Eq. (3.29) in the limit of K0 = 0. a The instability growth rate for σ1 = 0.00461, σ2 = 0.01, and three values of the frequency: fp = 445.634 kHz (the solid line), fp = 461.549 kHz (the dashed line), fp = 477.465 kHz (the dashed-dotted line). b The instability growth rate for fp = 477.465 kHz, σ2 = 0.01, and three values of the dimensionless conductance: σ1 = 0.00461 (the solid line), σ1 = 0.00561 (the dashed line), σ1 = 0.00661 (the dashed-dotted line). c The instability growth rate for fp = 477.465 kHz, σ1 = 0.00461, and three values of the dimensionless conductance: σ2 = 0.01 (the solid line), σ2 = 0.015 (the dashed line), σ2 = 0.02 (the dashed-dotted line). Here, kc = 0.1, χc = 10−4 , and the network parameters are taken as per Eq. (3.4). Reprint from Ref. [17], Copyright 2022, with permission from Springer

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3.1.4 Evolution of Chirped Lambert W-Kink Pulses in the Network of Fig. 3.1 Here, we seek for analytical solutions for chirped Lambert W-kink signals in the lossy network of Fig. 3.1. For this aim, we employ the ansatz ψ(ξ, T2 ) = ρ(ζ ) exp [i (ϕ(ζ ) − ωk T2 )] ,

(3.30a)

where ζ = α0 ξ − β0 T2 is the travelling coordinate, ρ and ϕ are real functions of ζ , ωk is a real constant, and ϕ(ζ ) − ωk T2 is a phase function. For ansatz (3.30a), υp = β0 /α0 and 1/α0 are, respectively, the group velocity and width of the pulse’s envelope, while the corresponding intensity of the propagating pulse is |ψ(ξ, T2 )|2 = ρ 2 (ζ ). With ansatz (3.30a), the chirping δph (ξ, T2 ) across the pulse is given as δph (ξ, T2 ) = − (∂/∂ξ ) (ϕ(ζ ) − ωk T2 ) = −α0 d ϕ(ζ )/d ζ , which is time-dependent, unless ϕ is a linear function of variable ζ . Assuming that the frequency chirp depends on the wave’s amplitude ρ through relation d ϕ(ζ )/d ζ = A2 ρ 2 + A0 and inserting Eq. (3.30a) into Eq. (3.13a), we obtain that ρ must simultaneously satisfy the following two equations " ! α0 Q3i + α0 Q4i − 4A2 α02 Pi 2 2A0 α02 Pi d ρ d 2ρ + ρ − dζ2 dζ Pr α02 Pr α02 +

ωk − Ri + A0 β0 − A20 α02 Pr ρ Pr α02

+

A2 β0 − Q1r − 2A0 A2 α02 Pr + A0 (α0 Q3r − α0 Q4r ) 3 ρ Pr α02

+

A2 (α0 Q3r − α0 Q4r ) − A22 α02 Pr − Q2r 5 ρ = 0, Pr α02

(3.31)

# $ 4A2 α02 Pr − α0 Q4r − α0 Q3r 2 2A0 α02 Pr − β0 d ρ Rr − A20 α02 Pi d 2ρ + ρ + ρ + dζ2 dζ Pi α02 Pi α02 Pi α02 + +

A0 (α0 Q3i − α0 Q4i ) − 2A0 A2 α02 Pi − Q1i Pi α02 A2 (α0 Q3i − α0 Q4i ) − A22 α02 Pi − Q2i Pi α02

ρ3

ρ 5 = 0.

(3.32)

Asking that Eq. (3.31) coincides with Eq. (3.32), we obtain A0 = A2 =

β0 , 2 2 |P| α02 Pr

Pr (Q3r + Q4r ) + Pi (Q3i + Q4i ) , 4α0 |P|2

(3.33) (3.34)

3.1 Chirped Lambert W-Kink Waves Propagation in a Lossy Electrical …

ωk =

Pr Rr + Pi Ri Pr − Pi 2 |P|2

β02 α02

59

,

(3.35)

β0 4 |P|2 (Pi Q1r − Q1i Pr ) , = α0 2Pr (Pi (Q3r − Q4r ) − Pr (Q3i − Q4i )) + Pi Pr (Q3r + Q4r ) + Pi2 (Q3i + Q4i )

(3.36) under the condition that coefficients of the amplitude equation (3.13a) satisfy the condition (Pr (Q3r + Q4r ) + Pi (Q3i + Q4i )) (Pi (Q3r − Q4r ) − Pr (Q3i − Q4i )) + 4 |P|2 (Pr Q2i − Pi Q2r ) = 0.

(3.37a) Under assumptions (3.33)–(3.37a), we arrive to the following second order ordinary differential equation for ρ  dρ d 2ρ  + N1 ρ + N3 ρ 3 + N5 ρ 5 = 0, + M2 ρ 2 + M0 dζ2 dζ

(3.38)

where M0 , M2 , N1 , N3 , and N5 are given as

N3 = N5 =



Pr (Q3i +Q4i )−Pi (Q3r +Q4r ) , N1 = α12 2 0 0  2 2|P| α0 2P +P (Q +Q )−P P (Q +Q ) − PQα1i2 + 2|P|Pr2 α2 ( r i ) 3i 2P 4i|P|2 r i 3r 4r i 0 i 0 α0 (Q3r −Q4r )A2 −α02 Pr A22 −Q2r . Pr α02

Pi β0 M0 = − |P| 2 2 , M2 = α

P2 β 2 Rr − 4|P|r 4 α02 Pi 0 β0 , α0



, (3.39)

Equation (3.38) admits a variety of solutions, such as periodic, kink, and solitonlike ones. When the network parameters are all fixed, except linear conductance G 1 , Eq. (3.37a) establishes relationship g0 (ω, σ1 ) = 0 between frequency fp = ω/2π and spatial linear parameter χ , g0 (ω, σ1 ) being the function that coincides with the lhs of Eq. (3.37a) times exp (4χ n), that is,  g0 (ω, σ1 ) = exp (4χ n) (Pr (Q3r + Q4r ) + Pi (Q3i + Q4i ))

× (Pi (Q3r − Q4r ) − Pr (Q3i − Q4i )) + 4 |P|2 (Pr Q2i − Pi Q2r ) .

To prove that equation g0 (ω, σ1 ) = 0 admits real solutions in σ1 , we depict in Fig. 3.5 the dimensionless conductance σ1 versus frequency ω, in an implicit form defined by g0 (ω, σ1 ) = 0 [any set of values (ω, σ1 ) on the curve of Fig. 3.5 satisfies the equation (3.37a)]. It is seen from Fig. 3.5 that for given propagating frequency ω, there exist more than one value of dimensionless conductance σ1 for which the set (ω, σ1 ) satisfies relationship (3.37a). To derive chirped kink solitonlike solutions of the CQ-CGL equation (3.13a), we impose to the first derivative d ρ/d ζ to be a polynomial of the third degree in ρ. The chirped Lambert W-kink solution of Eq. (3.13a) is expressed via the Lambert W-kink

60

3 Transmission of Dissipative Solitonlike Signals …

Fig. 3.5 The variation of dimensionless conductance σ1 as an implicit function defined by equation g0 (ω, σ1 ) = 0, showing (ω, σ1 ) that satisfy condition (3.37a). The network parameters used to generate different plots are given in the text. Reprint from Ref. [17], Copyright 2022, with permission from Springer

solitonlike solution of Eq. (3.38). To this end, we seek the solution to Eq. (3.38) in the form r , (3.40a) ρ(ζ ) = q + 1 + f [g(ζ )] where q and r are two real constants, and f and g are two real functions of variable ζ satisfying relationship f0 + f1 (q + r) + (f0 + f1 q) f (ζ ) df = , dg g(ζ ) (1 + f (ζ ))

(3.40b)

f0 and f1 being two real parameters. If we differentiate Eq. (3.40a) in respect with ζ and replace f by f = (r + q − ρ) / (ρ − q), we obtain 1 1 dg dρ = − (ρ − q)2 (f0 + f1 ρ) , dζ r g dζ

(3.40c)

where f0 and f1 are two real parameters. Asking to the rhs of Eq. (3.40c) to be a third-degree polynomial in ρ, we find that g −1 d g/d ζ must be a constant and f1 = 0. Hence, we arrive to g(ζ ) =  λ0 exp (μ0 ζ ), where  λ0 is a real constant. Replacing now g by this expression in Eq. (3.40b) and integrating the resulting equation, we find that f = f (ζ ) = 0 must be any solution of the algebraic equation 

rf1 +f 1+ f0 + qf1



# $ ! " 2 + qf f0 + qf1 (f ) 0 1 0 − exp − f = exp K μ0 ζ , (3.41a) rf1 rf1

3.1 Chirped Lambert W-Kink Waves Propagation in a Lossy Electrical …

61

0 being a constant of integration. Restricting ourselves to the special case when K f1 = −1/r and f0 = (r + q) /r, Eq. (3.41a) gives   0 , f (ζ ) = W exp μ0 ζ + K

(3.42)

where W = W (z) is the Lambert W function [10]. Inserting Eq. (3.42) in Eq. (3.40a) leads to r .   (3.43) ρ(ζ ) = q + 0 1 + W exp μ0 ζ + K Inserting now g =  λ0 exp (μ0 ζ ) into Eq. (3.40c), we find, after some algebraic manipulations, that  %  & 5M0 5M0 M0 ,∓ ,− , (3.44) (r, q, μ0 ) = ± 3M2 3M2 3 4 M0 M2 , N5 = if N1 = 29 M02 , N3 = − 15

2 M 2, 25 2

and M0 M2 > 0,

 & M0 M0 (r, q, μ0 ) = ∓3 − , ± − , 3M0 , M2 M2 %



(3.45)

if M0 M2 is negative and N1 = N3 = N5 = 0, and  & 5M0 5M0 ,± − , 4M0 , (r, q, μ0 ) = ∓2 − M2 M2 %



(3.46)

if M0 M2 < 0, N1 = 2M02 , N3 = (4/5)M0 M2 , andN5 = (2/25)M22 . Here, M0 , M2 , N1 , N3 , and N5 are given by Eq. (3.39). Hence, by taking μ0 , q, and r from Eqs. (3.44)–(3.46), equation (3.43) will be a solution of Eq. (3.38). Now, if we insert Eq. (3.43) together with (r, q, μ0 ) in Eq. (3.30a), we arrive to the following (rather cumbersome) chirped Lambert W-kink solution of the CQ-CGL equation (3.13a): %

&  r   exp i (ϕ(ξ, T2 ) − ωk T2 ) , ψ(ξ, T2 ) = q +  1 + W exp μ0 (α0 ξ − β0 T2 ) + K0

(3.47) with

62

3 Transmission of Dissipative Solitonlike Signals …

    2qrA2   0 ϕ(ξ, T2 ) = A0 + A2 q2 ζ + ln W exp μ0 ζ + K μ  0  ⎤' ⎡ 0 ' μ0 ζ +K 2 W e ' A2 r ⎦ ⎣  ''  + ln ;  μ ζ + K 0 0 μ0 1+W e '

(3.48)

ζ =α0 ξ −β0 T2

the corresponding chirp reads ⎡

%

δph (ξ, T2 ) = −α0 ⎣A0 + A2 q +

r

 1 + W eμ0 (α0 ξ −β0 T2 )+K0

&2 ⎤  ⎦.

(3.49)

0 is an arbitrary real constant, α0 = 0 and β0 = 0 are two arbitrary real Here, K parameters satisfying condition (3.36), A0 , A2 , and ωk are given by Eqs. (3.33)– (3.35), respectively, ϕ(ξ, T2 ) = ϕ(ζ )|ζ =α0 ξ −β0 T2 , and ( ϕ(ζ ) =

  A0 + A2 ρ 2 d ζ =

( (

= A0 ζ + A2 q ζ + 2qrA2 2

(

⎡

%

⎣A0 + A2 q +

r   1 + W eμ0 ζ +K0

&2 ⎤ ⎦ dζ

dζ   1 + W eμ0 ζ +K0

dζ    2 1 + W eμ0 ζ +K0 ( ( A2 r 2 2qrA2 dx dx + = A0 ζ + A2 q2 ζ + , μ0 x [1 + W (x)] μ0 x [1 + W (x)]2 +r 2 A2



with x = eμ0 ζ +K0 . Setting w = W (x), one has x = wew and dx = (1 + w) ew dw, which leads to Eq. (3.48). Equations (3.47) and (3.49) show that both the chirped Lambert W-kink solution of Eq. (3.13a) and the corresponding chirping are expressed in terms of the Lambert W function [10]. It is important to note that Eq. (3.38) also admits kink solitonlike solution, solution of the following first-order ODE for ρ, dρ = λ1 ρ + λ3 ρ 3 , dζ

(3.50)

where λ1 and λ3 = 0 are real parameters. A differentiation of Eq. (3.50) yields d 2ρ − 3λ23 ρ 5 − 4λ1 λ3 ρ 3 − λ21 ρ = 0. dζ2

(3.51)

3.1 Chirped Lambert W-Kink Waves Propagation in a Lossy Electrical …

63

Inserting now Eq. (3.50) in Eq. (3.38) and comparing the resultant equation with Eq. (3.51) determine the wave parameters,

λ1 =

−M0 ±

M02 − 4N1 2

, λ3 =

−M2 ±

M22 − 12N5 6

,

(3.52)

under the conditions that

6N3 − 2M0 M2 ± M2

M02 − 4N1 ≥ 0, M22 − 12N5)≥ 0,    2 M0 − 4N1 ∓ M0 M22 − 12N5 + 2 M02 − 4N1 M22 − 12N5 = 0.

(3.53) Under condition λ1 λ3 < 0, we integrate Eq. (3.50) to obtain the kink soliton solution  ρ(ζ ) = ± −

λ1 [1 + tanhλ (λ1 ζ )], with tanhλ (λ1 ζ ) ≡ 2λ3

−λ3 λ1 ζ e λ0 λ1 −λ3 λ1 ζ e λ0 λ1

− e−λ1 ζ

, + e−λ1 ζ (3.54) λ0 > 0 being a real constant. By inserting Eq. (3.54) into Eq. (3.30a), we arrive to the following chirped kink solution of Eq. (3.13a): 

λ1 (1 + tanhλ [λ1 (α0 ξ − β0 T2 )]) exp [i {ϕ(ξ, T2 ) − ωk T2 }] , 2λ3 (3.55) where ε0 is a real number satisfying condition |ε0 | = 1 and ψ(ξ, T2 ) = ε0 −

ϕ(ξ, T2 ) = A0 (α0 ξ − β0 T2 ) + ϕ0 −

" ! A2 λ3 2λ1 (α0 ξ −β0 T2 ) ; ln 1 − e 2λ3 λ0 λ1

(3.56)

the corresponding chirp being " ! A2 λ1 δph (ξ, T2 ) = −α0 A0 − (1 + tanhλ [λ1 (α0 ξ − β0 T2 )]) . 2λ3

(3.57)

Here, A0 , A2 , and ωk are given by Eqs. (3.33)–(3.35), respectively, λ1 and λ3 are given by Eq. (3.52), α0 = 0 and β0 = 0 being real parameters satisfying condition (3.36). Now, we turn to the transmission of chirped Lambert W-kink pulses in the lossy network of Fig. 3.1. For this aim, we go back to dimensional parameters and original variables and use the decomposition (3.5) of voltage Vn (t). In expansion (3.5), we ' ' neglect all harmonics, in comparison with the fundamental one u, 'ujl '  |u| for all j and l, so that the shape of the electrical pulse will be well approximated by the shape of the fundamental harmonic, the expression for which is

64

3 Transmission of Dissipative Solitonlike Signals …

Fig. 3.6 The real and imaginary parts of complex group velocity υg versus frequency ω for different values of dissipative parameter σ1 . (a): The variation of real part υgr of the group velocity for σ1 = 0.461 × 10−2 (the solid line), σ1 = 0.04 (the dashed line), and σ1 = 0.08 (the dashed-dotted line). (b): The variation of imaginary part υgi of the group velocity for σ1 = 0.461 × 10−2 (the solid line), σ1 = 0.561 × 10−2 (the dashed line), and σ1 = 0.661 × 10−2 (the dashed-dotted line). Different plots are produced for network parameters (3.4). Reprint from Ref. [17], Copyright 2022, with permission from Springer

  " ! √ β0 ε Pr (Q3r + Q4r ) + Pi (Q3i + Q4i ) 2 P t Vn (t) = 2 ε exp −χ n + υ + q r gi α0 2 2 |P|2 ⎛ ⎞ ⎜ ⎜ ×⎜ ⎜q + ⎝

%

  1 + W cos εα0 μ0 υgi t × e

× cos [L (n, t) + NL (n, t)] .

⎟ ⎟

r

   & ⎟ ⎟ β ⎠ εα0 μ0 n− υgr +ε α0 t+φ0W 0

(3.58)

In this equation (3.58), L = L (n, t) and NL = NL (n, t) are given respectively by        L = k + εα0 A0 + A2 q2 n − ω + ωk ε2 + ε A0 + A2 q2 α0 υgr + εβ0 t, ! 

  "   εα μ n− υgr +ε αβ0 t+φ0W 2qrA2 0 (3.59) NL = ln W cos εα0 μ0 υgi t e 0 0 μ0 

   ⎤ ⎡   εα0 μ0 n− υgr +ε αβ0 t+φ0W 0 W cos εα μ υ t e 0 0 gi ⎥ A2 r 2 ⎢  + ln ⎢

   ⎥ , ⎦ ⎣ β0   μ0 εα μ n− υgr +ε α t+φ0W 0 1 + W cos εα0 μ0 υgi t e 0 0 where k and ω are the wavenumber and angular frequency of the carrier wave, χ is the spatial linear parameter given by Eq. (3.7b), α0 = 0 and β0 = 0 are real parameters satisfying equation (3.36), φ0W and φ0K are two arbitrary real parameters, 0 < ε  1 is a small dimensionless parameter, υgr and υgi are, respectively, the real and the imaginary parts of complex group velocity υg defined in Eq. (3.10), A0 , A2 , and ωk are given, respectively, by Eqs. (3.33), (3.34), and (3.35), and (μ0 , q, r), taken from

3.1 Chirped Lambert W-Kink Waves Propagation in a Lossy Electrical …

65

Eqs. (3.44)–(3.46), are given by Eqs. (3.44)–(3.46). In general, υgr is positive, while υgi is negative. This behavior is seen in Fig. 3.6, in which υgr and υgi are plotted versus the angular frequency ω for different values of dimensionless conductance σ1 . As we can see from Fig. 3.6a, b, υgr and vgi severally increases and decreases with the increase of σ1 . The presence of factor exp [−χ n] in Eq. (3.58) indicates that the dissipative effects, introduced by conductance G 1 in the series branch of the network of Fig. 3.1, lead to a decay of the amplitude as the Lambert W-kink pulses propagate in the lattice. Depending on the sign of the parameter  λ given by   β0 Pr (Q3r + Q4r ) + Pi (Q3i + Q4i ) 2  λ = υgi Pr q , + α0 2

the presence of factor exp

ε λ t 2|P|2



in Eq. (3.58) shows that dissipative effects introduced by G 1 lead to either decrease (for  λ < 0) or increases (when  λ > 0) of the amplitude in the course of the propagation of the Lambert W-kink signals through the λ must be neganetwork. Because G 1 accounts for the loss in the network of Fig. 3.1,  tive so that the wave amplitude decreases. One then concludes that the self-steepening term Q3 and self-frequency shift Q4 can be used to modulate the amplitude of the Lambert W-kink signals propagating in the dissipative network. As a consequence of the presence of the dissipation, the wave’s phase is a superposition of linear and nonlinear phases L (n, t) and NL (n, t), given by Eq. (3.59). Equation (3.58) reveals that the center of the chirped Lambert W-kink pulses moves with a constant speed, υc = υgr + εβ0 /α0 , which is different from the group velocity. Equation (3.58) also indicates that the width of the W-kink pulses is 1/ (εα0 μ0 ), and, according to the expression for μ0 [see Eqs. (3.44)–(3.46)], it is proportional to α0 /β0 . Therefore, the width of the W-kink pulse remains constant during its propagation. With the use of network parameters (3.4) with σ1 = 0.766 × 10−2 and σ2 = 1.152 × 10−2 , and using (r, q, μ0 ) given by Eq. (3.45), a relevant set of values of the solution parameters is chosen to depict, in Fig. 3.7, the temporal and spatial (left and right panels, respectively) profiles of the chirped Lambert W-kink signals, propagating at frequency fp = 1171.38 kHz. For these data, condition (3.21) is violated, meaning that the propagating frequency fp = 1174.05 kHz does not belong to the domain of the baseband MI. It is important to notice that for the signals shown in Fig. 3.7d–f, the center of the pulse is located (approximately) at n = 800, n = 809, and n = 813, respectively, meaning that the pulse moves with a positive speed. Thus, Fig. 3.7 demonstrates that the dynamics of the Lambert W-kink signal is different from that of the well-known kink soliton, cf. Ref. [22]. In conclusion, we have studied analytically in this Section the results for the baseband MI and the transmission of chirped Lambert W-kink signals through a lossy electrical network. First, we have derived a generalized CQ CGL equation with self-steepening and self-frequency shift that governs the dynamics of slowly modulated waves propagating in our network system. The spatial-decay rate χ is found in the analytical form, including its dependence on the wave’s frequency. The

66

3 Transmission of Dissipative Solitonlike Signals …

Fig. 3.7 The propagation of chirped Lambert W-kink signals in the network at frequency fp = 1174.05 kHz. Plots in the left panels show the evolution of the chirped Lambert W-kink signal at different cells of the network for φ0W = 40: a n = 0, b n = 5, and c n = 10. Plots (d)–(f) in the right panels show the spatial shape of the chirped Lambert W-kink signal with φ0W = −800 at different times, t = 0.0 μs, t = 0.18 μs, and t = 0.26 μs, respectively. Different curves are produced for  voltage (3.58) and parameters ε = 10−3 , kc = 2.75, χc = 10−4 (note that 2π fp , σ1 satisfies Eq. (3.37a)). Other parameters are given in the text. Reprint from Ref. [17], Copyright 2022, with permission from Springer

derived amplitude equation predicts the linear stability of slowly modulated Stokes’ waves, showing how the dissipative elements of the network affect the baseband MI of the system. Further, our investigations have revealed that the model under the consideration supports the propagation of a variety of chirped pulses including the chirped Lambert W-kink signals. Analytical expressions for the chirped phase, ϕ(ξ ) − ωk t, and nonlinear frequency change δph (x, t) across the pulse are obtained as functions of the intensity of the signal, while its amplitude can be modulated by varying parameters Q3 and Q4 of the self-steepening and self-frequency shifts. The derived chirped Lambert W-kink solution of the amplitude equation (3.13a) is then used to investigate analytically the effects of the dissipation on the amplitude of

3.2 Spatiotemporal Modulation of Damped Solitonlike Wave …

67

the Lambert W-kink soliton; we have obtained that its width and velocity remain constant as the wave propagates in the network. Our results show that the effects of dissipative losses in the series branch of the network are stronger than those resulting from the dissipative losses in the shunt branch. Results of the present Section can be useful for exploring the dynamics of Lambert W-kink pulses in other complex systems, which, in the first approximation, may be modelled by equations of the CQ-CGL type, including the self-steepening and self-frequency shift.

3.2 Spatiotemporal Modulation of Damped Solitonlike Wave Signals in a Nonlinear RLC Transmission Network 3.2.1 Introduction Focusing our ourselves on lossy nonlinear transmission networks, we present in this Section a method to investigate analytically spatiotemporal nonlinear modulated waves with damped amplitude (in both time and space) in dissipative NLTNs . Although device and transmission network losses can degrade the performance of NLTNs [23] and cannot be discarded in realistic circuit simulations, we believe that in real nonlinear dispersive media, the coexistence of the dissipative effect with the nonlinear and dispersive effects may play some roles in a wave generation as well as in its propagation. Thus, the model studied in this Section is a distributed model of a lossy NLTN which can be viewed as a multiple combination of small RLC circuit segments shown in Fig. 3.8 [24–26]. In this model, the series inductance is due to magnetic field effects, while the capacitance is due to electric field coupling between the lines. The losses in this model are depicted by the series and the shunt resistors which represent the finite conductivity of the conductors and the dielectric insulator between the conductors, respectively. The linear resistances R1 and R2 that account for the network losses, the linear inductance L, and the voltage-dependent capacitance C are the relevant network parameters. In this study, we assume that the model under consideration is formed of N identical cells, each of which consists of a linear inductive element of inductance L and a linear resistor with resistance R1 in the series branch constituting the linear dispersive element, and a nonlinear capacitor of voltage-dependent capacitance C and a linear resistor with resistance R2 in the shunt branch. The nonlinear element of this model is the voltage-dependent capacitor which is assumed to be biased by an equilibrium constant voltage V0 (alias bias voltage of the capacitor) and depends on the voltage Vn across the n-th capacitor. A complex cubic GL equation governing slowly modulated wave propagation through our model is derived. Considering linear wave propagating in the network

68

3 Transmission of Dissipative Solitonlike Signals …

Fig. 3.8 Schematic representation of the one cell of the lossy NLTL with the linear inductor in series branch and the nonlinear capacitor in parallel branch. Conductance (dissipative elements) are connected in series on the two elements. Reprint from Ref. [27], Copyright 2022, with permission from Springer

under consideration, we derive in terms of the propagating frequency the spatial decreasing rate alias linear dissipation parameter and show that its must important contribution comes from the dissipative element of the shunt branch. The MI criterion of modulated Stokes wave propagating in the network is investigated and the analytical expression of the MI gain is derived. We show that in the case of weak dissipation, there are no significant changes for the bandwidth frequency where the network may exhibit MI. Exact and approximative envelope solitonlike solutions of the amplitude equation are presented and used to investigate analytically the dynamics of spatiotemporal modulated damped signals along our network system. We also show that the solution parameters can be used for managing the evolution of the envelope soliton signals along the network. Our investigations show that the signal amplitude decays in both space and time, while the velocity remains constant when the envelope soliton signal propagates along the dissipative network under consideration. In order to investigate the dynamics of spatiotemporal modulated damped signals in the network model of Fig. 3.8, we assume, for low voltage at cell n, the voltagedependent capacitance of the form [25, 27–34] C(V0 + Vn ) =

  dQn ≈ C0 1 − 2αVn + 3βVn2 , dVn

(3.60)

where C0 = C(V0 ), and α and β are nonlinear coefficients that determine the electric charge Qn stored in the n- th capacitor in the line, and Vn is the voltage across the n- th capacitor. Here, the subscript n stands for the cell number in the network. According to the sign of α and β, Eq. (3.60) can be the second order curve fitting for the diode characteristics or the MOS varactor characteristics; for the diode, both α and β are positive, while for the MOS, they are negative [16]. For the network system under consideration, we assume α and β to be positive. During computation, we follow Sekulic et al. [24] and use the below values of the network parameters

3.2 Spatiotemporal Modulation of Damped Solitonlike Wave …

69

Fig. 3.9 Linear dispersion curve of the network showing the angular frequency ω as a function of the wavenumber k defined implicitly by Eq. (3.66) for different sets of the dimensionless resistances. a σ1 = σ2 = 0 (solid line) and σ1 = 1.152 × 10−2 and σ2 = 0.461 × 10−2 (dashed line); b σ1 = σ2 = 0 (solid line), σ1 = 0.461 × 10−2 and σ2 = 0 (dashed line), and σ1 = 0.15 and σ2 = 0 (dotted line); c σ1 = σ2 = 0 (solid line), σ1 = 0 and σ2 = 0.461 × 10−2 (dashed line), and σ1 = 0 and σ2 = 00.15 (dotted line); d σ1 = σ2 = 0 (solid line), σ1 = 0.461 × 10−2 and σ2 = 0.461 × 10−2 (dashed line), and σ1 = 0.15 and σ2 = 00.15 (dotted line). Reprint from Ref. [27], Copyright 2022, with permission from Springer

L = 10µH, α = 0.21V−1 , β = 0.0197V−2 , V0 = 2V, C0 = 10pF.

(3.61)

Assuming voltage across the capacitor to be Vn + V0 and applying the Kirchhoff laws, we arrive to the following difference-differential system

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3 Transmission of Dissipative Solitonlike Signals …

d 2 Vn dVn d + 2u0 σ2 (2Vn − Vn−1 − Vn+1 ) + u02 (2Vn − Vn−1 − Vn+1 ) + 2u0 σ1 dt 2 dt dt   d2  d  2 αVn − βVn3 = 2 αVn2 − βVn3 + 2u0 σ1 dt dt    d   2 2 2 3 3 α 2Vn − Vn−1 − Vn+1 − β 2Vn3 − Vn−1 . (3.62) − Vn+1 + 2u0 σ2 dt √ Here, u0 = /1 LC0 is the characteristic angular frequency of the network, and σ1 and σ2 are two dimensionless resistances, related to resistances R1 and R2 as follows R1 R2 = 2u0 σ1 , = 2u0 σ2 . L L

(3.63)

3.2.2 Linear Dispersion Relation and Spatial Decreasing Rate Following Yemélé et al. [32], the expression of propagating linear waves of initial amplitude Vm and angular frequency ω can be written as    kn − ωt + c.c., Vn (t) = Vm exp i 

(3.64)

where c.c. stands for the complex conjugation and  k = k + iχ , k and χ being respectively the real wavenumber and the real spatial decreasing rate (alias linear dissipation parameter) to be determined. Inserting Eq. (3.64) into Eq. (3.62) and ignoring nonlinear terms yield the linear algebraic system ⎛ ⎝

u02 −σ2 u0 ω ωσ2

u0

⎞

   2 1 2  u0 − 2 ω ⎠ cos [k] cosh [χ ] = . sin [k] sinh [χ ] ω (σ1 + 2σ2 )

(3.65)

It is obvious that system (3.65) has a mathematical sense when k = 0 only if ω(0) = χ (0) = 0; we also notice that it is incompatible when k = π . Therefore, it is reasonable for us to work with the wavenumber k taken from the zone 0 ≤ k ≤ π0 < π . Solving system (3.65) in the region 0 < k ≤ π0 for cos [k] cosh [χ ] and sin [k] sinh [χ ] yields respectively the following linear dispersion relation and spatial decreasing rate cos2 [2k] + z1 (ω) cos [2k] + z2 (ω) = 0 and

! χ = ln z3 (ω, k) +

" 1 + z32 (ω, k) ,

(3.66)

(3.67)

3.2 Spatiotemporal Modulation of Damped Solitonlike Wave …

71

Fig. 3.10 Evolution plot of the spatial decreasing rate χ (cell−1 ) versus the angular frequency ω for different values of the dimensionless resistances σ1 and σ2 : a σ1 = 0 and σ2 = 0.01; b σ1 = 0.01 and σ2 = 0; and c σ1 = 1.152 × 10−2 and σ2 = 0.461 × 10−2 . Reprint from Ref. [27], Copyright 2022, with permission from Springer

where  2   2 u02 2ω2 σ2 (σ1 + 2σ2 ) − ω2 + 2u02 + 2ωu02 (σ1 + 2σ2 ) + ωσ2 ω2 − 2u02 z1 (ω) = − ,  2 2u02 u02 + ω2 σ22 !     1 2 2 2 2 2 2 2 2 2 2 2 z2 (ω) =  2 2u0 2ω σ2 (σ1 + 2σ2 ) − ω + 2u0 − 2u0 u0 + ω σ2 − u0 2u02 u02 + ω2 σ22 "

(3.68)

2   2 , × 2ω2 σ2 (σ1 + 2σ2 ) − ω2 + 2u02 − 2ωu02 (σ1 + 2σ2 ) + ωσ2 ω2 − 2u02 

  2 2 2 ω 2u0 (σ1 + 2σ2 ) + σ2 ω − 2u0   z3 (ω, k) = . sin [k] 2u u2 + ω2 σ 2 0

0

2

Because of the non-negativity of z3 (ω, k) for k ∈ [0, π0 ], the linear dissipation parameter χ (k, ω(k)) is positive for all k ∈]0 , π0 ]. Therefore, the amplitude of linear waves defined by Eq. (3.64) is an exponentially decreasing function of the network cell number n (this is clearly seen if we replace in Eq. (3.64)  k by its expression  k = k + iχ ). In the special case of a lossless network (σ1 = σ2 = 0), the linear dissipation parameter (3.67) gives χ (ω) = 0, while the linear spectrum (3.66), under the condition that ω(0) = 0, reduces to the well-known dispersion relation of a band-pass filter ! " 2 2 2 k , (3.69) ω = 4u0 sin 2

72

3 Transmission of Dissipative Solitonlike Signals …

which possesses a gap ω0 = 0 at k = 0 and is also limited by the cut-off angular frequency ωc = 2u0 sin π20 when k = π0 . Therefore, relation (3.66) establishes the relation between the wavenumber k and the angular frequency ω in the general case of dissipative network of Fig. 3.8, and the propagation of plane waves and modulated waves in our dissipative system will take place in the frequency range of the allowed band, that is, fp ∈ [f0 , fc [, where f0 = 0 and fc = ωc /(2π ). Throughout this Section, we use, without loss of generality, the value π0 = π/2 so that the propagating frequency fp = ω/2π will be an increasing function of wavenumber k in [0, π0 ]. As one can see from plots of Fig. 3.9, to different sets of the dimensionless resistances σ1 and σ2 correspond different linear dispersion curves. It is seen from Fig. 3.9a, that the linear dispersion curve of the network associated with Eq. (3.66) can be well approximated by Eq. (3.69) for small enough σ1 and σ2 . We can see from plots of Fig. 3.9b that the introduction of the series dissipative elements in our system decreases the propagation frequency fp = ω/2π . Figure 3.9c reveals that the introduction of the dissipative elements in the shunt branches decreases [increases] the propagating frequency associating with low [high] wavenumber k. In order to analyze the contribution of the two resistances in the resulting magnitude of spatial decreasing rate χ , we have displayed, for the network parameters (3.61) and different values of σ1 and σ2 , χ given by Eq. (3.67) in Fig. 3.10 as a function of the angular frequency ω. It is seen from different plots of Fig. 3.10 that the spatial decreasing rate χ is more manifested for frequencies close  to f c = ωc /2π and is an increasing function of the frequency in the operating range f0 , fc = [0, ωc /2π ] of the network. Therefore, the wave amplitude will decrease more rapidly when the frequency of input wave increases. When comparing Fig. 3.11a, b, we observe that the linear dissipative parameter χ associated to σ2 only (Fig. 3.10a) is greater than that induced by σ1 only (Fig. 3.10b). Hence, the must important contribution to χ comes from the resistance R2 in the shunt branch of the network and consequently, this resistance R2 cannot be neglected in the analytical study of wave propagation in the network under consideration. Plots of Fig. 3.10 also reveal that for frequencies in  the operating range f0 , fc of the network, the spatial decreasing rate χ is very small (0 ≤ χ ≤ 0.03) so that its contribution to the linear waves (3.64) can be neglected. As we will see in the next Section, its presence can be crucial in the derivation of the model equations governing the dynamics of nonlinear modulated waves propagating in the dissipative nonlinear electrical network under consideration.

3.2.3 Amplitude Equation and Modulated Damped In this subsection, we first derive the amplitude equation that governs the dynamics of spatiotemporal modulated damped waves propagating in the network system of Fig. 3.8. Then we investigate the linear stability of plane wave with time-varying amplitude propagating in our network system.

3.2 Spatiotemporal Modulation of Damped Solitonlike Wave …

3.2.3.1

73

Mathematical Considerations

Now, we focus our attention to modulated damped waves with a slowly varying envelope in time and space with regard to a given carrier wave with angular frequency ω = ωp = 2π fp and wavenumber k = kp . To derive the amplitude equation, that is, the nonlinear partial differential equation that governs spatiotemporal modulated damped waves in the network of Fig. 3.8, we use the results of the preceding subsection that lead to the following statements: 1. Following Marquié et al. [30], we introduce, through a perturbative small parameter ε that measures the smallness of the modulation frequency and the amplitude of the input waves, two envelope variables x and τ beside the space variable n and   the time t, namely x = ε n −  υg t , and τ = ε2 t. These two envelope variables will help us to apply the reductive perturbation method in the semi-discrete limit [35]. 2. In order to take into account the dissipation effects of our network system during applying the reductive perturbation method, we will use, instead of a real k = kp + iχ , χ being the spatial wavenumber k = kp , the complex wavenumber  dissipation parameter defined in Eq. (3.67). Therefore  k and ω satisfy the complex linear dispersion relation    k = 0. 2u02 − ω2 − 2iu0 (σ1 + 2σ2 ) ω − 2 u02 − 2iu0 σ2 ω cosh i

(3.70)

The following complex group velocity is derived from Eq. (3.70)    2u0 σ2 ω + iu02 sinh i k dω  ,  υg ≡ υgr + iυgi = =−  dk ω + iu0 (σ1 + 2σ2 ) − 2iu0 σ2 cosh i k

(3.71)

where u0 Dr υgr =  2  2 , ω − 2u0 σ2 sinh [χ ] sin kp + u02 σ1 + 2σ2 − 2σ2 cosh [χ ] cos kp (3.72a) u0 Di υgi =   2 ,  2  ω − 2u0 σ2 sinh [χ ] sin kp + u02 2σ2 cosh [χ ] cos kp − σ1 − 2σ2 (3.72b) with

74

3 Transmission of Dissipative Solitonlike Signals …

   Dr = ω − 2u0 σ2 sinh [χ ] sin kp     × 2σ2 ω sinh [χ ] cos kp + u0 cosh [χ ] sin kp     + u0 2σ2 ω cosh [χ ] sin kp − u0 sinh [χ ] cos kp    × 2σ2 cosh [χ ] cos kp − σ1 − 2σ2 ,     Di = 2σ2 ω cosh [χ ] sin kp − u0 sinh [χ ] cos kp    × 2u0 σ2 sinh [χ ] sin kp − ω     + u0 2σ2 ω sinh [χ ] cos kp + u0 cosh [χ ] sin kp    × 2σ2 cosh [χ ] cos kp − σ1 − 2σ2 .

(3.73a)

(3.73b)

3. The solution of the difference-differential system (3.62) is assumed to have the following general form  Vn (t) = εψ(x, τ ) exp [iθ ] + ε2 ψ1 (x, τ ) + ψ2 (x, τ ) exp [2iθ ] + c.c., (3.74) where θ =  kn − ωt is the rapidly varying phase and ψ1 and ψ2 are respectively the dc term and the second-harmonic term added to the fundamental term ψ(x, τ ) in order to take into account the asymmetry of the charge-voltage relation given by Eq. (3.60).

3.2.3.2

The Amplitude Equation [36] for the Dynamics of Modulated Waves

Under assumptions (a)–(c), we present here the nonlinear partial differential equation that governs the dynamics of modulated waves propagating in the network of Fig. 3.8. Substituting Eq. (3.74) into Eq. (3.62) and taking into account conditions (a), (b), and (c), we obtain the dc term and the second-harmonic terms ψ1 and ψ2 as, ψ1 (x, τ ) = 2α |ψ|2 ,

    4αω ω + iu0 (σ1 + 2σ2 ) − iu0 σ2 e−2k + e2k ψ2 (x, τ ) =  ψ 2,    2 −2 k 2 k 4ω − 2u0 [u0 − 2iω (σ1 + 2σ2 )] + u0 (u0 − 4iσ2 ω) e +e and the following amplitude equation ∂ψ ∂ 2ψ + (Pr + iPi ) 2 + (Qr + iQi ) |ψ|2 ψ = 0. (3.75) ∂τ ∂x   In Eq. (3.75), Pr = Re[P], Pi = Im[P] , Qr = Re Q , and Qi = Im Q , P and Q being given by in terms of the complex wavenumber  k and angular frequency ω as i

3.2 Spatiotemporal Modulation of Damped Solitonlike Wave …

75

 υg2   , (3.76a) 2 −ω − iu0 (σ1 + 2σ2 ) + 2iu0 σ2 cosh  k   k 2α 2 ω ω + 2iu0 (σ1 + 2σ2 ) − 4iu0 σ2 cosh   Q=  −ω − iu0 (σ1 + 2σ2 ) + 2iu0 σ2 cosh k   k 3βω 1 + 2u0 (σ1 + 2σ2 ) − 4u0 σ2 cosh   −   2 −ω − iu0 (σ1 + 2σ2 ) + 2iu0 σ2 cosh k   k 4α 2 ω2 ω + 2iu0 (σ1 + 2σ2 ) − 4iu0 σ2 cosh   +  −ω − iu0 (σ1 + 2σ2 ) + 2iu0 σ2 cosh k  k ω + iu0 (σ1 + 2σ2 ) − 2iu0 σ2 cosh 2  . (3.76b) × 2 4ω − 2u0 [u0 − 2iω (σ1 + 2σ2 )] + 2u0 (u0 − 4iσ2 ω) cosh 2 k P=

k = kp + iχ , χ Here,  υg is the complex group velocity given by Eq. (3.71) and  being the dissipation coefficient given by Eq. (3.67). The nonlinear partial differential equation (3.75) is a complex cubic Ginzburg-Landau (GL) equation that governs the propagation of modulated damped waves in the network of Fig. 3.8. In Eq. (3.75), the complex parameters P and Q are respectively the dispersion and the nonlinear coefficients of the amplitude equation. It is seen from expressions (3.76a) and (3.76b) that Pr , Pi , Qr , and Qi depend on the dissipation coefficient χ and can take any sign. When Pi and Qi are respectively negative and positive, they designate the dispersion dissipation coefficient and the nonlinear dissipation coefficient, respectively. In the regime when Pi is positive and Qi is negative, the resistance R1 and R2 act as if the driven field Pi and Qi are taken separately. It should be noted that in certain frequency range, it can occur the situation when Pi Qi > 0. In this situation, the effect of resistances R1 and R2 in the dynamics of modulated waves depends on the balance between the two coefficients [32].

3.2.4 Linear Stability We now turn to the analysis of the uniform wave train propagating along the lossy nonlinear electrical transmission network under small perturbations. We start by seeking exact plane wave solutions of the GL equation (3.75) of the form ψ(x, τ ) = a0 (τ ) exp [i0 x + iϕ0 (τ )] ,

(3.77)

where 0 is a real number and a0 (τ ) and ϕ0 (τ ) are two real functions of τ . For equation (3.77) to satisfy the complex GL equation (3.75), a0 (τ ) and ϕ0 (τ ) must satisfy the equation d ϕ0 da0 − a0 − 20 Pa0 + Qa03 = 0, i dτ dτ

76

3 Transmission of Dissipative Solitonlike Signals …

which leads to the following differential system of ordinary differential equations for a0 (τ ) and ϕ0 (τ ) : d ϕ0 da0 = Pi 20 a0 − Qi a03 , = Qr a02 − Pr 20 . dτ dτ

(3.78)

Integrating the first equation of system (3.78) yields a02 (0) , ifPi = 0, 1 + 2Qi a02 (0)τ a0 (0) a0 (τ ) = ) , ifPi = 0.    2 Qi a0 (0) Qi a02 (0) 2 exp −2P + 1 −  τ i 0 P 2 P 2

a02 (τ ) =

i 0

(3.79a) (3.79b)

i 0

Another important special solution of system (3.78) under the condition Pi Qi > 0 is a0 (τ ) = a00 , ϕ0 (τ ) =

Pi Qr − Pr Qi 2 Qi 2 a00 τ, with 20 = a , Pi Pi 00

(3.79c)

where a00 = 0 is any real constant. Solution (3.79c) of system (3.78) corresponds to a plane wave solution (3.77) with a constant real amplitude. To investigate the linear stability of the plane wave solution (3.77), we consider the ansatz of the form ψ(x, τ ) = [a0 (τ ) + δψ] exp [i0 x + iϕ0 (τ )] ,

(3.80)

where δψ is a small perturbation on the wave amplitude a0 . Substituting Eq. (3.80) into the GL equation (3.75) and linearizing the resulting equation with respect in δψ yield the following equation that describes the dynamics of the perturbation  ∂ 2 δψ ∂δψ  ∂δψ +P + Qr a02 + 2iQi a02 − iPi 20 δψ + Qa02 δψ ∗ = 0. + 2iP0 2 ∂τ ∂x ∂x (3.81) Let us denote respectively by K and  the real wavenumber and the complex frequency of the modulation waves and seek nonzero solutions of Eq. (3.81) in the form i

⎡ δψ = b1 exp ⎣iKx − i

(τ 0

⎤

⎡

(ξ )d ξ ⎦ + b∗2 exp ⎣−iKx + i

(τ

⎤ ∗ (ξ )d ξ ⎦ ,

(3.82)

0

where b1 and b2 are two complex constants satisfying the condition |b1 | + |b2 | > 0. Inserting Eq. (3.82) into Eq. (3.81) and asking that |b1 | + |b2 | > 0, we obtain the linear modulation dispersion relation

3.2 Spatiotemporal Modulation of Damped Solitonlike Wave …

   2     − 2Pr 0 K + i 2Qi a02 − Pi 20 + K 2 + 4iPi 0 K Qr a02 − Pr K 2   −Pr2 K 4 + 2 Pr Qr a02 + 2Pi2 20 K 2 + Qi2 a04 = 0,

77

(3.83)

leading to    (τ ) = ± (τ ) = 2Pr 0 K ± ξ + i Pi 20 + K 2 − 2Qi a02 ± ζ ;

(3.84a)

here, 

 √ √ X2 + Y2 −X + X 2 + Y 2 ξ = ,ζ = , (3.84b) 2 2     X = Pr2 K 4 − 2 Pr Qr a02 + 2Pi2 20 K 2 − Qi2 a04 , Y = 4KPi 0 K 2 Pr − a02 Qr . X+

(3.84c) For the phenomenon of the modulational instability to occur, at least one of the frequencies of modulation ± must possess a positive imaginary part [37–39], meaning that    (3.85) Im [] = Im ± = Pi 20 − 2Qi a02 + Pi K 2 ± ζ > 0, 2 Pi 20 − 2Q  i a0 and  of −2 and + . When K → +0,2 the quantities for at2 least one 2 2 Pi 0 − 2Qi a0 + Pi K have the same sign. If Pi 0 − 2Qi a0 > 0, then Im + will be positive as K → +0, and the corresponding perturbation (3.82) will increase exponentially when τ → +∞, meaning that the system remains  unstable under modulation. In the case where Pi 20 − 2Qi a02 < 0, we have Im − < 0 as K → +0, and the modulational instability will occur only when

   Im + = Pi 20 − 2Qi a02 + Pi K 2 + ζ > 0.  By expanding Im + as 

√   4Pi2 20 K 2 + Qi2 a04 + X 2 + Y 2 1 Pr Qr a02 − Pr2 K 2 K 2 + 2 2 )   2 1 2 2 > Pi 0 − 2Qi a0 + Pi K + K Pr Qr a02 − Pr2 K 2 , 2

   Im + = Pi 20 − 2Qi a02 + Pi K 2 +

 we can see that the condition Im + > 0 is satisfied as soon as 1 Pr Qr a02 − Pr2 K 2 > 0 and 2 )  2  1 2 2 Pi 0 − 2Qi a0 + Pi K + K Pr Qr a02 − Pr2 K 2 > 0. 2 These last inequalities lead to

78

3 Transmission of Dissipative Solitonlike Signals …

Fig. 3.11 Plots of a: Real part of the dispersion coefficient Pr as a function of the angular frequency ω ; b Imaginary part of the dispersion coefficient Pi as a function of the angular frequency ω; c Real part of the nonlinear coefficient Qr as a function of the angular frequency ω ; d Imaginary part of the nonlinear coefficient Qr as a function of the angular frequency ω; e Product Pr Qr as a function of the angular frequency ω; f Product Pi Qi as a function of the angular frequency ω. Plots (b1 ) and (f1 ) show the behavior of respectively Pi and Pi Qi far from ω = 0, while plots (e1 ) and (f2 ) show how behave the products Pr Qr and Pi Qi near ω = 0, respectively. Plot (f3 ) shows the behavior of the imaginary parts of the dispersion coefficient Pi and nonlinear coefficient Qi far from ω = 0 and ω = ωc . Different plots are generated with the network parameters (3.61) with the dimensionless resistances σ1 = 25 × 10−4 and σ2 = 5 × 10−3 associated with R1 = 5 and R2 = 10 (value close to the numerical one [24]). Reprint from Ref. [27], Copyright 2022, with permission from Springer

Pr Qr >

2  2Qi a02 − Pi 20 − Pi K 2 a02 K 2

+

Pr2 K 2 > 0. 2a02

Therefore, for the perturbation (3.82) to increase exponentially as τ → +∞, it necessary that Pr Qr > 0 and sufficient that 0 ≤ K 2 < 2 QPrr a02 . The condition of the modulational instability of the system under consideration is thus Pr Qr > 0;

(3.86)

the corresponding local growth rate (gain) of MI reads    Im [] = Im + = Pi 20 + K 2 − 2Qi a02 +  Qr = |a0 | 2 , Pr

 −X +

√ X2 + Y2 , 0 ≤ K < Kmax 2

(3.87)

where X and Y are given by Eq. (3.84c). Relation (3.86) is the modulational instability criterion for the plane wave with time-varying amplitude in the dissipative nonlinear transmission network of Fig. 3.8.

3.2 Spatiotemporal Modulation of Damped Solitonlike Wave …

79

Using Eq. (3.75) with a nonzero right-hand side as the basic model, Lange and Newell [40] studied the modulational instability of harmonic waves by means of the cumulative momentum method. They showed that plane waves with constant amplitude are unstable (stable) under modulation for Pr Qr + Pi Qi > 0 (Pr Qr + Pi Qi < 0); in the physics literature, the relation Pr Qr + Pi Qi > 0 is known as the Lange and Newell’s criterion of the MI [40]. When examining the MI of slowly modulated waves in physical systems described by a higher-order GL equation, Pelap and Faye [41] generalized the Lange and Newell’s criterion with a corrective term which depends on the characteristic parameters of the carrier wave. The condition (3.86) of the MI of Stokes waves with time-varying amplitude thus differs from the Lange and Newell’s criterion for plane waves with constant amplitude. It is important to note that MI criterion (3.86) is in agreement with the Benjamin–Feir criterion for Stokes waves of NLS equation [42–45]. With the use of the network parameters given in Eq. (3.61), we display in Fig. 3.11 the behavior of coefficients P and Q through their real and imaginary parts, as well as products Pr Qr and Pi Qi as functions of angular frequency ω. For generating different plots, we use the dimensionless resistances σ1 = 25 × 10−4 and σ2 = 5 × 10−3 that correspond respectively to R1 = 5 and R2 = 10 (values close to the numerical one [24]). As it is seen from Fig. 3.12a, b and d, Pr , Pi , and Qi change their sign in the whole frequency range of the allowed band; as we can see from Fig. 3.11c, the real part Qr of the nonlinear coefficient Q is always negative for the network parameter (3.61). It is also seen from Fig. 3.11f that product Pi Qi changes its sign when ω varying in the segment [0, ωc ]. Plots (b), (d), and (f3 ) reveal that different scenarios described after Eqs. (3.76a) and (3.76b) occur for the network parameters (3.61) and the dimensionless parameters σ1 = 25 × 10−4 and σ2 = 5 × 10−3 . Especially, in some given range of angular frequency, Pi is negative while Qi is positive, as it is clearly seen from Fig. 3.11 (f3 ); this means that in such a region of variation of the angular frequency ω, Pi and Qi designate respectively the dispersion dissipation coefficient and the nonlinear dissipation coefficient. Also, it is seen from Fig. 3.11 (f3 ) that the situation where either Pi > 0 or Qi > 0, the resistances act as if the driven field Pi and Qi are taken separately. The situations where Pi < 0 and Qi < 0 and Pi > 0 and Qi > 0 can be also observed in Fig. 3.11 (f3 ); in the corresponding frequency range, the effect of resistances in the dynamics of modulated waves depends on the balance between the two coefficients. Lastly, Fig. 3.11e reveals that equation Pr Qr = 0 admits only one solution, ωnull = 156106.5, which corresponds to the propagating frequency fnull = 24. 85 kHz. According to condition (3.86) of the modulational instability, Fig. 3.11e establishes the existence of two regions concerning the MI of plane wave with time-varying amplitude and possible soliton solutions of the complex Ginzburg-Landau equation (3.75): (i) Region

80

3 Transmission of Dissipative Solitonlike Signals …

associating with fp ∈ [0, fnull [ for which Pr > 0 and Qr < 0, leading to Pr Qr < 0 and corresponding the modulational stability and hole soliton, and (ii) region for fp ∈ [fnull , fc [ for which Pr < 0 and Qr < 0, leading to Pr Qr > 0 and corresponding to the modulational instability and envelope soliton. Because Pr , Pi , Qr , and Qi depend on the dissipative elements R1 and R2 , the instability growth rate (3.87) and the domain of the variability of the wavenumber K of the modulation depend on R1 and R2 . To show how R1 and R2 affect the MI growth rate , we have depicted in Fig. 3.12 the MI growth rate according to Eq. (3.87) for the carrier wave with parameters (3.79c) for a00 = 1 and for different values of resistances R1 and R2 . Figure 3.12a shows the MI gain when only R2 varies, while Fig. 3.12(b) shows the MI growth rate when only R1 varies. It is seen from these plots that the dissipative elements R1 and R2 enhance the instability region. For the wavenumber of modulation K near zero (small K = 0), the MI growth rate decreases [increases] when R2 increases [R1 increases], while for wavenumber of modulation K near Kmax , the MI growth rate increases [decreases] when R2 increases [R1 increases]; this situation is well seen in Fig. 3.12c. Also, we can see from Fig. 3.12 that Kmax increases with R2 as we can seen from plots (a), while increasing R1 decreases the value of Kmax , as we can well observe from plots (b). By comparing plots of Fig. 3.12a obtained with a fixed value of R1 and those of Fig. 3.12b obtained with a fixed value of R2 , it appears that the values of the growth rate of the MI showed in Fig. 3.13a is higher that those showed in Fig. 3.12b. This means that the must important contribution to the MI gain comes from the resistance R2 in the shunt branch of the network. In conclusion, the effects of dissipative losses in shunt branch are more manifested than those resulting from the dissipative losses in the series branch. Remark It is important to notice that in the case of damping modulated plane waves, the amplitude of the propagating wave may decrease rapidly to zero and the waves vanish after few cells. In such situations, the phenomena of the MI would not be experimentally observed if the processing of instability takes place after many cells. Consequently, it should not be experimentally possible to predict the domain in which the electrical network may support the propagation of envelope solitons.

3.2.5 Spatiotemporal Modulated Signals Propagating Through the Network of Fig. 3.8 We start this subsection by presenting exact and approximative solitonlike solutions of the complex GL equation (3.75). Then, these exact and approximate solutions are used to investigate analytically the transmission of spatiotemporal damped envelope soliton signals though the network system of Fig. 3.8. Without loss of generality, we limit our study to the exact and approximative bright solitonlike solutions. When Pi = Qi = 0, Eq. (3.75) coincides with the standard nonlinear Schrödinger equation and admits, under the condition Pr Qr > 0, the bright soliton solutions of the form

3.2 Spatiotemporal Modulation of Damped Solitonlike Wave …

81

Fig. 3.12 Modulational instability growth rate Im[] according to Eq. ( 3.87) for the carrier wave with parameters (3.79c) for different values of resistances R1 and R2 . a MI gain for R1 = 5 and three values of resistance R2 , R2 = 5 (solid line), R2 = 15 (dash-dotted line), and R2 = 23 (dash line); b Mi growth rate for R2 = 5 and three values of R1 , R1 = 5 (solid line), R1 = 15 (dash-dotted line), and R1 = 23 (dash line); c MI gain for R1 = 5 and R2 = 23 (solid line) and for R1 = 23 and R2 = 5 (dashing line line). In different plots (a)–(c), the dotted lines show the variation of the MI growth rate in the absence of the dissipative elements, that is, when R1 = R2 = 0. Different plots are generated with the network parameters (3.61) and are associated with the linear wave propagating at frequency fp = 19.10 MHz for which the conditions Pi Qi > 0 and Pi 20 − 2Qi a02 < 0 are satisfied. Reprint from Ref. [27], Copyright 2022, with permission from Springer

" !  2Pr 20 − Qr As

exp i 0 x − τ , ψ(x, τ ) = Qr 2 cosh 2P A τ − υ ) (x s 0 r As

(3.88)

where As is the wave amplitude and υ0 = 2Pr 0 is the wave speed. In the situation when |Pi | + |Qi | > 0, Eq. (3.75) is not integrable and does not admit envelope soliton solutions of form (3.88). In this situation, we can build either exact or approximative

82

3 Transmission of Dissipative Solitonlike Signals …

Fig. 3.13 a Plot of the dimensionless resistance σ2 versus the angular frequency ω obtained as an implicit function defined form equation Pr Qi − Pi Qr = 0 for the network parameters (3.61) with σ1 = 25 × 10−4 . b Three-dimensional plot of the product Pi Qi . Reprint from Ref. [27], Copyright 2022, with permission from Springer

bright solitonlike solutions with time-varying parameters when Pr Qr > 0. In what follows, we focus ourselves to the exact and approximative envelope solitonlike solutions of Eq. (3.75) under the condition that coefficients P and Q of the GL Eq. (3.75) satisfy the condition (3.86) of the MI and |Pi | + |Qi | > 0.

3.2.5.1

Bright Solitonlike Solution with Time-Varying Parameters

To find analytical bright solitonlike solutions with time-varying parameters of the GL equation ( 3.75), we introduce the ansatz ψ(x, τ ) =

2 exp [i {2 x + h4 (τ )}] , coshN [1 x + h3 (τ )]

(3.89a)

where 1 > 0 and 2 are two free real parameters called solution parameters, h3 (τ ) ] ,N and h4 (τ ) are two real functions of variable τ , and coshN [X ] = N exp[X ]+exp[−X 2 being any positive parameter. Imposing to (3.89a) to satisfy the complex cubic GL equation (3.75) yields

3.2 Spatiotemporal Modulation of Damped Solitonlike Wave …

   h3 (τ ) = Pi 22 − 21 − 2Pr 1 2 τ + 3 ,    h4 (τ ) = Pr 21 − 22 − 2Pi 1 2 τ + 4 , 1 Qr N = 2 , Pr Qi − Pi Qr = 0, 21 Pr

83

(3.89b) (3.89c) (3.89d)

where 3 and 4 are real parameters. The solution parameters 1 > 0 and 2 , as we will see in the following, play an important role in the management of spatiotemporal envelope signals propagating through the network of Fig. 3.8. The positivity of N in Eq. (3.89b) follows from that of product Pr Qr . It is important to notice that the exact solitonlike solution (3.89a) with parameters (3.89b)–(3.89d) is valid only when P and Q satisfy the additional condition Pr Qi − Pi Qr = 0; for Qi = 0, condition (3.86) of the MI under the condition Pr Qi − Pi Qr = 0 becomes Pi Qi > 0. In the practice, condition Pr Qi − Pi Qr = 0 can be used as follows: For the given network parameters (3.61) and, let us say, for a given dimensionless resistance σ1 , one can use equation Pr Qi − Pi Qr = 0 to express σ2 in terms of the angular frequency ω (here, taking into account the fact the most contribution to the spatial decreasing rate comes from R2 , we have preferred to solve Pr Qi − Pi Qr = 0 in σ2 ); in other words, for a given propagating frequency fp = ω/2π , one can find the appropriate dimensionless resistance σ2 = σ2 (ω) that leads to Pr Qi − Pi Qr = 0. If condition Pi Qi > 0 is satisfied for such σ2 (ω), then Eq. (3.89a) with parameters (3.89b)–(3.89d) will give an envelope solitonlike solution of the complex GL equation (3.75) for the given network parameters (3.61) with dimensionless resistances σ1 and σ2 = σ2 (ω). For a better understanding, we show in Fig. 3.13a the plot of the dimensionless resistance σ2 versus the angular frequency ω as an implicit function derived from the equation Pr Qi − Pi Qr = 0 for the network parameters (3.61) and the dimensionless resistance σ1 = 25 × 10−4 (this value of σ1 corresponds to the numerical value R1 = 5  [24]). As we can see from Fig. 3.13b, product Pi Qi is positive for σ2 around the numerical value σ2 = 5 × 10−3 . It is seen from Fig. 3.13a that the numerical value σ2 = 5 × 10−3 corresponds to the propagating frequency fp = 10.135 MHz. Now, if we insert Eq. (3.89a) in the decomposition (3.74) and ignore the dc term ψ1 and the second harmonics term ψ2 , we will obtain that the shape of the network soliton signal can be well approximated by the shape of the first harmonic whose expression is given as  cos [z2 ] cos ε1 υgi t coshN [z1 ]   − sin ε1 υgi t sin [z2 ] sinhN [z1 ]  ; Vn (t) = 4ε exp 2 ευgi t − χ n sinh2N [z1 ] + N cos2 ε1 υgi t  here, z1 , z2 , and coshN z1,2 as defined as

(3.90a)

84

3 Transmission of Dissipative Solitonlike Signals …

Fig. 3.14 Evolution plots of the real part υgr (a) and the imaginary part υgi (b) of the complex group velocity  υg as functions of the angular frequency ω defined by Eqs. (3.72a) and ( 3.72b), respectively. Plot (a) shows that the group velocity υgr is nonnegative, while plot (b) shows that υgi changes its sign in the whole range of the propagating frequency. Different plots are generated with the network parameters (3.61) with the dimensionless parameters σ1 = 25 × 10−4 and σ2 = 5 × 10−3 . Reprint from Ref. [27], Copyright 2022, with permission from Springer

    z1 = z1 (n, t) = 1 εn + ε ε Pi 22 − 21 − 2Pr 1 2 − 1 υgr t + 3 ,  

    z2 = z2 (n, t) = kp + 2 ε n + ε ε Pr 21 − 22 − 2Pi 1 2 − 2 υgr − ω t + 4 ,

(3.91) N exp [z1 ] + exp [−z1 ] N exp [z1 ] − exp [−z1 ] coshN [z1 ] = , sinhN [z1 ] = . 2 2

In general, the real part υgr of group velocity υg is a nonnegative quantity, while its imaginary part υgi can take any sign, negative, null, or positive. For a better understanding, we depict in Fig. 3.14 the variation of υgr and υgi versus the angular frequency ω for the network parameters (3.61); here, we use the dimensionless resistances σ1 = 25 × 10−4 and σ2 = 5 × 10−3 corresponding respectively to R1 = 5  and R2 = 10  (values close to the numerical one [24]). As we can see from different plots of Fig. 3.14, υgr is nonnegative (Fig. 3.14a) and υgi changes its sign (Fig. 3.14b). we can see that the wave amplitude is proportional to exp  From Eq. (3.90a), ε2 υgi t − χ n so that if the free parameter 2 is taken from the condition 2 υgi < 0, the envelope soliton signal associated with Eq. (3.90a) will be damped in both time and space. That is, the signal amplitude will decrease in both the time t and the cell number n (since χ is a positive quantity). The expression of z1 (n, t) given in Eq. (3.91) reveals that the soliton width is ωdth = 1/ε1 , while its speed is defined as    2 υsol = υgr + ε Pi 1 − −1 1 2 + 2Pr 2 and coincides with the group velocity if the free parameters 1 and 2 are chosen from the condition (Pr ± |P|) 1 − Pi 2 = 0. Therefore, the situation when 1 and 2 satisfy the relationship (Pr ± |P|) 1 − Pi 2 = 0 corresponds to the case where the electrical network is excited by a soliton signal with initial velocity corresponding to

3.2 Spatiotemporal Modulation of Damped Solitonlike Wave …

85

the group velocity υgr of the wave packet. The centre of the envelope soliton in such a situation will move with the constant speed υsol = υgr , while its amplitude will vary proportionally to exp 2 ευgi t − χ n . If 1 and 2 are taken from the condition (Pr ± |P|) 1 − Pi 2 = 0, we obtain a situation in which our network system is excited by a bright soliton with a speed υsol different from the group  velocity υgr of the wave packet, and its amplitude varies proportionally to exp 2 ευgi t − χ n . It should be to noticed that in the case when N = 1, both 1 and 2 can be used to control the wave amplitude in the propagating regime where υgi = 0. It follows from the expression ωdth = 1/ε1 of the wave width, that parameter 1 can also be used to manage the soliton width. Hence, the two solution parameters 1 and 2 can be used to manage the spatiotemporal bright soliton signals propagating through the network of Fig. 3.8.

3.2.5.2

Approximative Bright Solitonlike Solutions of the GL Equation (3.75)

In the following, we assume that P and Q satisfy the condition of the MI (3.86) and intend to find the approximative bright solitonlike solution of Eq. (3.75). For this aim, we consider the ansatz ψ(x, τ ) = (τ )ρ(ξ ) exp [i (λx + (τ ))] , ξ = Ax + B(τ ), A = 0,

(3.92)

where A and λ are two real constants, and (τ ), ρ(ξ ), (τ ), and B(τ ) are real functions to be determined. Inserting Eq. (3.92) into Eq. (3.75) and setting the real and imaginary parts of the resulting equation equal to zero yield the differential system of two equations in ρ, , B, , and  :   d 2ρ dρ d 2 − P ρ + Qr 2 ρ 3 = 0, − 2P λA λ + i r dξ2 dξ dτ (3.93a)     2 d ρ dB dρ d Pi A2 2 +  + 2Pr λA + − Pi λ2  ρ + Qi 3 ρ 3 = 0. dξ dτ dξ dτ (3.93b) Pr A2

To find the approximative bright solitonlike solution of Eq. (3.75), we ask that A and B satisfy the equation dB + 2Pr λA = 0. (3.94a) dτ For simplicity, we assume that ρ(ξ ) and its various order derivatives vanish at |ξ | = +∞, and limit ourselves to the localized solutions of the field equation. If we multiply both sides of Eq. (3.93a) by ρd ξ and integrate the resulting equation with respect to ξ from ξ = −∞ to ξ = +∞, we arrive to

86

3 Transmission of Dissipative Solitonlike Signals …

 (+∞  (+∞ (+∞ 2 dρ d 4 2 2 2 ρ d ξ − Pr λ + ρ d ξ − Pr A d ξ = 0. (3.94b) Qr  dτ dξ 2

−∞

−∞

−∞

Multiplying now the both sides of Eq. (3.93b) by equation under the condition (3.94a) yield  Pi A2

dρ dξ

2

 +

dρ dξ

and integrating the resulting

 d 1 − Pi λ2  ρ 2 + Qi 3 ρ 4 = 0. dτ 2

(3.94c)

Let μ0 = μ0 (τ ) and ρ0 = ρ0 (τ ) be two real functions of τ that satisfy the relationship μ20 = −

1 Qi 2 2 ρ (implying that Pi Qi < 0). 2 Pi A2 0

(3.95)

Asking that  satisfies the equation d 1 − Pi λ2  + Qi ρ02 3 = 0, dτ 2 it is easily seen that ρ (ξ ) =

ρ0 (τ ) cosh [μ0 (τ )ξ ]

(3.96)

(3.97)

is a special solution of Eq. (3.94c). Using now Eqs. (3.97) and (3.94b), we obtain that d 4Pi Qr + Pr Qi 2 2 ρ0  + Pr λ2 = 0. (3.98) − dτ 6Pi Gathering Eqs. (3.94a), (3.96), and (3.98) yields the following ordinary differential system for the determination of the solution parameters B(τ ), (ξ ), and (τ ), assuming that λ and A are two arbitrary real constants dB + 2Pr λA = 0, dτ d 1 − Pi λ2  + Qi ρ02 3 = 0, dτ 2 4Pi Qr + Pr Qi 2 2 d − ρ0  + Pr λ2 = 0. dτ 6Pi

(3.99a) (3.99b) (3.99c)

We can easily see that solutions of system (3.99a)–(3.99c) depend on the choice of the functional parameter ρ0 (τ ) from Eq. (3.95). In the special case when ρ02 = 2, Eq. (3.99b) coincides with the first equation in Eq. (3.78) when λ2 is replaced by 20 and  is replaced by a0 . In this special case, a special solution of system (3.99a)–(3.99c) is given as

3.2 Spatiotemporal Modulation of Damped Solitonlike Wave … B(τ ) = A (−2Pr λτ + B0 ) , 0 (τ ) =  ,   2 2   Q Qi 0 i 0 2 exp −2Pi λ τ 2 P 2 + 1− i λ

87

(3.100)

Pi λ

⎡ ⎤ Pi λ2 20 2λ2 (2Pi Qr − Qi Pr ) Pr Qi + 4Pi Qr ⎣ ⎦,  (τ ) = 0 + τ+ ln  3Qi 6Pi Qi Pi λ2 − Qi 20 2 (τ )

if λ  = 0,

B = AB0 , 20 , 220 Qi τ + 1 (4Pi Qr + Pr Qi )  = ln 1 + 220 Qi τ + 0 , if λ = 0. 6Pi Qi

2 =

(3.101)

Here, B0 = B(0)/A, 0 = (0) and 0 = (0) are three real constants of integration. To make (τ ) and (τ ) defined as τ → +∞, Pi must be negative (this leads, from condition Pi Qi < 0, to Qi > 0). Inserting Eqs. (3.97) and (3.100) into Eq. (3.92), we arrive to the following approximative bright solitonlike solution of the complex GL Eq. (3.75) ψ(x, τ ) =

cosh

as (τ ) = 

as (τ ) ϕ (x, τ )] , exp [i Qi − 2P as (τ ) (x − 2Pr λτ + B0 ) i

2 Qi 0 Pi λ 2 +

 1−

√ 20  2 Qi 0 Pi λ 2

 exp −2Pi λ2 τ

,

(3.102)

⎡ ⎤ 2Pi λ2 20 2λ2 (2Pi Qr − Qi Pr ) Pr Qi + 4Pi Qr ⎣ ⎦ + 0 .   ϕ (x, τ ) = λx + τ+ ln  3Qi 6Pi Qi Pi λ2 − Qi 20 as2 (τ )

Here, λ = 0, 0 = 0 and 0 are three arbitrary real constants, as (τ ) and ϕ(x, τ ) are respectively the soliton amplitude and the soliton phase. Henceforth, the free parameter λ will be referred to as “the solution parameter” for the approximative bright solitonlike solution of the GL equation (3.75). Inserting now Eqs. (3.97) and (3.101) into Eq. (3.92), we obtain, for λ = 0, the following approximative bright solitonlike solution of Eq. (3.75)

   2 r Qi ln 1 + 2Q  τ exp i λx + 0 + 4Pi Q6Pr +P i 0 Q i i " ! , 2 Qi 0 2Qi 0 τ + 1 √ cosh − Pi (x + B0 ) 2 √

ψ(x, τ ) =

20

2Qi 0 τ +1

(3.103a) 0 = 0 and 0 being two arbitrary real constants.

88

3 Transmission of Dissipative Solitonlike Signals …

It is important to note that Eqs. (3.102) and (3.103a) are solutions in the averaged sense. They can be obtained with the help of the adiabatic perturbation theory developed for bright solitons [46, 47]. It is seen from Eqs. (3.102) and (3.103a) that the soliton amplitude decays in time τ as the wave progresses. In the special case when λ = 0, solution (3.102) corresponds to the situation where the electrical network is excited by a soliton signal with initial velocity corresponding to the group velocity υgr of the wave packet. The centre of the envelope soliton in such a situation moves with the speed dx/d τ = 0, and this means that the soliton speed remains constant while its amplitude as (τ ) varies according to the second equation in Eq. (3.102). Therefore, the network electrical bright soliton propagates in the lossy network of Fig. 3.8 with a constant speed which coincides with the group velocity of the carrier wave. In the situation when λ = 0, the initial amplitude as (τ ) = 0 and speed υs = 2Pr λ = 0 of the bright soliton satisfy the relationship as (τ ) = )

√ ± 20

4Pr2 Qi 20 Pi υs2

 + 1−

4Pr2 Qi 20 Pi υs2



, Pi υs2 exp − 2P2 τ r

and solution (3.102) in this case λ = 0 corresponds to the situation when the network under consideration is excited by an envelope solitons with a velocity υs different to the group velocity of the wave packet. To use the above approximate soliton solution for the transmission of bright solitonlike signal through our network system, we proceed as in the case of the exact soliton solution of the amplitude equation. We use the decomposition (3.74) of the network voltage Vn (t) and the approximative envelope solitonlike solutions of Eq. (3.75), and return back to dimensioned parameters variables. We then obtain, after ignoring the dc and the second harmonics, that the shape of the network envelope soliton signal can be well approximated by the shape of the first harmonic whose expression is given by either  cosh [z] cos [φ] cos [ϕ] − sinh [z] sin [φ] sin [ϕ] , Vn (t) = 2εas (t) exp ελυgi t − χ n sinh2 [z] + cos2 [ϕ] (3.104) √ as (t) = 

2 Qi 0 Pi λ 2

 + 1− 

 20  2

Qi 0 Pi λ 2



exp −2Pi λ2 ε 2 t



, ϕ = ϕ(t) = ευgi t −

Qi as (t), λ  = 0, 2Pi

" !   Qi B0 , as (t) n − υgr + 2Pr λε t + 2Pi ε & %   2ε 2 λ2 (Qi Pr − 2Pi Qr ) t φ = φ(n, t) = kp + λε n − ω + λευgr + 3Qi ⎤ ⎡ 2Pi λ2 20 Pr Qi + 4Pi Qr ⎣ ⎦.   +0 + ln 6Pi Qi P λ2 − Q 2 a2 (t) z = z(n, t) = ε −

i

i

0

s

(3.105)

3.2 Spatiotemporal Modulation of Damped Solitonlike Wave …

89

Fig. 3.15 Evolution of the damped envelope soliton signal in the network at frequency fp = 10.135 MHz for the network parameters (3.61) with R1 = 5 , R2 = 10 , and ε = 2 × 10−3 . Different plots are generated with the help of the shape of the first harmonic (3.90a) for (a): 1 = 0.2, 2 = 3, 3 = 10, and 4 = 0; b 1 = 0.2, 2 = 0.1, 3 = −2.2, and 4 = 0. Plots of panels (a) show that the soliton amplitude decreases as the cell number n increases, while those of column (b)  show that the wave amplitude decays as time t increases. c, d show the wave amplitude factor exp 2 ευgi t − χn as function of time t for n = 500 with the same data as in plots (a) and function of the cell number n for t = 4 μs with the same data as in plots (b), respectively. Reprint from Ref. [27], Copyright 2022, with permission from Springer

for the approximative solution (3.102) with λ = 0, or Vn (t) = 2ε exp [−χn]

z = z(t) =

0 ε2 Qi 20 t

+



ε0 −2ε2 Pi 20 t −

1 2

Pi Qi

φ = φ(n, t) = kp n − ωt + 0 +

cosh [z] cos [φ] cos [ϕ] − sin [φ] sinh [z] sin [ϕ] , sinh2 [z] + cos2 [ϕ]

(3.106)



n − υgr t + ε−1 B0 , ϕ = ϕ(t) =





4Pi Qr + Pr Qi ln 1 + 2ε2 Qi 20 t , 6Pi Qi

for the approximative solution (3.103a) with λ = 0.

ευgi 0 Pi −Q − 2ε2 Pi 20 t i

t,

(3.107) (3.108)

90

3 Transmission of Dissipative Solitonlike Signals …

It follows from Eqs. (3.104) and (3.105) that:  (i) The soliton amplitude is proportional to 2ε exp λευgi t − χ n , meaning that the soliton amplitude decreases as the cell number n increases; (ii) Because Pi < 0, the soliton amplitude as (t) decreases as the wave propagates [lim as (t) = 0 as t → +∞]; (iii) The soliton speed υ = υgr + 2εPr λ is constant and depends on the free parameter λ; (iv) The soliton width is  ωdth = ε

−1

−

2Pi 1 Qi |as (t)|

and tends to +∞ as t → +∞; (v) As we can see from the expression of φ [see Eq. (3.105 )], the angular frequency of the dissipative bright soliton voltage (3.104 ) is formed of two terms, a linear term ωp = ω + λευgr + and a nonlinear term ωn =

2ε2 λ2 (Qi Pr − 2Pi Qr ) , 3Qi

Pr Qi + 4Pi Qr 1 das , 3Pi Qi as (t) dt

while the wavenumber is k = kp + λε and depends on the free parameter λ. The above analysis can be done for the approximate shape of the network soliton (3.106)– (3.108). As we will see in the following, the free parameter λ plays an important role in the transmission of spatiotemporal damped envelope soliton signals through the network of Fig. 3.8.

3.2.6 Transmission of Spatiotemporal Modulated Damped Envelope Signals Through a Lossy Network Our purpose now is to investigate analytically the transmission of spatiotemporal modulated envelope signals through the RLC network of Fig. 3.8. For this aim, we use the above shapes of the first harmonic (3.90a), (3.104), and (3.106) obtained with the help of exact and approximate solutions of the amplitude equation (3.75). Our first example is obtained with the use of the shape of the first harmonic (3.90a) corresponding to the exact bright solitonlike solution (3.89a). Here, the network parameters (3.61) with σ1 = 25 × 10−4 corresponding to R1 = 5  are used; using a value of σ2 that corresponds to the numerical value R2 = 10  used in Ref. [24], we find the propagating frequency to be fp = 10.135 MHz. Computing υgr and υgi , we find that υgr is positive, while υgi is negative (this can be seen from Fig. 3.15a, b when ω ≈ 6.368 × 107 ). We then use these data to depict in Fig. 3.15 the evolution of spatiotemporal damped bright soliton signals propagating at frequency fp = 10.135

3.2 Spatiotemporal Modulation of Damped Solitonlike Wave …

91

Fig. 3.16 Plot of the voltage Vn (t) as a function of the cell number n, at given times showing the evolution of damped soliton signals in the network of Fig. 1 at frequency fp = 7.96 MHz. Plots of the left and right columns show the soliton voltage signals obtained with the approximative shape of the network soliton (3.104) for respectively λ = −5 and λ = 5, while the plots of the middle column show the evolution of the soliton voltage signals obtained with the help of the approximative shape of the network soliton (3.106) associated with λ = 0. Different plots are generated with the network parameters (3.61) for ε = 4 × 10−3 and the dimensionless resistances σ1 = 25 × 10−5 and σ2 = 5 × 10−3 . Other parameters appearing in solutions (3.104) and (3.106) are B0 = 0, 0 = 0, and 0 = 70. It is important to notice that the propagating frequency fp = 7.96 MHz and the above dimensionless resistances σ1 and σ2 lead to a positive υgi . Reprint from Ref. [27], Copyright 2022, with permission from Springer

MHz through our network system. From plots of Fig. 3.15, we can see that the soliton amplitude decays as either the cell number increases [column (a)] or the time t increases [column (b)]. Also, plots of Fig. 3.15 reveal that the centre of the bright soliton moves with both the cell number n [column (a)] (b)].  and time t [column Fig. 3.15c and 8d show how the exponential factor exp 2 ευgi t − χ n of the wave amplitude maximum decays respectively versus time t and versus cell number n. Our second example is generated with the shape of the first harmonic (3.104) and (3.106) associated with the approximative envelope solitonlike solutions of the complex GL equation (3.75). This example demonstrates how the solution parameter λ affects the transmission of the spatiotemporal damped envelope soliton signals propagating through the lossy network of Fig. 3.8. In order to examine the effects parameter λ on the soliton signals, we have depicted in Fig. 3.16 the evolution plots

92

3 Transmission of Dissipative Solitonlike Signals …

of damped bright soliton signals for three values of λ. The data used in Fig. 3.16 lead to a positive υgi so that λυgi > 0 for positive λ and λυgi < 0 for negative λ. From plots of Fig. 3.16, we conclude that 1. in the situation where the electrical network is excited by a soliton with initial velocity corresponding to the group velocity υg = υgr of the wave packet (that is, λ = 0), the soliton amplitude is higher than in the general situation where the electrical network is excited by the envelope solitons with initial velocity different to the group velocity of the wave packet (that is, λ = 0); 2. the soliton amplitude for λ > 0 is higher than that for λ < 0 (this is well seen when comparing plots of the left column with those of the right column, especially for large time); 3. the soliton speed decreases as λ increases; this behavior is a consequence of the fact that the soliton speed, as we can see from Eqs. (3.105) and (3.107), is υ = υgr + 2εPr λ and Pr < 0 for the data used in Fig. 3.16. We can then conclude that parameter λ of the solution can be used to modulate the damped soliton amplitude and velocity. In conclusion, we have investigated in this Section the effects of losses of a RLC nonlinear electrical transmission network on the slowly modulated wave dynamics. We have derived the spatial decreasing rate χ in terms of the propagating frequency and showed that the amplitude of linear waves is an exponentially decreasing function of the network cell number n. Our studies reveal that the introduction of the losses in the network increases the propagating frequency in the operating frequency range of the network. We have showed that the dynamics of modulated waves in our network model are governed by a complex cubic GL equation whose coefficients depend on the spatial decreasing rate χ . We have investigated the linear stability of slowly modulated plane wave with time-varying amplitude and established the criterion of their MI. We found that for the modulated damped plane waves, the phenomena of the MI would not be experimentally observed if the processing of instability takes place after many cells. Parametric analytical exact and approximate bright solitonlike solutions of the amplitude equation (3.75) are presented and used to investigate the effects of losses on the electrical envelope soliton propagation through the network. Our investigation showed that the amplitude of the envelope soliton decreases in both time t and cell number n, the width of the soliton depends on the spatial decreasing rate χ as the soliton propagates along the system. More interestingly, we found that both the soliton speed in the moving frame at the group velocity of carrier wave and the group velocity of carrier wave remain constant during the soliton propagation. Comparing the contribution of dissipative losses in the series branch to that in the shunt branch of the network on the modulational instability, we found that in the domain where the network may support the envelope soliton propagation, effects of dissipative losses in shunt branch are more manifested than those resulting from the dissipative losses

3.3 Modulated Wavetrains in a Dissipative Bi-Inductance Transmission Network

93

in the series branch. Also, our study has showed that the solution parameters 1 and 2 or λ can be used to manage the soliton amplitude and speed.

3.3 Modulated Wavetrains in a Dissipative Bi-Inductance Transmission Network 3.3.1 Introduction and Circuit Equations We consider in this Section a lossy nonlinear bi-inductance transmission network with dispersive linear elements. The envelope modulation is reduced to a generalized cubic-quintic Ginzburg-Landau (CQ-CGL) equation. Using this equation, we analyze the MI phenomenon and derive a generalized Lange-Newell criterion [48]. The nonlinear coherent shapes are demonstrated by formulating the analytical solutions for the envelope equation via appropriate methods. To make possible the study of coherent structures, the found CQ-CGL equation is first reduced to a third-order ordinary differential system. The electrical network considered in this Section is the discrete nonlineardispersive bi-inductance transmission network with losses shown in Fig. 3.17. Applying the Kirchhoff’s laws to this network, we arrive to the following set of network equations ∂I1,n ∂I1,n+1 = Vn−1 − Vn , L1 = Vn − Vn+1 , ∂t ∂t 1 ∂ ∂Qn + Vn = In − In+1 , In − I1,n = CS (Vn−1 − Vn ) , ∂t ∂t G L2

(3.109)

where Qn = Qn (Vn ) is the charge carried by the nonlinear capacitance C(Vn ), Vn = Vn (t) is the voltage on it, I1,n is the current passing through the n-th linear inductor L2 .

Fig. 3.17 One section of the discrete nonlinear-dispersive dissipative bi-inductance transmission network composed of N such identical sections. Reprint from Ref. [2], Copyright 2022, with permission from Springer

94

3 Transmission of Dissipative Solitonlike Signals …

Eliminating the currents from system (3.109) and assuming a voltage-dependence nonlinear capacitor of the form   V C(V ) ≈ C0 1 − V0

(3.110)

yield the following difference-differential equations for the voltage (network equations) C0

d 2 Vn 2 d 2V 2 n −CS dtd 2 (Vn−1 − 2Vn + Vn+1 ) + G dV − C0 b dt 2n dt 2 dt = L12 (Vn−1 − Vn ) − L11 (Vn − Vn+1 ) ,

(3.111)

where n = 1, 2, . . . , N (N being the total number of cells in our network system), C0 and V0 are the capacitance and voltage scales, and V  V .

3.3.2 Generalized Cubic-Quintic Complex Ginzburg-Landau Equation for a Lossy Network To analyze short-wavelength modulated nonlinear waves propagating in our network system, a semi-discrete approximation is used [49, 50]. We introduce, for analyzing the propagation of the group of waves centered around wavenumber k (0 ≤ k ≤ π ) and frequency ω, two slow time scales T1 = t and T2 =  2 t , in addition to the original one T0 = t, along with the spatial long scale, and one slow spatial variable X1 = x, in addition to the original x = n, where   1 is a small dimensionless parameter. Following Taniuti and Yajima [51], the voltage Vn (t), solution of system (3.111) is sought in the general form Vn (t) =  1/2 v11 eiθ + v22 e2iθ +  3/2 v33 e3iθ  + 2 v40 + v42 e2iθ + v44 e4iθ + c.c.,

(3.112)

where θ = kn − ωt, and vjk = vjk (X1 , T1 , T2 ) are complex amplitudes. (3.112) into (3.111) and equating of like powers of   coefficients   Substituting , eiθ , one obtains many equations. At order  1/2 , eiθ , the following linear dispersion relation is derived     sin k 2 2k 2 k 2 = 0, sin − C0 + 4CS sin ω − i Gω − L 2 2 L0 with

1 1 1 + , = L L1 L2

1 1 1 = − . L0 L2 L1

(3.113)

3.3 Modulated Wavetrains in a Dissipative Bi-Inductance Transmission Network

95

Separating the linear dispersion relation (3.113) into real and imaginary parts and solving the resulting equations in 1/G = R(k) and ω = ω(k) yield L0 1 ≡R= G 2 cos k2



2

k  , ω = sin  2 L C0 + 4CS sin2 k2



2 ,  L C0 + 4CS sin2 k2 (3.114)

with 0 ≤ k ≤ π. Note that, for C0 < 4CS , relation R = R(k) exhibits one minimum for 0 ≤ k ≤ π/2, which corresponds to the leading unstable mode, as we can see from Fig. 3.18 for the given network parameters. The minimization of the dispersion relation R = R(k) leads to a numerical problem for the critical wavenumber k = kC , all parameters evaluated at kC being referred to as the “critical network parameters”. In particular, to the critical network parameters C0C = CSC = 540 pF, L1C = 2L2C = 28 μH, bC = 0.16 V−1

(Lp)

there correspond the critical wavenumber k = kC ≈ 1.318, angular frequency ω = ωc ≈ 4.45 × 106 , and RC = R(kC ) ≈ RC = 315. In Fig. 3.18a, the horizontal dotted line is obtained with the  critical  value R = RC ≈ 315. Equation of order  5/2 , eiθ gives the following amplitude equation that governs the evolution of the wave packet i

∂ |v11 |2 ∂v11 ∂v11 ∂ 2 v11 2 4 |v | |v | + Q v + Q v + γ v + Q v +P + Q2 |v11 |2 = 0. 11 11 3 11 11 11 1 11 ∂t ∂x2 ∂x ∂x

(3.115) Here, P = Pr + iPi , Q = Qr + iQi , γ = γr + iγi , and Qj = Qjr + iQji , (j = 1, 2, 3) depend on the wavenumber k and are given in terms of the network parameters as P Q γ Q1

   + A2 C0 − 4CS sin2 k2  , =− 2C0 ω + 8CS ω sin k2 + iG  4 2 C0 bω2 Re(Q) =− , 2 k 2C0 ω + 8CS ω sin 2 + iG * 4C0 bω2 Re(Q)  =− , 2 k 2C0 ω + 8CS ω sin 2 + iG      C B (1 + A) C0 − 4CS sin2 k2 − 4BCS ω sin [k] + 4iC0 bAω , =− 2C0 ω + 8CS ω sin2 k2 + iG (3.116) CS ω2 − 4CS Aω sin [k] −

e−ik 2L2

−

eik 2L1 2

2C0 bω2 (A3 + AA2 )  + Q1 , 2C0 ω + 8CS ω sin2 k2 + iG Cω2 2 2 C0 bA2 B  − 2 Q3 = 2C0 ω + 8CS ω sin2 k2 + iG

Q2 = −

96

3 Transmission of Dissipative Solitonlike Signals …

Fig. 3.18 Dependency a R = R(k) and b ω = ω(k) given by Eq. (3.114) for critical values C0C = CSC = 540 pF and L1C = 2L2C = 28 μH. The unique minimum of R = R(k) corresponds to the most unstable mode. Reprint from Ref. [2], Copyright 2022, with permission from Springer

' '2  2     ∗  ' C ' B C0 − 4CS sin2 k2 + 4iC0 bBω + 2C0 bω2  C D + A1  . × 2C0 ω + 8CS ω sin2 k2 + iG The nonlinear partial differential equation (3.115) is often called the modified or generalized CQ CGL equation. Its equivalent form reads i

∗ ∂ 2 v11 ∂v11 2 ∂v11 + (Q + Q ) |v |2 ∂v11 = 0 +P + Q |v11 |2 v11 + Q3 |v11 |4 v11 + γ v11 + Q1 v11 2 1 11 ∂t ∂x ∂x ∂x2

(3.117) ∗ is the complex conjugate of v11 . Equation (3.117) is most general and is where v11 usually called a generalized CQ CGL equation with derivative terms |v11 |2 ∂v11 /∂x 2 ∗ ∂v11 /∂x [52]. Deissler and Brand [16] showed numerically that these two and v11 additional terms can significantly reduce the speed of pulses and break the symmetry of their shape [31].

3.3.3 Linear Analysis and Modulational Instability Here, we address the MI of the Stokes-wave solution of the generalized CQ CGL equation (3.115). We first note that Eq. (3.115) admits Stokes-wave solution v11 (x, t) = a0 exp (ik0 x − iω0 t) ,

(3.118)

where a0 , k0 and ω0 are related by ω0 − Pk02 + γ + Q |a0 |2 + ik0 |a0 |2 Q2 + Q3 |a0 |4 = 0. Splitting equation (3.119) into real and imaginary parts yields

(3.119)

3.3 Modulated Wavetrains in a Dissipative Bi-Inductance Transmission Network

+

97

ω0 = Pr k02 − γr + (k0 Q2i − Qr ) |a0 |2 − Qr |a0 |4 , Q3i |a0 |4 + (Qi + k0 Q2r ) |a0 |2 − Pi k02 + γi = 0.

(3.120)

Solving the second equation of system (3.120) in |a0 |2 yields |a0 |2 =

−Qi − Q2r k0 ±

 2  Q2r + 4Pi Q3i k02 + 2Qi Q2r k0 + Qi2 − 4γi Q3i 2Q3i

.

(3.121) It follows from Eq. (3.121) that the amplitude |a0 | will be bounded as a func2 tion of k0 only when the global existence condition, Q2r + 4Pi Q3i < 0, is adopted. Moreover, the square root in Eq. (3.121) will make sense only under the condition Qi2 − 4γi Q3i > 0 so that the above Stokes-wave solution exists for k0 in a bounded interval (k0 )lower , (k0 )upper , where

(k0 )lower

=

(k0 )upper =

2 2 −Qi Q2r + 2 γi Q3i Q2r + 4Pi γi Q3i − Pi Qi2 Q3i 2 Q2r + 4Pi Q3i 2 2 −Qi Q2r − 2 γi Q3i Q2r + 4Pi γi Q3i − Pi Qi2 Q3i 2 Q2r + 4Pi Q3i

, .

The CW solutions (3.118)–(3.119) are parameterized by the wavenumber k0 . Now we study the MI of the carrier CW solution (3.118) by letting v11 = (a0 + δa) exp [i (k0 x − ω0 t)] ,

(3.122)

where δa is a small perturbation. Substituting (3.122) in (3.115), linearizing it with respect to the perturbation δa and using (3.119), we obtain i

 ∂δa ∂ 2 δa  ∂δa + P 2 + 2ik0 P + |a0 |2 (Q1 + Q2 ) ∂t ∂x ∂x ∗   ∂δa + |a0 |2 Q + ik0 Q2 + 2 |a0 |2 Q3 δa + a02 Q1  ∂x  + a02 Q + ik0 Q2 + 2 |a0 |2 Q3 δa∗ = 0.

(3.123)

Non trivial solutions of Eq. (3.123) is then looked in the form of    a = b1 exp [i (Kx + t)] + b∗2 exp −i Kx + ∗ t ,

(3.124)

where K and  are the modulation wavenumber and frequency, b1 and b2 are two complex constants. Substituting (3.124) into (3.123) and using the fact that |b1 | + |b2 | > 0 yield (3.125) ( − αr − iαi )2 = β = βr + iβi ,

98

3 Transmission of Dissipative Solitonlike Signals …

where αi and βr are as   αi = |a0 |2 Qi + k0 Q2r + 2 |a0 |2 Q3i − Pi K 2 , ⎡ 2 2 ⎤ 4k0 |P| − |a0 |4 |Q1 |2 ⎢ ⎥ ⎢ − 4k0 |a0 |2 (Pi Q1i − Pr Q1r + Pi Q2r − Pr Q2r ) ⎥ 2 ⎥ βr = |P|2 K 4 − ⎢ ⎢ + 2 |a |2 (P Q + P Q + k (P Q2i) − P Q )⎥ K 0 r r i i 0 r i 2r ⎦ ⎣ + |a0 |4 |Q1 + Q2 |2 + 4 |a0 |4 (Pr Q3r + Pi Q3i ) ⎡ ⎤ 4k0 (Pi Q2r − Pr Q2i ) ⎢ ⎥ ⎢ − 2 |a0 |2 (Qi (Q1r + Q2r ) − Qr (Q1i + Q2i )) ⎥ 2⎢ ⎥K − |a0 | ⎢ ⎥ 2 ⎣ − 2k0 |a0 | (Q2r (Q1r + Q2r ) + Q2i (Q1i + Q2i ))⎦ − 4 |a0 |4 (Q3i (Q1r + Q2r ) − Q3r (Q1i + Q2i ))  2 + 2k0 Pr + |a0 |2 (Q1i + Q2i ) K 2   2  − |a0 |2 Qi + k0 Q2r + 2 |a0 |2 Q3i − Pi K 2 ⎤ ⎡ − 2k0 (Q2i Qr − Q2r Qi ) ⎥ ⎢ ⎥ ⎢ |a |2 4 ⎢ − 4k0 0 (Q2i Q3r − Q2r Q3i )⎥ − |a0 | ⎢ ⎥. ⎦ ⎣ + 2k0 (Qi Q2r − Qr Q2i ) 2 − 4k0 |a0 | (Q3i Q2r − Q3r Q2i ) Solving Eq. (3.125) in  yields )  = αr ±

% & ) βr + |β| −βr + |β| + i αi ± . 2 2

(3.126)

The MI sets in provided that the imaginary part of the complex angular frequency  is negative, that is, ) −βr + |β| < 0. (3.127) Im []± = αi ± 2 For condition (3.127) to be satisfied, it is necessary that αi should be negative:   |a0 |2 Qi + k0 Q2r + 2 |a0 |2 Q3i − Pi K 2 < 0. Under condition (3.128), we arrive to Im []− = αi −

 (−βr + |β|) /2 < 0.

For Im[]+ to be negative under the condition (3.128), it is sufficient that

(3.128)

3.3 Modulated Wavetrains in a Dissipative Bi-Inductance Transmission Network

Pr Qr + Pi Qi + 2 |a0 |2 (Pr Q3r + Pi Q3i ) +

99

σ < 0, 2 |a0 |2 K 2

(3.129)

where σ is the real number given by # σ =

− |a0 |4 |Q1 |2 − 4k0 |a0 |2 (Pi Q1i − Pr Q1r + Pi Q2r − Pr Q2r )

$

+ 2 |a0 |2 k0 (Pr Q2i − Pi Q2r ) ⎡ ⎤ 4k0 (Pi Q2r − Pr Q2i ) ⎢ ⎥ ⎢ − 2 |a0 |2 (Qi (Q1r + Q2r ) − Qr (Q1i + Q2i )) ⎥ ⎥ + |a0 |2 ⎢ ⎢ − 2k |a |2 (Q (Q + Q ) + Q (Q + Q ))⎥ K 0 0 2r 1r 2r 2i 1i 2i ⎦ ⎣ − 4 |a0 |4 (Q3i (Q1r + Q2r ) − Q3r (Q1i + Q2i ))

K2

+4 |a0 |4 k0 (Q2r Qi − Q2i Qr ) 2  − 2k0 Pr + |a0 |2 (Q1i + Q2i ) K 2 − 2αi2 − |P|2 K 4 . Relation (3.129), in which K verifies condition (3.128), represents the MI criterion associated to the generalized CQ CGL equation (3.115) or its equivalent form (3.117). This result generalizes the well-known Lange-Newell criterionfor the stability of  Stokes waves [48] by the presence of the additional term σ/ 2 |a0 |2 K 2 , which   depends on the amplitude and wavenumber of the carrier k0 , |a0 |2 , as well as on perturbation wavenumber K.

3.3.4 Coherent Structures Now, we turn our attention to the coherent structures in our network system. Throughout this subsection, ω and k are not the same angular frequency and wavenumber used in the derivation of the amplitude equation; they have the same meaning as ω0 and k0 used in the Stokes-wave solution of the amplitude equation. The temporal evolution of coherent structures in the modified/generalized CQ-CGL equation amounts to the uniform propagation with velocity υ and overall phase oscillations with frequency ω : v11 (x, t) = a(z) exp [i(z) − iωt] ,

z = x − υt.

(3.130)

This ansatz covers all possible traveling wave solutions. The substitution of Eq. (3.130) in the modified/generalized CQ-CGL equation (3.115) gives rise to an ordinary differential system for a and :

100

3 Transmission of Dissipative Solitonlike Signals … ⎧! "  2 ⎪ d 2 d 2a d d 2 da ⎪ a − 2Pi ddz da − P − P + γ ω + υ ⎪ r i r 2 dz dz dz + (Q2r + 2Q1r ) a dz + Pr dz 2 ⎪ dz ⎪ ⎪   ⎪ ⎪ d 3 5 ⎪ ⎪ ⎨ + Qr − Q2i dz a + Q3r a = 0 ! ⎪  2 "   ⎪ ⎪ d 2 d ⎪ + P − P a + 2Pr ddz − υ γ ⎪ i r dz 2 i dz ⎪ ⎪ ⎪   ⎪ ⎪ ⎩ + Qi + Q2r ddz a3 + Q3i a5 = 0.

da dz

2

+ Pi ddz2a + (Q2i + 2Q1i ) a2 da dz

(3.131) In the special case of the plane wave, that is, when a(ξ ) = a0 and (z) = kz, 2 2 + 4Pi Q3i ≥ 0, Q2r + 4Pi Q3i < 0, system (3.131) yields, under the conditions Q2r 2 and Qi − 4γi Q3i > 0 : ω = Pr k 2 − υk − γr + (Q2i − Qr ) a02 − Q3r a04 ,  2  −Qi − Q2r k ± Q2r + 4Pi Q3i k 2 + 2Qi Q2r k + Qi2 − 4γi Q3i 2 a0 = . 2Q3i that plane waves exist for k taken from a bounded interval We have thus found (k0 )lower , (k0 )upper , where

(k0 )lower = (k0 )upper =

2 2 −Qi Q2r + 2 γi Q3i Q2r + 4Pi γi Q3i − Pi Qi2 Q3i 2 Q2r + 4Pi Q3i 2 2 −Qi Q2r − 2 γi Q3i Q2r + 4Pi γi Q3i − Pi Qi2 Q3i 2 Q2r + 4Pi Q3i

, .

0 Seeking (z) in the form  = y(z)dz, we obtains, after some manipulations, the following three-dimensional dynamical system: da = q, dz

(3.132a)

⎧ ⎡  ⎤ 2 1 ⎨ ⎣ Pr y − υy − ω − γr a + 2Pi yq dq ⎦ = Pr dz |P|2 ⎩ − (Q2r + 2Q1r ) a2 q + (Q2i y − Qr ) a3 − Q3r a5 # $1 (Pi y − γi ) a + (υ − 2Pr y) q +Pi (3.132b) , − (Q2i + 2Q1i ) a2 q − (Qi + Q2r y) a3 − Q3i a5 1 q

dy 2y = P y) − 2P − 2P (υ r r i dz |P|2 a 1 2

+ 2 Pr Pi y − γi − (Q2i + 2Q1i ) aq − (Qi + Q2r y) a2 − Q3i a4 |P|

3 −Pi Pr y2 − υy − ω − γr − (Q2r + 2Q1r ) aq + (Q2i y − Qr ) a2 − Q3r a4 .

(3.132c)

3.3 Modulated Wavetrains in a Dissipative Bi-Inductance Transmission Network

101

It is clear that a = 0 is singular point for system (3.132a)–(3.132c). To overcome this difficulty, we can use the so-called ”blowup” transform, or the “σ -process" [53], by setting x = q/a. (3.133) The singularized (a, q, y) system (3.132a)–(3.132c) under transformation (3.133) reduces to the following desingularized (a, x, y) system a = ax,

(3.134a)

⎧ # $⎫ ⎪ Pr y2 − υy − ω − γr − (Q2r + 2Q1r ) a2 x ⎪ ⎪ ⎪ ⎪ ⎪ P ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ r + (Q y − Qr ) a2 − Q a4 1 3r 2i  2 , (3.134b) x = −x + # $ ⎪ |P|2 ⎪ Pi y − γi + υx − (Q2i + 2Q1i ) a2 x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ + Pi − (Qi + Q2r y) a2 − Q3i a4 ⎫ ⎧

P (υ − 2Pr y) − 2Pi2 y x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ r ⎪ ⎪ # $ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ P y − γ − + 2Q x a (Q ) ⎪ ⎪ i i 2i 1i ⎬ ⎨ 1 + Pr  2 4 . y = (3.134c) + Q y) a − Q a − (Q 2 2r i 3i ⎪ ⎪ |P| ⎪ ⎪ # $ ⎪ ⎪ ⎪ ⎪ ⎪ Pr y2 − υy − ω − γr − (Q2r + 2Q1r ) a2 x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ − Pi 2 4 + (Q2i y − Qr ) a − Q3r a

The desingularized (a, x, y) system (3.134a)–(3.134c) has the invariant plane a = 0. Thus, the σ -process has transformed the former singular point into an invariant plane. On the invariant plane a = 0, the system (3.134a)–(3.134c) reduces to Pr2 2 Pi υ P 2 − Pr υ Pr (ω + γr ) + Pi γi y + x+ i y− , (3.135a) 2 2 |P| |P| |P|2 |P|2 Pi Pr Pr υ Pi (Pr + υ) Pi (ω + γr ) − Pr γi y = − 2 y2 − 2xy + x+ y+ (3.135b) . 2 2 |P| |P| |P| |P|2 x = −x2 +

It is clear that system (3.135a)–(3.135b) does not depend on Qi , Qr , Q1r , Q1i , Q2r , Q2i , Q3r , Q3i . Following the results of Ref. [54], we can easily show that system (3.135a)–(3.135b) characterizes the behavior of solutions of the singularized system (a, q, y) system at a → 0. In the following, system (3.134a)–(3.134c) is addressed in the case when ω = −γr and γi = 0.

(3.136)

 We should remember that γi = Im (γ ) and γr = Re γ are free parameters so that one may indeed set γi = 0 and choose ω as a function of the independent free parameter γr . Under these conditions, the desingularized (a, x, y) system (3.134a)– (3.134c) becomes a dynamical system in the standard form. This system gives rise

102

3 Transmission of Dissipative Solitonlike Signals …

Fig. 3.19 The phase portrait of system (3.134a)–(3.134c) for variables (a, x, y) in the invariant plane a = 0 when Pi (Pr + 2υ) ≥ 0 : (a1 ) υ = −2.210 1 × 107 < υmin (unstable focus); (a2 ) υ = 2.2101 × 107 > υmax (unstable focus); (a3 ) υ = 91313 > 0 ∈ (υmin , υmax ) (unstable node); (a4 ) υ = 2.2101 × 106 > υmax with Pi (Pr + 2υ) = 0 (center); (a5 ) υ = −99541 ∈ (υmin , υmax ) (unstable node); (a6 ) υ = 0 ∈ (υmin , υmax ) (unstable node); (a7 ) υ = υmin = −99542 (unstable node); (a8 ) υ = υmax = 91317 (unstable node). Reprint from Ref. [2], Copyright 2022, with permission from Springer

Fig. 3.20 The phase portrait of system (3.134a)–(3.134c) for variables (a, z) in the invariant plane a = 0 with z = x − υt, when Pi (Pr + 2υ) < 0 and υ = 2.210 1 × 107 > υmax (a stable focus). Reprint from Ref. [2], Copyright 2022, with permission from Springer

3.3 Modulated Wavetrains in a Dissipative Bi-Inductance Transmission Network

103

Fig. 3.21 Behavior of |v11 (x, t)| = |a(z = x − υt)| corresponding to (a1 ) , (a2 ) , (a3 ) , (a4 ) , (a5 ) , (a6 ) , (a7 ) , and (a8 ), respectively, from Fig. 26. Plots show the dependence of v11 on x and t. Waves shown in panels (b1 ), (b5 ) and (b7 ) travel to the left, while other waves travel to the right. Reprint from Ref. [2], Copyright 2022, with permission from Springer

Fig. 3.22 Behavior of |v11 (x, t)| = |a(z = x − υt)| corresponding to Fig. 3.21. This figure shows the dependence of v11 on x and t for the wave traveling to the right. Reprint from Ref. [2], Copyright 2022, with permission from Springer

104

3 Transmission of Dissipative Solitonlike Signals …

to a number of critical points, (0, 0, 0) being one of them, which corresponds to the zero amplitude, a = 0. If the coefficients of the amplitude equation (3.115) satisfy the conditions Qr Q3i − Qi Q3r = 0and (Pr Qi − Pi Qr ) (Pi Q3r − Pr Q3i ) > 0, then

   ± (Pr Qi − Pi Qr ) / (Pi Q3r − Pr Q3i ), 0, 0

are also critical points of system (3.134a)–(3.134c) under conditions (3.136). Using the linearization in the (a, x, y) space, the stability of the fixed point (0, 0, 0) is addressed here. In similar way, we can address the stability of the other critical points. The result is that point (0, 0, 0) has a two-dimensional unstable manifold when Pi (2υ + Pr ) ≥ 0, and a one-dimensional one, when Pi (2υ + Pr ) < 0. The respective linearized system has real eigenvalues for υ ∈ [υmin , υmax ] , and complex eigenvalues when υ ∈ / [υmin , υmax ] , where  Pi2 Pr + Pi Pr |P| Pi2 Pr − Pi Pr |P| , = min , 2Pr2 2Pr2  2  Pi Pr + Pi Pr |P| Pi2 Pr − Pi Pr |P| = max , . 2Pr2 2Pr2 

υmin υmax

The results are summarized by means of phase portraits shown in Fig. 3.19 for Pi (2υ + Pr ) ≥ 0, and in Fig. 3.20, when Pi (2υ + Pr ) < 0, for the critical-line parameters Lp . For Pi (2υ + Pr ) ≥ 0, we depict |v11 (x, t)| = |a(z)|, with z = x − υt in Fig. 3.21 for Pi (2υ + Pr ) ≥ 0, and in Fig. 3.22 for Pi (2υ + Pr ) < 0. To analyze the propagation of the wave, we here take  = 10−2 . As initial conditions, we use (a0 , x0 , y0 ) = (a(0), x(0), y(0)) = (0.01, 0.001, −0.001). We can see from panels (b1 ), (b5 ) and (b7 ) of Fig. 3.21 that waves are absorbed somewhere to the left of z = 0 (and to the right in that of Fig. 3.22). It is also seen from panels Fig. 3.21 (b2 ) , (b3 ) , (b6 ) and (b8 ) that waves are emitted from somewhere to the left of z = 0. The wave shown in panel (b4 ) of Fig. 3.21 has a uniform profile. In conclusion, modulated wavetrains in the discrete nonlinear-dispersive dissipative transmission network sketched in Fig. 3.17 have been addressed in this Section. With the use of the semi-discrete approximation and the reductive perturbation method, it is proven that the evolution of nonlinear excitations in the network is governed by the modified/generalized CQ CGL equation with the cubic derivative terms. The MI analysis of the Stokes waves is included, along with the MI criterion. Also, a real-valued system of ordinary differential equations was derived to describe coherent structures of the amplitude equation.

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Chapter 4

Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks

Abstract In this chapter, we analytical discuss the transmission of modulated rogue waves through electric nonlinear transmission networks when the losses are either neglected or taken into consideration. For a given electrical model of the nonlinear transmission network, we employ the reductive perturbation method in the semidiscrete limit to reduce the dynamics of modulated waves propagating along the network to nonlinear partial differential equations (NPDEs) such as equations of NLS type either without or with a dissipative term or complex cubic/quintic GinzburgLandau equation with derivative terms. Using the derived NPDE of a given physical system, we investigate the phenomenon of baseband modulational instability for the physical model to be studied. For each physical model to be studied, the integrable conditions under which rational polynomial solutions of the derived NPDE exist are obtained and first-order or/and second-order rational polynomial solutions are built. Using these rational polynomial solutions, we study in details the emission of first-order or/and second-order modulated rogue waves along the given electrical network.

4.1 Emission of Rogue Wave Signals Through the Modified Noguchi Electric Transmission Network 4.1.1 Introduction Firstly observed in the ocean [1–3], rogue waves (alias killer, giant, or extreme storm waves) are waves that are about twice or thrice higher than expected for the ocean state. Rogue waves can appear out of nowhere [4]. Mathematically, rogue waves are rational polynomial solutions of a class of NPDEs such as equations of NLS type [4–8]. In the physical context, rogue waves are located both in space and time, propagate on a nonzero continuous wave background, and concentrate the energy of the CW background into a small region [3, 6, 9–17]. As physical and mathematical phenomena, rogue waves occur in many fields of nonlinear science including optical fibers [18], super fluids [19], quantum mechanics [20], BECs [21], NLTNs [22], atmospherics [23], plasmas [24], etc. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 E. Kengne and W. Liu, Nonlinear Waves, https://doi.org/10.1007/978-981-19-6744-3_4

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4 Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks

The present Section deals with the propagation of RWs in the modified Noguchi electrical NLTN depicted in Fig. 2.1. These RWs are produced as solutions of a NLS equation with an additional time-varying linear-potential term [25]. The RWs propagate at bandwidth frequencies at which the network may exhibit baseband MI. Effects of network parameters on characteristics of the RW parameters are considered. We shown how to use parameters and the propagation frequency to maintain or suppress the RWs in a network such as that of Fig. 2.1.

4.1.2 Modulated Waves and Linear Analysis 4.1.2.1

Network Equations

The physical model to be studied in this Section is the 1D modified Noguchi electrical NLTN shown in Fig. 2.1 made of N identical cells. Assuming the capacitance-voltage (C − V ) characteristics to be of the form C (Vb + Vn ) =

  d Qn  C0 1 − 2αVn + 3βVn2 , d Vn

(4.1)

we employ the Kirchhoff’s laws and obtain the following network system  d2  Vn + λ (2Vn − Vn+1 − Vn−1 ) − αVn2 + βVn3 2 dt + ω02 Vn + u 20 (2Vn − Vn+1 − Vn−1 ) = 0,

(4.2)

where n = 1, 2, ..., N , Vb and C0 = C(Vb ) are respectively the equilibrium voltage and the capacitance scale, α and β are nonlinearity coefficients that determine the electric charge Q n stored in the n-th capacitor in the network, Vn is the voltage across the n-th capacitor, u 20 = (L 1 C0 )−1 and ω02 = (L 2 C0 )−1 are the characteristic frequencies of the network, and λ = C S /C0 is the dimensionless dispersive parameter. In the following, we focus our attention on nonlinear modulated waves centered at (ω, k), where ω = ω p = 2π f p and k = k p are respectively the angular frequency and the wavenumber. We assume that the perturbation voltage |Vn /Vb |  1 and apply the reductive perturbation method in the semi-discrete limit [26]. We then introduce one slow space and one slow time variables, x = ε(n − υg t) and τ = ε2 t, where 0 < ε  1 is a small ordering parameter, υg is the group velocity defined as  2  u 0 − λω02 sin k dω =  υg = 2 , dk ω 1 + 4λ sin2 (k/2)

(4.3)

4.1 Emission of Rogue Wave Signals Through the Modified …

111

and the angular frequency ω is related to the wavenumber k through the following linear dispersion relation     ω2 1 + 4λ sin2 (k/2) − ω02 + 4u 20 sin2 (k/2) = 0.

(4.4)

Following Marquié et al. [27], we seek the solution of system (4.2) in the general form [27, 28]   Vn (t) = εψ(x, τ ) exp (iθ ) + ε2 ψ10 (x, τ ) + ψ20 (x, τ ) exp (2iθ ) + c.c., (4.5) where θ = (kn − ωt) is the rapidly varying phase, ψ(x, τ ), ψ10 (x, τ ), and ψ20 (x, τ ) are respectively the fundamental, dc and second-harmonic terms. During the numerical computations, the wavenumber k will be taken from the first Brilloum zone (0 ≤ k ≤ π ). In the following, we assume the group velocity (4.3) to be nonnegative in the first Brilloum zone, which is possible only when the dispersive parameter λ satisfies the following restriction [29]: 0≤λ
0. The phenomenon of the MI for network system of Fig. 2.1 is investigated by perturbing the plane wave solution (4.12) as follows, ⎡ ⎤  2 T x + σx 1 ψ0 + δψ ⎦, (y)dy + i exp ⎣−i  ψ(x, τ ) = √ 4P τ + τ0 0 (τ + τ0 )

(4.13)

T0

where δψ is a small perturbation. Using the linear stability analysis and denoting by respectively K and  the modulation wavenumber and the complex modulation angular, we arrive to the following time-dependent dispersion relation:

2  2Q ψ02 2 σ K − K 2 40 K 2 − (T ) − 0 (τ + τ0 ) = 0. P 2 P 0

(4.14)

The modulational instability may set in provided that τ0 < 0 and Im[] = 0, whence the following necessary condition for the MI

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113

Q 2 τ + τ0 ψ < 0. P 0 0

(4.15)

K2 − 2

Since 0 (τ + τ0 ) > 0, for condition (4.15) to be satisfied, it is necessary that P Q > 0 : this MI criterion of the plane wave solution (4.12) coincides with the MI criterion P Q > 0 for plane-wave solutions of the standard NLS equation. The corresponding local MI growth rate (gain) reads  |Im ()| = 20 K 2 2

Q 2 τ + τ0 ψ − K 2. P 0 0

(4.16)

It is important to note that the CW solution (4.12) will be stable under modulation as soon as P Q < 0. Although parameter χ (τ ) of the linear potential does not affect the MI criterion, it affects implicitly the local MI growth rate. Indeed, the dependence of the growth rate (4.16) on “time” τ is a consequence of the presence of χ (τ ) in the amplitude equation (4.8) of the network. Employing the network parameters [30] L 1 = 220 µH,L 2 = 470 µH,C0 = 370 pF,C S = 56 pF, α = 0.21V−1 , β = 0.0197V−2 ,

(4.17)

we display, respectively, in Fig. 4.1a–c linear dispersion curves showing the propagating frequency f = ω/ (2π ), the dispersion coefficient P, the nonlinearity coefficient Q, and the product P Q versus the wavenumber k. Figure 4.1a shows four different regions concerning the MI and possible soliton solutions (regions of positive P Q correspond to envelope solitons, while region of negative P Q lead to hole solitons): • • • •

Region (1): Region (2): Region (3): Region (4):

P P P P

> 0 and > 0 and < 0 and < 0 and

Q Q Q Q

< 0 (here, > 0 (here, > 0 (here, < 0 (here,

PQ PQ PQ PQ

< 0); > 0); < 0); > 0).

4.1.3 Construction of Rogue Waves in the Lossless Network Under the Condition P Q > 0 We now turn to the construction of RWs in the lossless network of Fig. 2.1. For this purpose, we first need to find rogue wave solutions of the amplitude equation (4.8).

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Fig. 4.1 a The linear-dispersion curve showing frequency f = ω/(2π ) as a function of wavenumber k. The allowed band [ f 0 , f c ] is divided in four regions concerning the instability of the system, depending on the sign of P Q. b Plots of the dispersion and nonlinearity coefficients, in the form of 0.2P × 10−6 and Q × 10−5 , versus wavenumber k. c Product P Q as a function of k. The network parameters used in plots (a)–(c) are given in the text. Reprint from Ref. [25], Copyright 2022, with permission from American Physical Society

4.1.3.1

First-Order and Second-Order RW Solutions of Eq. (4.8)

To build the first- and the second-order RW solutions of Eq. (4.8), we use the following seed solutions   (4.18) ψ(x, τ ) = ± exp i (β0 x + γ0 + Qτ ) , where β0 and γ0 satisfy the differential system dγ0 dβ0 − χ = 0, + Pβ02 = 0. dτ dτ

(4.19)

Using the seed solution (4.18) and applying the direct method, we seek for the firstorder and second-order RW solutions of Eq. (4.8) in the form

  F(ξ, τ ) exp i (β0 x + γ0 + Qτ ) , ψ1,2 (x, τ ) = ±1 + G(ξ, τ )

(4.20)

where F(ξ, τ ) is a complex polynomial and G(ξ, τ ) is a real polynomial satisfying the conditions deg F < deg G and G(ξ, τ ) = 0 for all ξ and τ , and ψ1 and ψ2 stand for respectively the first- and second-order rational polynomial solution of Eq. (4.8). For the first-order and the second-order RWs, we ask that deg G = 2 and deg G = 6, respectively. Under these assumptions, we insert Eq. (4.20) into Eq. (4.8) and arrive to √

  4 + 8i Q/2Pξ exp i (β0 x + γ0 + Qτ ) ,(4.21) ψ1 (ξ, τ ) = −1 + 2 2 2 1 + 4Q τ + 2(Q/P)ξ

  g2 (ξ, τ ) + i h 2 (ξ, τ ) exp i (β0 x + γ0 + Qτ ) , (4.22) ψ2 (ξ, τ ) = 1 + d2 (ξ, τ )

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115

where  Q 2 3 , ξ + Q2τ 2 + 2P 4 2  15 3Q 2 Q 2 2 2 2 2 h 2 (ξ, τ ) = Qτ − + Q τ − , ξ +2 ξ +Q τ 8 2P 2P     1 3 6Q 2 Q 2 2 2 2 2 2 1 + 44Q τ + ξ + 3Q τ − ξ d2 (ξ, τ ) = 64 P 4 2P   1 Q 2 3 ξ + , Q2τ 2 + 3 2P d X1 + 2Pβ0 = 0. ξ = x + X 1 (τ ), dτ 3 g2 (ξ, τ ) = − + 4 

4.1.3.2



3 Q 2 ξ + 5Q 2 τ 2 + 2P 4



(4.23a) (4.23b)

(4.23c) (4.23d)

Propagation of RWs Through the Lossless Network

In the following, we employ the above first- and second-order rational polynomial solutions of the amplitude equation (4.8) to investigate the dynamics of firstand second-order modulated rogue waves propagating along the network system of Fig. 2.1. Such investigation is carried out via typical physical example, based on the form of the magnitude χ (τ ) of the linear potential term of the amplitude equation (4.8). The case of constant χ (τ ) (time-independent linear density profile) As the first example, we consider the time-independent linear density profile and set χ (τ ) = χ = constant. Solving then Eqs. (4.19) and (4.23d) yields β0 = χ τ + β00 , γ0 (τ ) = −(3χ )−1 P (χ τ + β00 )3 + γ00 , and ξ(τ ) = x − (P/χ ) (χ τ + β00 )2 + X 10 if χ = 0, and β0 = γ0 = 0 and ξ = x, if χ = 0, where β00 , γ00 , and X 10 are three free real constants. For the numerical value k = 0.75 rad/cell of the wavenumber and for the network parameters (4.17), we display in Fig. 4.2 the typical structure of the first-order (left panels) and second-order (right panels) modulated rogue waves obtained with respectively the rational polynomial solutions (4.21) and (4.22). As we can see from plots of Fig. 4.2, the first-order [second-order] modulated rogue wave is formed of one hump [four humps] located around the centre (0, 0).

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4 Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks

Fig. 4.2 RWs (rogue waves) |ψ(x, τ )| in the network of Fig. 2.1 for a time-independent lineardensity profile with χ(τ ) = 1. The structure of the a first-order and b second-order RWs is provided by Eqs. (4.21) and (4.22), respectively. Contour plot of the c first-order and d second-order RWs, plotted as per Eqs. (4.21) and (4.22). respectively. Different plots are generated with wavenumber k = 0.75 rad/cell and network parameters (4.17). Solution parameters used in these plots are β00 = X 10 = 0. Reprint from Ref. [25], Copyright 2022, with permission from American Physical Society

The case of the temporal periodic modulation of the linear density profile Our second example corresponds to a temporal periodic modulation of the linear potential whose magnitude is taken as   χ (τ ) = (2P)−1 2 + 452 cos (τ ) , with the constant frequency . From Eqs. (4.19) and (4.23d), we obtain for this example that β0 (τ ) = (2P)−1 (2τ + 45 sin (τ ) + β00 ) , ξ(τ ) = x − τ 2 − 2β00 τ + 45 cos (τ ) + X 10 , where β00 and X 10 are two constants of integration. A typical example of the evolution of the first-order (left panels) and secondorder (right panels) modulated rogue waves, obtained with the use of the above found rational polynomial solution (4.21) and (4.22), and propagating through the network of Fig. 2.1. is depicted in Fig. 4.3 for the network parameters (4.17). Different

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117

plots of Fig. 4.3 show RWs moving along an oscillating trajectory. Also, plots of Fig. 4.3 reveal that the first-order [second-order] RW features only one extremum [two extrema]. The case of an exponentially decaying/growing linear density profile Our last example is obtained when the strength of the linear-potential undergoes a temporal modulation, that is, χ (τ ) = χ0 exp (χ1 τ ), χ0 and χ1 being nonzero constants. In this special case, we obtain from Eqs. (4.19) and (4.23d) that β0 (τ ) = (χ0 /χ1 ) exp (Pτ ) + β00 ,    ξ(τ ) = x − 2P χ0 /χ12 exp (χ1 τ ) + β00 τ + X 10 , where β00 and X 10 are two constants of integration. For the present example, we show respectively in Figs. 4.4 and 4.5 the dynamics of the first-order and the second-order modulated RWs generated with the rational polynomial solutions (4.21) and (4.22) and showing the evolution of modulated rogue waves along the network system of Fig. 2.1. As we can see from plots of Fig. 4.4,

Fig. 4.3 The dynamics of the first-order and second-order RWs, |ψ(x, τ )|, of the network of Fig. 2.1 for a temporal periodic modulation of the linear density profile, χ(τ ) =  (2P)−1 2 + 452 cos (τ ) with  = 104 for wavenumber k = 0.75 rad/cell and network parameters (4.17). The evolution plot are shown for a the first-order |ψ1 (x, τ )| and b the second-order |ψ2 (x, τ )| RWs produced by Eqs. (4.21) and (4.22), respectively. Contour plot are shown for the c first-order and d second-order RWs described by Eqs. (4.21) and (4.22) respectively. The solution parameters used in these plots are β00 = 0 and X 10 = −100. Reprint from Ref. [25], Copyright 2022, with permission from American Physical Society

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4 Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks

Fig. 4.4 The dynamics of the first-order RW |ψ(x, τ )| of the network of Fig. 2.1 described by Eq. (4.21) for an exponentially decaying or growing linear density profile, χ(τ ) = χ0 exp (χ1 τ ) with k = 0.75 rad/cell and the network parameters (4.17). a χ0 = 1, χ1 = −P < 0; b χ0 = 1, χ1 = P > 0; c χ0 = −1, χ1 = −P < 0; d χ0 = −1, χ1 = P > 0. The solution parameters used in these plots are β00 = 0 and X 10 = 0. Reprint from Ref. [25], Copyright 2022, with permission from American Physical Society

the first-order rogue wave is reduced to a dark-bright-dark soliton. Plots of Fig. 4.5 show a second-order RW with four peaks. Plots of Figs. 4.4 and 4.5 also show how much the solution parameters χ0 and χ1 affect the evolution of the modulated RWs propagating through the network of Fig. 2.1. Although these two parameters have no effect on the wave shape, they affect the wave behavior. Plots of Figs. 4.4 and 4.5 reveal that modulated RWs of nonlinear electric transmission networks may either take birth from dark [bright] solitonlike waves and “disappear without a trace”, or “appear from nowhere” and disappear as dark [bright] solitonlike waves [4].

4.1.4 Emission of Rogue Wave Signals Through the Network of Fig. 2.1 Based on the exact rational polynomial solutions (4.21) and (4.22) of the amplitude equation (4.8), we turn to the emission of RW signals through the network of Fig. 2.1. For this purpose, it is convenient to go back to the original variables n and t and use expansion (4.5):

4.1 Emission of Rogue Wave Signals Through the Modified …

119

Fig. 4.5 The dynamics of the second-order RE |ψ(x, τ )| of the network of Fig. 2.1 described by Eq. (4.22) for an exponentially decaying or growing linear density profile, χ(τ ) = χ0 exp (χ1 τ ) with k = 0.75 rad/cell and the network parameters (4.17). a χ0 = 1, χ1 = −P < 0; b χ0 = 1, χ1 = P > 0; c χ0 = −1, χ1 = −P < 0; d χ0 = −1, χ1 = P > 0. The solution parameters used in these plots are β00 = 0 and X 10 = 0. Reprint from Ref. [25], Copyright 2022, with permission from American Physical Society

  Vn (t) = 2εRe ψ(x, τ ) exp [i (kn − ωt)] + ε {ψ10 (x, τ ) + ψ20 (x, τ ) exp [2i (kn − ωt)]} ,

(4.24) where x = ε(n − υg t), τ = ε2 t, and ψ(x, τ ) = ψ1,2 (ξ(x, τ ), τ ). For the simplicity, we focus ourselves to the case when the linear potential is absent in Eq. (4.8), that is, χ (t) = 0; in this case, we take β0 = 0, γ0 = 0, and X 1 = 0 so that ξ = x. In the following, we aim to address effects of various parameters such as the wavenumber k, the linear dispersive element C S , the linear capacitance C0 , as well as the linear inductances L 1 and L 2 on the evolution of first- and second-order modulated RWs propagating along our network system. For the computational purpose, we use the numerical value of ε = 10−3 .

4.1.4.1

Effects of Network and Equation Parameters on the Waveform

To study the effect of the wavenumber k on the modulated RWs propagating along our network system, we show, for different values of the wavenumber k, the waveform at time t = 0 of the first- and second-order RWs in Fig. 4.6. It is seen from different plots of Fig. 4.6 that the amplitudes and the widths of both the RW and its background decreases with the increase in the values of the wavenumber k. Since the propagating

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4 Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks

Fig. 4.6 The waveforms of the first-order (a–f) and the second-order (g–l) RW voltage with different values of the wavenumber k for C S = 0.9999 (L 2 /L 1 ) C0 and other network parameters given by Eq. (4.17). a, g k = 0.35 rad/cell; b, h k = 0.365 rad/ cell; c, i k = 0.38 rad/cell; d, j k = 2.85 rad/cell; e, k k = 2.95 rad/cell; f, l k = 3.10 rad/cell. Reprint from Ref. [25], Copyright 2022, with permission from American Physical Society

frequency f p is an increasing function of the wavenumber k, we conclude that the wave amplitude and the wave width decrease as the propagating frequency increases. To observe how much the linear dispersive parameter C S and the capacitance scale C0 affect the motion of modulated RWs during their propagation along our network system, we have displayed in respectively Figs. 4.7 and 4.8 the waveforms of the first-order (top panels) and the second-order (bottom panels) RW voltages, for different numerical values of C S and C0 . Plots of Fig. 4.7 (Fig. 4.8) show that the wave amplitude and the wave width decreases (increases) with the increase in the values of parameter C S (C0 ). This means that linear capacitances C S and C0 have opposite effects on the wave amplitude and width. Lastly, we investigate the effects of the linear inductances L 1 and L 2 on the modulated RWs moving along the network of Fig. 2.1. For this aim, we have, for different values of L 2 and L 1 , displayed in respectively Figs. 4.9 and 4.10 the waveforms of the first-order (top panels) and second-order (bottom panels) modulated RWs propagating along our network system. Plots of Fig. 4.9 (Fig. 4.10) reveal that any increase in the values of L 2 (of L 1 ) leads to an increase (decrease) in the wave amplitude. Observing different plots of Figs. 4.9 and 4.10, it is clearly seen that the linear inductances L 1 and L 2 have no effect on the wave amplitude. Therefore, L 1 and L 2 have opposite effects on the wave width.

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121

Fig. 4.7 The waveforms of the first-order (top panels) and second-order (bottom panels) RW voltage for different values of the dispersive element C S for the network parameters (4.17) with k = 0.365 rad/cell. a, d C S = 0.98 (L 2 /L 1 ) C0 , b, e C S = 0.99 (L 2 /L 1 ) C0 , c, f: C S = 0.999(L 2 /L 1 )C0 . Reprint from Ref. [25], Copyright 2022, with permission from American Physical Society

Fig. 4.8 The waveforms of the first-order (top panels) and second-order (bottom panels) RW voltage with different values of the linear capacitance C0 for the network parameters (4.17) with C S = 0.999 (L 2 /L 1 ) C0 and k = 0.365 rad/cell. a, d C0 = 370 pF, b, e C0 = 373.7 pF, c, f C0 = 377.4 pF. Reprint from Ref. [25], Copyright 2022, with permission from American Physical Society

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4 Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks

Fig. 4.9 The waveforms of the first-order (top panels) and second-order (bottom panels) RW voltage with different values of the linear inductance L 2 for the network parameters (4.17) with C S = 0.98 (L 2 /L 1 ) C0 and k = 3.1 rad/cell. a, d L 2 = 470 mH; b, e L 2 = 517 mH; c, f L 2 = 564 mH. Reprint from Ref. [25], Copyright 2022, with permission from American Physical Society

Fig. 4.10 The waveforms of the first-order (top panels) and second-order (bottom panels) RW voltage with different values of the inductance of linear element L 1 , for network parameters (4.17) with C S = 0.98 (L 2 /L 1 ) C0 and k = 3.1 rad/cell. a, d L 1 = 176 mH; b, e L 1 = 198 mH; c, f L 1 = 220 mH. Reprint from Ref. [25], Copyright 2022, with permission from American Physical Society

4.1.4.2

Analytical Emission of RWs Voltages Via Exact RW Solutions (4.21) and (4.22)

Using now the exact rational polynomial solutions (4.21) and (4.22) of the amplitude equation (4.8), we can experience the emission electrical modulated RWs in the network of Fig. 2.1. The typical spatial and temporal evolution of the first- and second-order modulated RWs are showed in respectively Figs. 4.11 and 4.12. Plots of Fig. 4.11 (Fig. 4.12) show the spatial [temporal] evolution of the first-order (top panels) and second-order (bottom panels) modulated RWs propagating at frequency f p = 382.074 kHz along the network system of Fig. 2.1. Different plots reveal that

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123

Fig. 4.11 Voltage Vn as a function of the cell number n, at given times (t0 , t1 , t2 , and t3 ), showing the propagation of the first-order (top panels) and second-order (bottom panels) RWs in the network at frequency f p = 930.524 kHz for parameters (4.17). a, e t0 = 0; b, f t1 = 0.033; c, g t2 = 0.066; d, h t3 = 0.1. Reprint from Ref. [25], Copyright 2022, with permission from American Physical Society

Fig. 4.12 The propagation of the first-order (top panels) and second-order (bottom panels) RWs in the network at frequency f p = 930.524 kHz for parameters (4.17). a, e n = 0; b, f n = 1000; c, g n = 2000; d, h n = 3000. Reprint from Ref. [25], Copyright 2022, with permission from American Physical Society

second-order [first-order] modulated RWs can be emitted without essential distortions [are emitted with distortions].

4.1.4.3

Effect of the Carrier Phase on the RW Shape

Although the carrier phase ϕ(x, τ ) = β0 (τ )x + γ0 (τ ) + Qτ has no affect on the wave amplitude, it can affect the shape of the RW peaks and holes. For a better understanding, we consider a carrier phase with a nonzero parameter β0 (t). Integrating Eq. (4.19) yields τ β0 (τ ) = χ (s)ds + β0 (0). 0

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4 Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks

Therefore, studying the effect of β0 (τ ) on the motion of the modulated RWs through the network of Fig. 2.1 is tantamount to studying the effect of parameter χ (t) on the RWs. On the basis of Figs. 4.6, 4.7, 4.8, 4.9, 4.10, 4.11 and 4.12, which pertain to different values of k, can be interpreted as the effect of the carrier phase on the modulated RWs propagating along our the network system. To investigate the effect of the carrier phase originating from parameter χ (t) only, one may vary χ , keeping all the network parameters and wavenumber k constant.

4.1.5 Conclusion and Discussions We have considered in this Section the lossless one-dimensional modified Noguchi nonlinear NLTN with dispersive elements, displayed in Fig. 2.1. The dynamics of nonlinear modulated wave of this network are reduced to a NLS equation with an external linear potential. The criterion of the MI is presented and exact first- and second-order rational polynomial solutions of the amplitude equation are built. Using these solutions, we have, for various physical linear potential, studied the dynamics of modulated RWs propagating through our network system. The effects of the network parameters as well as those of the solution parameters and the carrier wavenumber are investigated too. We found that the linear capacitances C S and C0 and linear inductances L 1 and L 2 have opposite effects on the wave amplitude/width. More interestingly, we found that suitably choosing the linear capacitances C S and C0 and the propagating frequency f p may lead to either the wave amplification or the wave suppression. We have also found that the propagating frequency f p , the linear dispersive element C S , the linear inductance L 1 , the linear capacitance C0 , and the linear inductance L 2 can be chosen adequately for the wave compression. Our studies also revealed that modulated RWs in electric nonlinear transmission network can either appear from nowhere and disappear without a trace, or take birth from dark/bright solitonlike waves and disappear without a trace, or appear from nowhere and disappear as a dark/bright solitonlike waves. The results found in the present Section may be useful to realize the RW dynamics in various physical systems, in addition to NLTNs, such as nonlinear fibers, BECs, plasmas, etc.

4.2 Generation of Network Modulated Rogue Waves Under the Action of the Quintic Nonlinearity The present Section deals with the transmission of non-autonomous rogue waves along the network system with dispersive elements illustrated in Fig. 2.1. Here, we use the same capacitance-voltage dependence (4.1) as in the previous Section. We show that the amplitude equation is reduced to a cubic-quintic NLS equation with cubic derivative terms. One-parameter first-order rational polynomial solutions of

4.2 Generation of Network Modulated Rogue Waves Under …

125

this equation are presented and used to investigate analytically the generation of firstorder non-autonomous modulated RWs propagating in the network system of Fig. 2.1. We study the effects of the quintic nonlinearity and nonlinear-dispersion parameters on the first-order non-autonomous modulated RWs during their transmission through our network system.

4.2.1 The Kundu–Eckhaus Model for Non-autonomous Modulated Rogue Waves in a Lossless Electric Network Applying the Kirchhoff’s laws to the network in Fig. 2.1 yields the system of nonlinear discrete equations (4.2). Asking to the linear waves propagating along the network of Fig. 2.1 to be proportional to the expression exp [i (kn − ωt)] yields the linear dispersion relation (4.4) of a typical passband filter. In the following, we use the following network parameter for the computational purpose [22, 25]: L 1 = 28µH, L 2 = 14µH, C0 = 540 pF,C S ∈ [0, C0 /2[pF,α ≤ 0.21 V−1 , β ≤ 0.0197 V−2 , V0 = 1.5 V.

(4.25)

In order to carry out efficient investigation of the dynamics of non-autonomous modulated RWs propagating along the network of Fig. 2.1, it is reasonable to first derive the amplitude equation. To accomplish this, we employ the reductive perturbation method in the semidiscrete approximation, considering the interaction of any wave packet centered at (ω, k), where ω = 2π f is the carrier frequency and k is the carrier wavenumber. Following Marquie et al. [25], we introduce the following slow variables (4.26) ξ = ε(n − υg t) and τ = ε2 t, where ε  1 is a small parameter, υg =

  2 u − λω02 sin k dω =  0 2 dk ω 1 + 4λ sin2 (k/2)

(4.27)

is the group velocity, and the frequency ω is related to the wavenumber k via the following linear dispersion relation ω2 =

  ω02 + 4u 20 sin2 k2   . 1 + 4λ sin2 2k

(4.28)

Next, we seek for the solution of the difference-differential system (4.2) in the general form [25]

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4 Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks

  Vn (t) = εψ(ξ, τ ) exp [iθ − i(ξ, τ )] + ε2 V02 (ξ, τ ) + V12 (ξ, τ ) exp (2iθ ) + c.c., (4.29) where c.c. stands for the complex conjugation, θ = kn − ωt is the rapidly varying phase, (ξ, τ ) is an additional imprinted phase, and ψ, V02 , and V12 are respectively the fundamental, dc, and second harmonics. Under all the above assumptions, we find that V02 (ξ, τ ) =

2α 2 ωυg2   |ψ(ξ, τ )|2 , α υg2 − u 20

(4.30a)

 2 4αω2  2 ψ(ξ, τ ) exp [−i(ξ, τ )] ,  2 2 2 2 4ω − ω0 + 4 4λω − u 0 sin k (4.30b) ∂ = β0 |ψ(ξ, τ )|2 , (4.30c) ∂ξ   ∂ψ ∂ψ ∗ ∂ = −iβ0 P ψ − ψ∗ + 2β02 P |ψ(ξ, τ )|4 (4.30d) ∂τ ∂ξ ∂ξ   ∂ 2ψ ∂ψ ∂ |ψ|2 + P 2 + Q |ψ|2 ψ + β02 P |ψ|4 ψ − 2iβ0 P ψ = 0, (4.30e) i ∂τ ∂ξ ∂ξ V12 (ξ, τ ) =

where β0 is an arbitrary real constant, P and Q are real parameters, function of the wavenumber k, defined as    2

 u0 λ 2 k 1 + 4λ sin + − ω cos k − 2λυg sin k, (4.31) P=− 2ω 2 2ω 2 2 2 2 3 2α ωυ 3β 4α ω g   ω+ 2 . (4.32) Q= − 2 u 0 − υg2 4ω2 − ω02 + 4 4λω2 − u 20 sin2 k υg2

 = Q/P, and ψ (ξ,  Letting  τ = Pτ, Q τ ) = ψ(ξ, τ ), Eq. (4.30e) takes the simpler form,   ∂ 2ψ  2  4 ∂ψ  ψ  ψ  ψ  + β02 ψ  − 2iβ0 + +Q i ∂ τ ∂ξ 2

  2   ∂ ψ =0 ψ ∂ξ

(4.33)

 and β0 . and contains only two real parameters, Q The cubic-quintic derivative NSL equation (4.30e), or its simpler form (4.33) is known as an equation of Kundu–Eckhaus (KE) [32–37] type, and, when β0 = 0, coincides with the standard cubic NLS equation. It governs the dynamics of modulated waves in the network of Fig. 2.1. In Eq. (4.30e), parameter β02 = 0 is the relative magnitude of the quintic nonlinearity, while the last term represents the nonlinear dispersion, with relative magnitude β0 . Using now the phase-engineering transformation (PIT) [38]

4.2 Generation of Network Modulated Rogue Waves Under …

ψ = q exp (iϕ) ,

127

4Pβ0 + 2 2 ∂ϕ |q| , = ∂ξ 3P

∂q 1 + 2Pβ0 = −i ∂τ 3

  ∂q ∗ 4 (1 + 2Pβ0 )2 4 ∂q |q| + Q |q|2 q − − q∗ ∂ξ ∂ξ 9P

and results from [39], it is seen that the KE equation (4.30e) conserves the integral power, +∞ |ψ(ξ, τ )|2 dξ. P= −∞

The amplitude equation (4.30e) admits the CW solution     ψ0 (ξ, τ ) = q0 exp ik0 ξ − i k02 P − Qq02 − β02 Pq04 τ with the real constant amplitude q0 and wavenumber k0 (of the carrier wave). Carrying out the linear stability analysis and denoting by K and  the wavenumber and complex angular frequency of modulation, we arrive to the following MI growth rate   02 − K 2 , |Im ()| = |P K | 2 Qq (4.34)  = Q/P > 0. Therefore, the uniform wave propagating along the lossless netif Q work of Fig. 2.1 is stable against the modulation if P Q < 0, and unstable under modulation if P Q > 0.

4.2.2 Generation of First-Order Non-autonomous Modulated Rogue Waves (Alias Peregrine Solitons) for a Lossless Electric Network  = Q/P > 0, exact first-order ratioHere, we first present, under the condition Q nal polynomial solutions of the amplitude equation (4.30e) which are then used for investigating the generation and management of the first-order non-autonomous modulated RWs (alias Peregrine solitons [40, 41]) in the lossless electric network of Fig. 2.1.

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4 Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks

4.2.2.1

One-Parameter First-Order Rational Polynomial Solutions of the Amplitude Equation

To obtain exact first-order rational polynomial solutions of the amplitude equation (4.30e), we introduce the ansatz ψ(ξ, τ ) = r0 u(X, T ) exp [i (ξ − Pτ )] , X = r0 (ξ − 2Pτ ) , T = r02 τ,

(4.35)

under which the KE equation (4.30e) take a similar form, i

∂ 2u ∂u ∂ |u|2 2 2 2 4 |u| |u| +P = 0, + Q u + r β P u − 2ir β Pu 0 0 0 0 ∂T ∂ X2 ∂X

(4.36)

which admits the special CW solution     u 1 (X, T ; q1 , k1 ) = q1 exp ik1 X − i k12 P − Qq12 − r02 β02 Pq14 T ; here, r0 = 0 is an arbitrary real constant, referred to as the “solution parameter”, and q1 and k1 are real constants. As in the case of the KE equation (4.30e), equation (4.36) conserves the integral power: +∞ +∞ 2 −1 |u(X, T )| d X = |r0 | |ψ(ξ, τ )|2 dξ. P= −∞

−∞

Using the CW solution u 1 (X, T ; 1, 0) as the seed solution and following the method elaborated in Refs. [41, 42], we arrive to the following one parameter first-order rational polynomial solution (with parameter r0 = 0) of the amplitude equation (4.30e): ⎡

⎤

⎢ ⎢ ψ(ξ, τ ) = r0 ⎢−1 + ⎣





⎥ 4 ⎥  2 ⎥  ⎦  −1/2 3  Q 4 2 2  1 + 4r0 Q τ + 4 1 + r0 2 (ξ − 2Pτ ) + 2Q 2 Q r 0 β0 τ

  × exp i ξ + r02 P

 − Q

1 − r04 β02 r02

1 + 2i Qr02 τ

 τ

(4.37)

⎫⎤ ⎪ ⎪    ⎪ ⎬⎥  −1/2 Qτ 1 + r0 Q2 (ξ − 2Pτ ) + 2r03 β0 2 Q 8r0 β0 ⎥ +  2 ⎪⎥ .  ⎦   −1/2 2Q ⎪  Q 4 3 ⎪ 2 2  1 + 4r0 Q τ + 4 1 + r0 2 (ξ − 2Pτ ) + 2Qr0 β0 2 Q τ ⎭ 

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Next, we employ the first-order rational polynomial solution (4.37) to evaluate the wave phase: 

 1 − r04 β02 ϕ N (X, T ) Q− P τ+ ϕ(ξ, τ ) = ξ 2 ϕ D (X, T ) r0  2 (G − R) + arccos  , 4 (G − R)2 + Q 2 T 2 H 2 + r02

(4.38)

with X = r0 (ξ − 2Pτ ) , T = r02 τ. 4.2.2.2

Results and Discussions

Since lim

ψ(ξ, τ ) = r0 ,

|ξ |2 +|τ |2 →∞

we conclude that the one-parameter Peregrine soliton solution (4.37) of the KE equation (4.30e) has the form of a single-peak structure that merges into the plane wave at |ξ | + |τ | → +∞. It is evident that various parameters such as the quintic nonlinearity and nonlinear-dispersion parameter β0 , solution parameter r0 , the carrier wavenumber k, coefficients P and Q of the amplitude equation (4.30e), as well as on the network parameters impact the transmission of first-order modulated wave through the network system of Fig. 2.1. These parameters determine the wave characteristics. As well as we know, the ideal initial perturbation that corresponds to the exact Peregrine soliton solution of the amplitude equation will evolve into a modulated RWs [43, 44]. It follows from solution (4.37) that the maximal value of the wave intensity is reached at τ = 0 : |ψ(ξ, τ )|max = |ψ(ξ, 0)|max = 3 |r0 | . This means that the wave amplitude and that of its CW background increase with the increase as |r0 | increases; moreover, |ψ(ξ, τ )|max is independent of the network system’s parameters. Effects of parameter r0 and β0 on the Peregrine solitons propagating through the lossless network Now, we use the exact rational polynomial solution (4.37) to study the effects of parameters r0 and β0 on the first-order modulated rogue waves propagating along the network system of Fig. 2.1, we have displayed in respectively Figs. 4.13 and 4.14 intensity |ψ| for different values of respectively r0 and β0 . Smoothly increasing the numerical values of r0 from an optimal value, we can visualize, as shown plots of Fig. 4.13, the formation of crests and troughs with an increase in the amplitude wave, large amplitude wave corresponding to large values of r0 . Plots of Fig. 4.14 reveal that the sign and the magnitude of parameter β0 as well as the wavenumber k affect the wave speed and have no effect on the wave amplitude. Plots

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4 Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks

Fig. 4.13 Plots of amplitude |ψ(ξ, τ )| produced by rational solution (4.37), showing the formation of the first-order non-autonomous RW in the lossless nonlinear electric network of Fig. 2.1 with parameters (4.25) for α = 0.21 V−1 and β = 0.0197 V−2 . The parameters are λ = C S /C0 = 0.3, k = 0.9, β0 = 0.2 and a r0 = 0.3; b r0 = 0.5; c r0 = 0.7; and d r0 = 1.2. Reprint from Ref. [22], Copyright 2022, with permission from American Physical Society

of Fig. 4.14 also reveal that the carrier wavenumber k has no effect on the wave shape during the wave propagation through our network system; however, the wave shape may be affected by the variation of parameter β0 . This fact is clearly seen from plots of Figs. 4.15 and 4.16 showing the wave evolution for different values of parameter β0 . When β0 is taken around some critical value β0c , the shape of the RW remains unchange during the wave propagation, as one can see from plots of Fig. 4.15. It is seen from plots of Fig. 4.16 that when parameter β0 increases from some critical value β0c of parameter β0 (the same results are obtained for β0 < β0c ), Peregrine solitons get more delocalized in space. Therefore, parameter β0 can be used to control the Peregrine solitons of the network of Fig. 2.1. Effects of the dispersive element C S on the Peregrine solitons To show the effects of the network parameter C S on the network Peregrine solitons, we depict in Figs. 4.17 and 4.18 the evolution of Peregrine solitons for different values of the dimensionless dispersive parameter λ = C S /C0 and respectively k = 0.3 and k = 0.9. We can clearly see from plots of Figs. 4.17 and 4.18 how much the dispersive element C S through parameter λ affects both the wave’s speed and the localization of its main peak. Also, plots of Figs. 4.17 and 4.18 show that, depending on the value of the carrier wavenumber k, the wave speed increases [decreases] and the Peregrine

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131

Fig. 4.14 Density plots of the first-order non-autonomous RWs associated with rational solution (4.37) for r0 = 1.2 and λ = C S /C0 = 0.3. Relative magnitude β0 of the nonlinear dispersion takes the following values: a, d β0 = −0.3; b, e β0 = 0.7; c, f β0 = 1.2. Plots (a)–(c) are generated with wavenumber k = 0.9, while plots (d)–(f) are obtained with k = 0.8. Other parameters are shown in the text. It is seen from these plots that the density distribution is not affected by the variation of β0 , which, however, affects the RW’s velocity. Reprint from Ref. [22], Copyright 2022, with permission from American Physical Society

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4 Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks

Fig. 4.15 The dynamics of the first-order non-autonomous RW |ψ(ξ, τ )|, associated with rational solution (4.37), with the same parameters as in Fig. 4.14a–c. Values of parameter β0 are a β0 = −0.3; b β0 = 0.7; c β0 = 1.2. The RWs shown in plots (a)–(c) have the same maximum value of the main peak, |ψ(ξ, τ )|max = 3r0 . Reprint from Ref. [22], Copyright 2022, with permission from American Physical Society

soliton gets more [less] localized with the increase in the values of λ as shown in Fig. 4.17 [Fig. 4.18]. For better understanding, we display in Fig. 4.19 the amplitude profile |ψ(ξ, τ )| at time τ = 0 for two values of the carrier wavenumber and different values of parameter λ. Effects of nonlinear parameters α and β and wavenumber k on the Peregrine solitons For different values of α, β, and k, we depict respectively in Figs. 4.20, 4.21, and 4.22 the magnitude of amplitude profile |ψ(ξ, 0)| at time τ = 0 for different values of respectively α, β, and k. It follows from plots of Figs. 4.20, 4.21, and 4.22 that the wave speed decreases [increases] and the Peregrine soliton gets less√[more] localized with the increase of each α, β and k taken from the region where P/2Q is an increasing [a decreasing] function of the corresponding α, β and k. Comparing plots of Fig. 4.19 to those of Figs. 4.20, 4.21, and 4.22, we conclude that parameter λ and parameters α, β, and k produce opposite effects, while parameters α, β, and k have the same effect on the Peregrine solitons. Therefore, these four parameters are useful for controlling the motion of Peregrine solitons along the electrical network of Fig. 2.1.

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133

Fig. 4.16 Plots of evolution (top) and density (bottom) of the first-order non-autonomous RWs |ψ(ξ, τ )| produced by rational solution (4.37) for network parameters (4.25). The constants are λ = C S /C0 = 0.3 and r0 = 1.2. Parameter β0 of the nonlinear dispersion takes values a, d β0 = 0.7; b, e β0 = 3; c, f β0 = 9. Other parameters are given in the text. Reprint from Ref. [22], Copyright 2022, with permission from American Physical Society

Effects parameters β0 , λ, α, and β on the phase profile of the Peregrine solitons Although the wave phase does not affect the wave amplitude, the steepness of the first-order modulated RWs is strongly dependent on the wave phase profile [45]. For better understanding, the phase evolution of the Peregrine soliton is displayed in Fig. 4.23 for different values of β0 and in Fig. 4.24 for different values of λ (Fig. 4.24 a–c), α (Fig. 4.24d–f) and β (Fig. 4.24g–i). We can see from Fig. 4.23 that the symmetric shape of the Peregrine soliton is broken by parameter β0 and the phase distributions are almost opposite for β0 and −β0 . Plots of Fig. 4.24 reveal that the symmetry of the Peregrine soliton is maintained under the variation of parameter λ,

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4 Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks

Fig. 4.17 The density (top) and evolution (bottom) plots of the first-order non-autonomous RWs corresponding to solution (4.37) for network parameters (4.25) with α = 0.16 V−1 and wavenumber k = 0.3 of the voltage signal for different values of dispersive parameter λ = C S /C0 , namely, a, d λ = 0.46; b, e λ = 0.47; c, f λ = 0.49. Other parameters are given in the text. Reprint from Ref. [22], Copyright 2022, with permission from American Physical Society

4.2 Generation of Network Modulated Rogue Waves Under …

135

Fig. 4.18 First-order non-autonomous RWs associated with solution (4.37) for network parameters (4.25) with α = 0.21 V−1 and wavenumber k = 0.9 of the voltage signal for different dispersion parameter λ = C S /C0 , with values a, d λ = 0.27; b, e λ = 0.33; c, f λ = 0.40. Other parameters are shown in the text. Reprint from Ref. [22], Copyright 2022, with permission from American Physical Society

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4 Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks

Fig. 4.19 Magnitude of the rational solution, |ψ(ξ, 0)|, at time τ = 0, as given by Eq. (4.37) with r0 = 1.2 and β0 = 0.695 for network parameters (4.25) with β = 0.0197. At τ = 0, the amplitude of the rational solution (4.37) is real and its maximum value is |ψ(ξ, 0)|max = 3 |r0 |. Plot (a) shows √ P/2Q as a function of parameter λ = C S /C0 of the linear dispersive element C S . Plots (b) are generated with α = 0.16 , k = 0.3 and different values of λ, viz., λ = 0.46 [(the solid line), λ = 0.47 (the dashed line), and λ = 0.49 (the dotted line). Plots (c) are generated for α = 0.21, k = 0.9 and λ = 0.27 (the solid line), λ = 0.33 (the dashed line), and λ = 0.40 (the dotted line). Other parameters are the same as in Figs. 4.17 and 4.18. Plots (b) and (c) are produced for values of λ taken from Region (I) and Region (II), respectively. It is seen that λ affects both the wave’s shape and velocity. Reprint from Ref. [22], Copyright 2022, with permission from American Physical Society

α, and β. These three parameters just causes a shift of the phase evolution, as we can see from plots of Fig. 4.25 showing the phase profile at time τ = 0 for different values of λ, α, or β. It follows from different plots of Figs. 4.24 and 4.25 that parameter λ and parameter β produce the same effect on the wave phase, while λ and α, and α and β demonstrate opposite effects.

4.2.3 Conclusion and Discussions In this Section, we have considered the electric nonlinear transmission network of Fig. 2.1 and reduced the amplitude equation to a one-parameter KE equation. The criterion of the MI of Stokes waves is obtained and a one-parameter first-order rational

4.2 Generation of Network Modulated Rogue Waves Under …

137

Fig. 4.20 The absolute value of the rational solution, |ψ(ξ, 0)|, at time τ = 0 as given by Eq. (4.37) with r0 = 1.2 and β0 = 0.7 for √ network parameters (4.25) with λ = C S /C0 = 0.29 and α = 0.16. Plot (a) shows the variation of P/2Q as a function of nonlinearity parameter α. Wavenumber k and α are varied as follows. b k = 0.8 and α = 0.1 (the solid line), α = 0.15 (the dashed line), and α = 0.18 (the dotted line); c k = 0.9 and α = 0.196 (the solid line), α = 0.2 (the dashed line), and α = 0.21 (the dotted lines). Plots (b) and (c) are generated with nonlinearity parameter β taken from Region (I) and Region (II), respectively. Reprint from Ref. [22], Copyright 2022, with permission from American Physical Society

polynomial solution is built. Using this solution, we have investigated analytically the dynamics and the transmission of Peregrine soliton through our network system. The effects of various parameters including the network parameters on the dynamics of the Peregrine soliton of our network system are also investigated. We found that the built rational polynomial solution produces large-amplitude waves, localized in both space and time, predicting that the wave energy may be concentrated into a small region. We also found that the solution parameter can be used either to amplify or to suppress modulated RWs in the network of Fig. 2.1. Our results also reveal that the network parameters are useful for controlling the motion of the Peregrine solitons in our network system.

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4 Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks

Fig. 4.21 The absolute value of the rational solution, |ψ(ξ, 0)|, at time τ = 0 as given by Eq. (4.37) with r0 = 1.2 and β0 = 0.7 for√network parameters (4.25) with λ = C S /C0 = 0.3 and α = 0.16. Plot (a) shows the variation of P/2Q as a function of the nonlinearity parameter β. Wavenumber k and parameter β are varied as follows. b k = 2.0 and β = 0 (the solid line), β = 0.005 (the dashed line), and β = 0.01 (the dotted lines); c k = 0.8 and β = 0.015 (the solid line), β = 0.017 (dashed line), and β = 0.019 (the dotted lines). Plots (b) and (c) are generated with nonlinear parameter β taken from Region (I) and Region (II), respectively. Reprint from Ref. [22], Copyright 2022, with permission from American Physical Society

4.3 Chirped Super Rogue Waves Propagating Along a Lossless Nonlinear Electrical Network This Section focuses on the investigation of the chirped version of higher-order rogue waves with nonlinear chirping, termed super chirped rogue waves (SRWs) for their super high peak amplitude, within the framework of the nonlinear electric transmission network shown in Fig. 2.1. Such an investigation is carried out with the help of a generalized cubic-quintic NLS equation with a self-steepening term [46– 48]. The effects of the network parameters such as the linear dispersive parameter on the chirped SRWs will be analyzed.

4.3 Chirped Super Rogue Waves Propagating …

139

Fig. 4.22 The absolute value of rational solution, |ψ(ξ, 0)|, at time τ = 0, as given by Eq. (4.37) with r0 = 1.2 and β0 = 0.6 for network parameters (4.25) with λ = C S /C0 = 0.3, α = 0.21, and √ β = 0.0197. Panel (a) shows the variation of P/2Q as a function of wavenumber k. It is varied as follows. b k = 1.0 (the solid line), k = 1.05 (the dashed line), and k = 1.1 (the dotted lines); c k = 2.1 (the solid line), k = 2.3 (the dashed line), and k = 3.0 (the dotted lines). Plots (b) and (c) are generated with k taken from Region (I) and Region (II), respectively. Reprint from Ref. [22], Copyright 2022, with permission from American Physical Society

4.3.1 Generalized Nonlinear Schrödinger Equation for a Lossless Electric Network The model to be studied in the Section is the lossless electric transmission network depicted in Fig. 2.1 made of N identical cells. For this system, we use the same voltage-capacitance dependence as in the previous two Sections, that is, Q n (Vn ) =   C0 Vn − αVn2 + βVn3 , where C0 is the characteristic capacitance, and α and β are two nonlinear coefficients which are assumed to be positive. Applying Kirchhoff’s laws to the network system of Fig. 2.1, we arrive to the following network equations:    d2 d2  2 d 2 Vn 2 2 3 αV , + u + λ − V − V V = − βV + ω ) (2V n n+1 n−1 n 0 0 n n dt 2 dt 2 dt 2 (4.39) √ √ where n = 1, 2, . . . , N , u 0 = 1/ L 1 C0 and ω0 = 1/ L 2 C0 are the characteristic frequencies of the network, and λ = CC0S is the dimensionless dispersive parameter.

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4 Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks

Fig. 4.23 The evolution of the RW’s phase according to Eq. (4.38) for network parameters (4.25) with α = 0.21, β = 0.0197, λ = C S /C0 = 0.3, r0 = 1.2, and different nonlinear dispersion parameter β0 . a The evolution of the first-order non-autonomous RW’s phase with β0 = 0. b The evolution of the first-order non-autonomous RW’s phase with β0 = −0.7. c The evolution of the first-order non-autonomous RW’s phase with β0 = 0.7. d The phase curve of the first-order non-autonomous RW at time t = 0 for different values of parameter β0 , viz., β0 = 0 (the solid line), β0 = −0.7 (the dashed line), and β0 = 0.7 (the dotted line). Different plots are generated with wavenumber k = 0.9 of the voltage signal propagating in the network. It is seen that the high-order effects break the symmetry of the first-order non-autonomous RW. Reprint from Ref. [22], Copyright 2022, with permission from American Physical Society

In the linear approximation Vn ∼ exp [i (kn − ωt)], Eq. (4.39) leads to the following linear dispersion relation of a typical band-pass filter 

 

 k k − ω02 + 4u 20 sin2 = 0. ω2 1 + 4λ sin2 2 2

(4.40)

For values of wavenumber k chosen in the first Brilloum zone (0 ≤ k ≤ π ), the angular frequency ω defined  by Eq. (4.40) is limited by the cutoff angular frequencies ω2 +4u 2

0 0 ω0 at k = 0 and ωc = at k = π : ω0 ≤ ω ≤ ωc . Therefore, the propagating 1+4λ frequency f p will be limited by the lower cut-off frequency f 0 = ω0 /2π and the upper cut-off frequency f c = ωc /2π .

4.3 Chirped Super Rogue Waves Propagating …

141

Fig. 4.24 The evolution of the RW’s phase according to Eq. (4.38) for network parameters (4.25), with r0 = 1.2 and β0 = 0.7. a–c The evolution of the phase profile with different values of linear dispersive parameter λ = C S /C0 . viz., a λ = 0.3, b λ = 0.31, and c λ = 0.32. d–f The evolution of the phase profile for different values of the nonlinear parameter α, viz., d α = 0.205, e α = 0.2075, and f α = 0.21. g–i The evolution of the phase profile for different values of the nonlinearity parameter β, viz., g β = 0.0157, h β = 0.0177, i β = 0.0197. Different plots are generated with wavenumber k = 0.9 of the voltage signal propagating in the network of Fig. 2.1. It is seen that linear dispersive parameter λ = C S /C0 and nonlinear parameters α and β do not affect the symmetry of the first-order non-autonomous RW phase evolution. Reprint from Ref. [22], Copyright 2022, with permission from American Physical Society

Next, our attention is focused to wave packet centered around (ω, k) for the angular frequency ω = ω p = 2π f p and wavenumber k = k p . To derive the amplitude equation that governs the slowly modulated waves propagating along our network system, we follow Taniuti and Yajima [49] and employ the reductive perturbation method in the semidiscrete  limit. For this aim, we introduce the following two slow variables,  x = ε n − υg t and τ = ε2 t, where the small ε  1 measures the smallness of the modulation frequency and the amplitude of the input waves, and υg = dω/dk is the group velocity of the wave packet. Then, we seek the solution Vn (t) of Eq. (4.39) in the general form [49]

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4 Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks

Fig. 4.25 The evolution of the RW’s phase at time τ = 0 for a different values of the linear dispersion parameter λ = C S /C0 , λ = 0.3 (the solid line), λ = 0.31 (the dashed line), and λ = 0.32 (the dotted line); or b different values of the nonlinearity parameter α, viz., α = 0.205 (the solid line), α = 0.2075 (the dashed line), and α = 0.21 (the dotted line); or c different values of the nonlinearity parameter β, viz., β = 0.0157 (the solid line), β = 0.0177 (the dashed line), and β = 0.0197 (the dotted line). Other parameters are the same as those used in Fig. 4.24. It is seen that λ and β have the same effects on the phase evolution, while λ and β, or α and β have opposite effects. Reprint from Ref. [22], Copyright 2022, with permission from American Physical Society

  1 Vn (t) = ε 2 u(x, τ )eiθ + ε u 20 (x, τ ) + u 22 (x, τ )e2iθ  3  +ε 2 u 30 (x, τ ) + u 33 (x, τ )e3iθ   (4.41) +ε2 u 40 (x, τ ) + u 42 (x, τ )e2iθ + u 44 (x, τ )e4iθ  7  5  +ε 2 u 50 (x, τ ) + u 53 (x, τ )e3iθ + u 55 (x, τ )e5iθ + c.c. + O ε 2 , where θ = kn − ωt is the rapidly varying phase, and c.c. stands for the complex conjugation. Under all the above assumptions, we find that the fundamental harmonic u(x, τ ) of ansatz (4.41) must satisfy the following generalized NLS equation with cubic derivatives terms: i

∂u ∂ |u|2 u ∂ |u|2 ∂ 2u + P 2 + Q 1 |u|2 u + Q 2 |u|4 u + iγ1 + i (μ1 − 2γ1 ) u = 0, ∂τ ∂x ∂x ∂x (4.42)

4.3 Chirped Super Rogue Waves Propagating …

143

where

k 1  2 u 0 + λω02 cos [k] − 4λυg2 sin2 − 4ωυg sin [k] , (4.43) 2ω 2 3β − 2αℵ0 Q 1 = υg − 2αυg ℵ1 2   u 20 + 4λω2 2 2 + ℵ1 2υg 1 + 4λ cos [k] − sin [2k] ω      2 u 0 − λω2 cos [k] 4u 20 ωα 2 υg υg sin [k] (4.44) + υg − + 6λ   +    , 2 ω 1 + 4λ cos2 2k ω 1 + 4λ sin2 2k υg2 − u 20     6αβυg2 (ℵ0 + ℵ1 ) + u 20 − υg2 2α (ℵ2 + ℵ4 + ℵ1 ℵ3 ) − 3β ℵ3 + 2ℵ21  Q2 = ω (4.45) , 2 υg2 − u 20  " !   2αω 2αωυg + ℵ1 u 20 + 4λω2 sin [k] − 2ωυg 1 + 4λ cos2 [k]   γ1 = 2υg (αℵ0 − β) − , ω02 + 4u 20 − 4ω2 − 4 u 20 + 4λω2 cos2 [k] P=

(4.46)

!    " 2αω 2αωυg + ℵ1 u 20 + 4λω2 sin [k] − 2ωυg 1 + 4λ cos2 [k]   μ1 = 4υg (αℵ0 − β) − . ω02 + 4u 20 − 4ω2 − 4 u 20 + 4λω2 cos2 [k]

(4.47) Here, ω is the angular frequency given by the linear dispersion relation (4.40) and ℵ0 =

2αυg2 υg2 −

ℵ1 = −

ω02 αυg2 ℵ2 = 2 υg − u 20

u 20

−

4αω2   ,  ω02 + 4u 20 − 9ω2 − 4 u 20 + 9λω2 cos2 3k2

4αω2   ,  + − − 4 u 20 + 9λω2 cos2 3k2    4α 2 υg4 4αυg2 2β 2 + 3ℵ1 2ℵ1 +  , 2 − α υg2 − u 20 υg2 − u 20 4u 20

9ω2

9ω2     ω02 + 4u 20 − 9ω2 − 4 u 20 + 9λω2 cos2 3k2   8α 2 ω2   β+ 2 , ω0 + 4u 20 − 4ω2 − 4 u 20 + 4λω2 cos2 [k]    8ω2 υg2 − u 20 (3βℵ1 − αℵ3 ) + αυg2 (3β − 2αℵ1 )    , ℵ4 =  2 υg − u 20 ω02 + 4u 20 − 4ω2 − 4 u 20 + 4λω2 cos2 [k]   2 u 0 − λω2 sin [k]   . υg =  ω 1 + 4λ sin2 k2 ℵ3 =

(4.48)

Equation (4.42) is the amplitude equation that describes the dynamics of modulated waves propagating through the network system of Fig. 2.1. It is a generalized

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4 Emission of Rogue Wave Signals in Nonlinear Electrical Transmission Networks

Fig. 4.26 a Linear dispersive curve showing the evolution of the frequency f p = ω/2π as function of the wavenumber k for given three values of the dispersive parameter λ; b Behavior of the groupvelocity dispersion P as function of the wavenumber k for given three values of λ; c Plot of the Kerr nonlinearity Q 1 versus the wavenumber k for given three values of λ: λ = 0 (solid line), λ = 0.03 (dot-dashed line), and λ = 0.06 (dashed line). Different plots are generated with the network parameters (4.49). Reprint from Ref. [48], Copyright 2022, with permission from John Wiley and Sons

cubic-quintic NLS equation with cubic derivative terms that can be used to various context of nonlinear phenomena [47, 50–59] . Its coefficients have different meanings, depending on the physical context of its use. In the context of fiber optics for example, the u(x, τ ) designates the complex envelope of an optical pulse, variables τ and x are respectively the distance and retarded time, P, Q 1 , and Q 2 are respectively the GVD, the Kerr nonlinearity, and the quintic nonlinearity, γ1 and μ1 account respectively for the pulse self-steepening effect and the self-frequency shift coefficient. For the numerical computation purpose, we will use the following network parameters Vb = 2V, C0 = 320 pF, L 1 = 220 mH, L 2 = 470 mH, α = 0.21 V−1 , β = b = 0.0197 V−2 ,

(4.49)

and choose the linear dispersive element C S from the condition λ < LL 21 , so that the group velocity υg remains nonnegative in the first Brilloum zone 0 0; (iii) Case of BECs with time-varying external expulsive potential [87]: k(t) = −

μ20 [1 − tanh (μ0 t/2)] , with r0 > 0. 8

For the integrable condition (8.36) to be satisfied for each of cases (i), (ii), and (iii) we choose a(t) and k(t) as [86, 87]: a(t) = r0 exp(λt), r0 > 0,

  k(t) = − mω2 /4 m + sin(ωt) + m cos2 (ωt) [1 + m sin(ωt)]−2 , a(t) = r0 [1 + tanh (μ0 t/2)] , r0 > 0,

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8 Rogue Matter Waves in Bose-Einstein Condensates …

respectively. In the following, parameter r0 will be referred to as the amplitude parameter. Using the above enumerated cases, we intend here to analyze in detail how the first- and second-order chirped RWs get modified by a(t) and k(t). Case (i): Evolution of Chirped Rogue Waves in BECs with Time-Independent Expulsive Potential As the first example, we follow Liang et al. [38] and consider BECs with a time-independent expulsive (anti-trapping) harmonic potential, modeled by Eq. (8.26) with the potential magnitude k(t) = −λ2 /4, parameter a(t) of the two-body interactions being (8.55) a(t) = r0 exp(λt). −1 It from Eqs.   (8.35) and (8.37) that (t) = r0 exp (−λt) and T (t) =

2then follows r0 /2λ exp (2λt) − 1 . For the present example, we use the first-order RW solution (8.46) and the corresponding chirping (8.47) to show in Figs. 8.11 and 8.12 a typical evolution of respec-

Fig. 8.11 Left panels: spatiotemporal evolution of the first-order RW generated with the use of the rational solution (8.46) for different values of parameter β. a β = −1/3; b β = 0; c β = 1/3. Right panels: density distribution in the first-order obtained with the help of the rational polynomial solution (8.46) for different values of β. d β = −0.7; e β = −1/3; f β = 0; g β = 1/3; h β = 0.7. Other parameters used in generating different plots are r0 = 0.25 and λ = 0.02. Reprint from Ref. [10], Copyright 2022, with permission from Elsevier

8.2 Chirped Rogue Matter Waves in Bose-Einstein Condensates…

311

Fig. 8.12 Spatiotemporal evolution (left panels) and top-view distributions (right panels) of chirping (8.47) associated with the first-order RW for with different values of parameter β. a, d β = 01/3; b, e β = 0; c, f β = 1/3. Other parameters used in generating different plots are r0 = 0.25 and λ = 0.02. Reprint from Ref. [10], Copyright 2022, with permission from Elsevier

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8 Rogue Matter Waves in Bose-Einstein Condensates …

Fig. 8.13 Evolutive formation of the first-order RWs in BEC with the nonlinearity coefficient exponentially increasing as per Eq. (8.55) and the time-independent trap’s frequency, k(t) = −λ2 /4. Different plots are generated for β = 1/3, λ = 0.02, and different numerical values of the amplitude parameter r0 in Eq. (8.55): r0 = 0.05 (a); r0 = 0.1 (b); r0 = 0.15 (c); r0 = 0.3 (d). Reprint from Ref. [10], Copyright 2022, with permission from Elsevier

tively the first-order rogue matter waves and the distribution of the corresponding chirping for different numerical values of parameter β, β = 0 corresponding to the BEC Peregrine soliton. Different plots show of Fig. 8.11 feature one hump and two valleys around the center. We can see from plots of Figs. 8.11 and 8.12 how much parameter β affect the wave motion. For example, parameter β produces an essential skew angle relative to the ridge of the RW in the anti-clockwise [clockwise] direction for β > 0 [for β < 0], and the skew angle becomes larger with the increase of |β| (this is clearly seen from plots of the bottom panels). We can also see from plots of Figs. 8.11 and 8.12 that the shape of both the RW and the corresponding chirping does not change drastically for β = 0. Left panels of Figs. 8.11 and 8.12 reveal that the BECs RWs as well as the corresponding chirping in the present case are embedded on an increasing nonzero CW background. As we can see from plots of Fig. 8.12, the chirping is localized in both time and space, and exhibits two dark-bright doubly localized structures, with the same location as in the two valleys of the corresponding first-order RW. To investigate the effects of the amplitude parameter r0 on the wave evolution, we show in Figs. 8.13 and 8.14 the formation of respectively the first-order and the second-order rogue matter waves for different numerical values of the amplitude

8.2 Chirped Rogue Matter Waves in Bose-Einstein Condensates…

313

Fig. 8.14 Evolutive formation of the second-order rogue matter waves in the BEC models with the same a(t) and k(t) as in Fig. 8.12, the coefficients being β = 1/3 and λ = 0.02. Parameter r0 in Eq. (8.55) takes values r0 = 0.01 (a); r0 = 0.05 (b); r0 = 0.1 (c); r0 = 0.2 (d); r0 = 0.6 (e); r0 = 1.0 (f). Reprint from Ref. [10], Copyright 2022, with permission from Elsevier

parameter r0 . Plots of Fig. 8.13 reveal that the peak value of the first-order RW grows and its width compresses when parameter r0 increases. Plots of Fig. 8.14 show that depending on the numerical value of the amplitude parameter r0 , the second-order RW can either remain a single RW or split into two to three first-order RWs whose width decreases with the increase of r0 . It is also seen from plots of Figs. 8.13 and 8.14 that the wave amplitude increases with the increase of r0 . To show the effect of parameter β on the dynamics of the BEC second-order rogue matter waves and the corresponding chirping in the present case, we display in Figs. 8.15 and 7.16 the dynamics of the second-order BEC RW and the corresponding chirping associated with the exact rogue wave solution (8.52) and chirping (8.53) for different numerical values of β. It is seen from different plots that β has almost the same effects on the second-order BEC RWs as on the first-order ones and the corresponding chirping: it does not impact dramatically the wave shape, but produces a skew angle relative to the RW’s ridge or chirping’s ridge in the anti-clockwise [clockwise] direction for β > 0 [β < 0]. Plots of Fig. 8.16 reveal that the chirping

314

8 Rogue Matter Waves in Bose-Einstein Condensates …

Fig. 8.15 Effects of strength β of the three-body inter-atomic interactions on the second-order rogue matter waves moving in a BEC with the exponentially-varying atomic scattering length (see Eq. (8.55) trapped in an expulsive time-independent parabolic potential. Plots of the left panels and right panels are organized as in Fig. 8.11, but for the rational polynomial solution given by Eq. (8.52), for β = −1/3 (a, d), β = 0 (b, e), and β = 1/3 (c, f). Parameters r0 and λ, as well as the time-dependent ones, a(t) and k(t), are given in the text. Reprint from Ref. [10], Copyright 2022, with permission from Elsevier

8.2 Chirped Rogue Matter Waves in Bose-Einstein Condensates…

315

Fig. 8.16 Spatiotemporal plot (left panels) and top-view distributions (right panels) of the frequency chirp associated with the second-order RW solution (8.52), given by Eq. (8.53), for k = −λ2 /4 and a(t) taken as per Eq. (8.55), cf. Fig. 8.12 for the first-order RWs. To generate different plots, we have used the following numerical data: r0 = 0.25, λ = 0.02, and three values of β, viz., β = −0.7 (a, d), β = 0 (b, e), and β = 0.7.(c, f). Reprint from Ref. [10], Copyright 2022, with permission from Elsevier

associated with the second-order BEC RWs is localized both in time and space. Comparing plots of Figs. 7.15 and 8.16, we find that the second-order RW and the corresponding chirping exhibit respectively four valleys and four dark-bright doubly localized structures, located at the same position. Case (ii): Evolution of Chirped Rogue Matter Waves in BECs with TimePeriodic Modulation of the Scattering Length As the second example, we consider a BEC system with a temporally periodic variation of the s-wave scattering length [86]. For this special case, we assumed the parameter a(t) of the two-body interactions to be given as

316

8 Rogue Matter Waves in Bose-Einstein Condensates …

Fig. 8.17 Left panels: top-view distribution for the first-order rogue matter wave generated with the help of the exact rational solution (8.46) with r0 = 0.25, m = 0.4, and for different values of strength β of the three-body inter-atomic interactions, viz., β = −0.7 (a); β = 0 (b); β = 0.7 (c). Middle panels: Spatiotemporal evolution of rogue matter wave for the same rational solution with r0 = 0.25, β = 1/3, and three values of parameter m, viz., m = 0.1 (d); m = 0.4 (e); m = 0.8 (f). Right panels: Density plots for the same RW solution with β = 1/3, m = 0.4, and three values of the amplitude parameter r0 , viz., r0 = 0.05 (g), r0 = 0.12 (h); r0 = 0.25 (i). Different plots are generated with ω = 0.5. Reprint from Ref. [10], Copyright 2022, with permission from Elsevier

a(t) = r0 [1 + m sin (ωt)] , with 0 < m < 1, and r0 > 0.

(8.56)

The corresponding potential strength k(t) is found from the integrable condition (8.36) and reads k(t) = −

  mω2 m + sin (ωt) + m cos2 (ωt) 4 [1 + m sin (ωt)]2

.

(8.57)

To analyze how parameters a(t) and k(t) given by respectively Eq. (8.55) and (8.56) modify the dynamics of the first- and second-order chirped BEC RWs, we display respectively in Figs. 8.17 and 8.18 and 8.19 the evolution of first- and secondorder RWs and the distribution of the corresponding frequency chirps for different numerical values of parameters β (top panels), m (middle panels), and r0 (bottom panels), β = 0 corresponding to the BEC Peregrine solitons of the standard NLS equation. Different plots show that the first- and the second-order RWs and the corresponding chirping are embedded on a modulated increasing nonzero CW background. Plots of these three figures show that β produces a skew angle relative to the

8.2 Chirped Rogue Matter Waves in Bose-Einstein Condensates…

317

Fig. 8.18 Top panels: top-view distribution for the second-order rogue matter wave obtained with the use of the exact rational solution (8.52) with r0 = 0.25, m = 0.4, and for different values of strength β of the three-body interaction, viz., β = −1/3 (a); β = 0 (b); β = 1/3 (c). Middle panels: spatiotemporal evolution of the second-order rogue matter wave generated with the same rational solution with r0 = 0.25, β = 1/3, and three values of parameter m, viz., m = 0.1 (d); m = 0.5 (e); m = 0.9 (f). Bottom panels: Spatiotemporal evolution of the second-order rogue matter wave obtained with the help of the same rational solution for β = 1/3, m = 0.2, and three values of the amplitude parameter r0 , viz., r0 = 0.01 (g); r0 = 0.05 (h); r0 = 0.25 (i). Reprint from Ref. [10], Copyright 2022, with permission from Elsevier

ridge of the RWs and their corresponding chirping, in the clockwise [anti-clockwise] direction for β < 0 [β > 0]. It is clearly seen from different plots that the chirping associated to first- and second-order RWs exhibit respectively two- and four-peak dark-bright doubly localized structures, located at the same positions where valleys of the corresponding RWs are located. Plots of the middle and bottom panels of Figs. 8.17 and 8.18 demonstrate how much the amplitude and the structure of the firstand second-order RWs are affected by parameters m and r0 ; for example, the wave amplitude increases with the increase in each of m and r0 , and the best structure of the BEC RWs is obtained for small values of m and large values of r0 ; therefore, m and r0 have the same effect on the waves’ amplitudes and opposite effect on their structure. Case (iii): Chirped Rogue Waves in Under the Action of the Scattering Length and Expulsive Parabolic Potential Subjected to the Stepwise Temporal Modulation As our last example, we consider a BEC system with time-varying scattering length trapped in an expulsive parabolic potential subjected to the stepwise temporal modulation. For this example, we choose a stepwise modulation profile defined as [87, 88], we

318

8 Rogue Matter Waves in Bose-Einstein Condensates …

Fig. 8.19 Left panels: spatiotemporal distribution of the chirping (8.47) corresponding to the firstorder RW solution (8.46). Right panels: spatiotemporal distribution of the chirping (8.53) for the second-order RW solution (8.52). Different plots are generated with r0 = 0.25, m = 0.4, ω = 0.5, and three different values of strength β of the delayed nonlinear response, viz., β = −1/3 (a, d); β = 0 (b, e); β = 1/3 (c, f). Reprint from Ref. [10], Copyright 2022, with permission from Elsevier

k(t) = −

 μ  μ20

0 1 − tanh t , μ0 > 0. 8 2

(8.58)

The integrable condition (8.36) in the present case then leads to

 μ  0 t , a(t) = r0 1 + tanh 2

(8.59)

where r0 > 0 is a free constant. As we will see in the following, the wave amplitude will increase with the increase in the values of either μ0 or r0 . To analyze the effects of parameters a(t) and k(t) as well as that of parameters β on the BEC rogue matter waves and the corresponding chirping in the case of a stepwise temporal modulation profile, we show for different numerical values of r0 , μ0 , and β the evolution plots of the first- and second-order BEC RWs and the distribution of the corresponding chirping in respectively Figs. 8.20 and 8.21 and 8.22. It is important

8.2 Chirped Rogue Matter Waves in Bose-Einstein Condensates…

319

Fig. 8.20 Evolution of the the first-order rogue matter wave generated the exact rational solution (8.46) of Eq. ( 8.26) with the time-modulation format given by Eqs. ( 8.58) and (8.59). Top panels: top-view distribution for r0 = 0.25, μ0 = 0.05, and different values of strength β of the threebody interaction, viz., β = −1/3 (a); β = 0 (b); β = 1/3 (c). Middle panels: Spatiotemporal wave evolution for r0 = 0.25, β = −1/3, and three values of μ0 , viz., μ0 = 0.1 (d); μ0 = 0.4 (e); μ0 = 5 (f). Bottom panels: spatiotemporal plots for β = −1/3, μ0 = 0.05, and three values of the r0 , viz., r0 = 0.05 (g), r0 = 0.1 (h); r0 = 0.25 (i). Reprint from Ref. [10], Copyright 2022, with permission from Elsevier

to note that the situation when β = 0 corresponds to the Peregrine solitons of the standard NLS equation. It is seen from plots of the middle and bottom panels of Figs. 8.20 and 8.21 that the first- and second-order rogue matter waves propagate on top of a kink-shaped increasing nonzero CW background. Different plots of Figs. 8.20, 8.21, and 8.22 reveal that parameter β of the delayed nonlinear response does not strongly affect the shape, but produces a skew angle relative to the ridge of the RWs and that of the corresponding chirping in the clockwise [anti-clockwise] direction for β < 0 [β > 0]. It is seen from plots of the middle and bottom panels of Figs. 8.20 and 8.21 that the wave amplitude increases with the increase in either r0 or μ0 , and that depending on the numerical values of parameters μ0 and r0 , the first-order [secondorder] BEC rogue matter wave is composed of one hump and two valleys [features three humps] located around the center; moreover, the hump width decreases with the increase of either μ0 or r0 . The best structure of the first- and second-order BEC rogue matter waves, as we can see from plots of the middle and bottom panels of Figs. 8.20 and 8.21, is obtained with small values of μ0 and large values of r0 ; moreover, the wave width decreases with the increase of μ0 and r0 . Plots of Fig. 8.22 reveal that

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8 Rogue Matter Waves in Bose-Einstein Condensates …

Fig. 8.21 Evolution of the second-order rogue matter wave as per the exact rational solution (8.52) with the time-modulation format, defined by Eqs. ( 8.58) and (8.59). Top panels: top-view distribution for r0 = 0.25, μ0 = 0.05, and different values of strength β of the three-body interaction, viz., β = −1/3 (a); β = 0 (b); β = 1/3 (c). Middle panels: spatiotemporal plots for r0 = 0.25, β = −1/3, and different values of μ0 , viz., μ0 = 0.1 (d); μ0 = 0.4 (e); μ0 = 5 (f). Bottom panels: Spatiotemporal wave evolution for β = −1/3, μ0 = 0.05, and three values of r0 , viz., r0 = 0.03 (g), r0 = 0.1 (h); r0 = 0.25 (i). Reprint from Ref. [10], Copyright 2022, with permission from Elsevier

chirping associated with the first- and second-order BEC RWs is localized in time and space, and exhibits two and four dark-bright localized structures, respectively.

8.2.4 Conclusion and Discussions In this Section, we have considered a one-dimensional cubic GP equation which describes the evolution of the macroscopic wave-function for BEC systems with time-varying atomic scattering lengths trapped in an expulsive parabolic potential. The criterion of the baseband modulational instability is presented. The integrable condition of this model equation is derived and first- and second-order rational solutions with nonlinear chirping are built. With the use of these rational solutions, we have investigated analytically the dynamics of first- and second-order chirped rogue matter waves in various types of BEC systems. Effects of the potential magnitude and parameter of the two-body inter-atomic interactions on the formation and motion of BEC rogue matter waves are investigated. Also, the effects of the strength of the delayed nonlinear response on the chirped BEC rogue matter waves are studied.

8.2 Chirped Rogue Matter Waves in Bose-Einstein Condensates…

321

Fig. 8.22 Top-view distribution of (left panels) the chirping (8.47) associated with the first-order rational polynomial solution (8.46), and (right panels)the chirping (8.53) corresponding to the second-order rogue wave solution (8.52). The step-like modulation format is taken as per Eqs. (8.58) and (8.59). Different plots are obtained with r0 = 0.25, μ0 = 0.1, and three different values of strength β of the delayed nonlinear response, viz., β = −0.7 (a, d); β = 0, (b, c) and β = 0.7 (e, f). Reprint from Ref. [10], Copyright 2022, with permission from Elsevier

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8 Rogue Matter Waves in Bose-Einstein Condensates …

We have found that the chirping associated with the found rogue matter waves are localized in time and space and exhibit two dark-bright localized structures for the first-order rogue matter waves, and four dark-bright localized structures for secondorder rogue matter waves.

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Chapter 9

Dynamics of Matter-Wave Solitons in Multi-component Bose-Einstein Condensates

Abstract Multi-coupled equations of Gross-Pitaevskii type that model binary (twocomponent) BECs are considered. Employing various analytical techniques such as the technique of Darboux transform (DT), exact analytical soliton solutions are presented. With the help of these soliton solutions, we investigate analytically the soliton management and stability of binary BECs with time-varying intrinsic interactions trapped in an external time-varying HO potential. The effects of time-modulated HO potential and the modulated nonlinearity on the BECs stability are studied too.

9.1 Soliton Management in a Binary Bose-Einstein Condensate By means of an integrable system of coupled GP equations of the Manakov’s type [1], we carry out in this Section the analytical investigation of dynamics of matterwave solitons in a binary BEC. Using the DT, we build exact analytical solutions with/without the cross-phase modulation (XPM) interaction between the two components for soliton set on top of a plane-wave background. The effects of XPM interaction between the two components are investigated. We show that in the presence of XPM, the built solutions exhibit properties different from those in the singlecomponent GP.

9.1.1 The Model and Analysis The mathematical model to be used in this Section is the following coupled onedimensional GP equations which governs the dynamics of two-component selfattractive BECs with negative scattering lengths:

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 E. Kengne and W. Liu, Nonlinear Waves, https://doi.org/10.1007/978-981-19-6744-3_9

329

330

9 Dynamics of Matter-Wave Solitons in Multi-component …

  1 ∂ 2 φ1 ∂φ1 a21 2 2 |φ | | |φ φ1 , =− i −g 1 + 2 ∂t 2 ∂x2 a12   1 ∂ 2 φ2 a22 ∂φ2 2 2 |φ2 | + |φ1 | φ2 , =− −g i ∂t 2 ∂x2 a12

(9.1)

where a11 and a22 and a12 are respectively the negative scattering lengths of intraand inter-component atomic collisions, the spatial coordinate x is measured in units of x0 ∼ 1 µm, time t is measured in units of mx02 /, with atomic mass m, φ j are √ measured in units of n 0 with n 0 being the maximum density of the initial distribution of the condensate, and the nonlinearity parameter (interaction constant) g is defined as g = 4π n 0 x02 |a12 | . When a11 = a22 = a12 , the coupled GP equations (9.1) are integrable [1]. Exact two-component solution of system (9.1) accounting for a soliton interacting with a plane wave in the special case a11 = a22 = a12 can be generated by means of DT method:     2ηC j G 2 4iηG 1 i + √ exp ϕ ; (9.2) φ j = A j exp (iϕ) 1 + F g 2 here, j = 1, 2,   F = D exp θ1 + A2 D exp (−θ1 ) − 2i A2 L − L sin ϕ1 + C exp (−θ2 ) ,

(9.3a) (9.3b)

G 1 = L exp θ1 − A L exp (−θ1 ) − |L| exp (−iϕ1 ) + A exp (iϕ1 ) ,     1 1 i i G 2 = L exp − (ϕ1 + ϕ2 ) + (θ1 − θ2 ) + A2 exp (ϕ1 − ϕ2 ) − (θ1 + θ2 ) , (9.3c) 2 2 2 2   1  ϕ ≡ kx + g A21 + A22 − k 2 t, (9.3d) 2 2

2

2

with θ1 , θ2 , ϕ1 , ϕ2 , L , and D given as 1 [(ξ − k) M I + ηM R ] t − θ10 , 2 ≡ ηx + ηξ t + θ10 , 1 ≡ M R x + [(ξ − k) M R − ηM I ] t + ϕ10 , 2  1 2 ξ − η2 t − ϕ10 , ≡ ξx + 2

≡ M R + ξ + k + i (M I + η) , M ≡ (k + ξ + iη)2 + A2 = M R + i M I ,

  ≡ |L|2 + A2 , A ≡ 4g A21 + A22 , C = |C1 |2 + |C2 |2 .

θ1 ≡ M I x +

(9.4)

θ2

(9.5)

ϕ1 ϕ2 L D

(9.6) (9.7) (9.8) (9.9)

All parameters k, θ10 , ϕ10 , A1 , and A2 are free real constants, while C1 and C2 are two free complex constants to be chosen from the condition

9.1 Soliton Management in a Binary Bose-Einstein Condensate

A1 C1 + A2 C2 = 0.

331

(9.10)

In the case of a vanishing CW background, that is, when the background amplitude A1 = A2 = 0, solution (9.2) leads to  √  φ j = ηε j / g cosh θ exp (−iϕ2 ) , with θ = η (x + ξ ) − θ0 , j = 1, 2, (9.11) where θ0 is a free real constant, and ε1 and ε2 are arbitrary complex constants satisfying the relationship |ε1 |2 + |ε2 |2 = 1. Solution (9.11) is a stable two-component soliton; the soliton speed, width, and amplitudes associated with the

√exact solution (9.11) are respectively Vsol = −ξ , wsol = η−1 , and As1,2 = ηε1,2 / g and satisfy the relationship A2s1 + A2s2 = η2 /g. For the exact soliton solution (9.11), the proportional to the number Q of atoms in the binary condensate (total norm of the two-component soliton) is given as +∞

2   |φ1 |2 + |φ2 |2 d x = 2 |η| /g, with Q j = 2 |η| ε j /g. Q ≡ Q1 + Q2 = −∞

Computing the momentum M and Hamiltonian H of the system, we obtain respectively i M ≡ M1 + M2 = − 2

+∞     φ 1 φ1,x − φ1 φ 1,x + φ 2 φ2,x − φ1 φ 2,x d x = Vsol Q −∞

+∞ 2 2



   1

φ1,x 2 + φ2,x 2 − g |φ1 |2 + |φ2 |2 2 d x = M − g Q 3 . H ≡ 2 2Q 24 −∞

Now, let us consider the situation of vanishing soliton amplitudes, A1s,2s = 0 (for example, η = 0). In this special case, solution (9.2) reduces to the Stokes wave (alias CW), (φ1 , φ2 )T = (A1 , A2 )T exp [iϕ] , with amplitudes A1 and A2 , wavenumber k, and angular frequency  

= g A21 + A22 − k 2 /2. Therefore, the exact solution (9.2) describes a two-component soliton propagating on a non-vanishing CW background. The above condition (9.10) is the condition of the XPM interaction between the two components of the CW background. When C1 = C2 = 0, the second term in solution (9.2) vanishes, and each soliton component will be embedded in its own CW background (a more detailed consideration of this special case is given below). In the general case when C1 C2 = 0, the second term in

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9 Dynamics of Matter-Wave Solitons in Multi-component …

solution (9.2) does not vanish. We then conclude that condition (9.10) of the XPM interaction implies that coefficients C1 and C2 depend on amplitudes A1 and A2 of the CW. For a better understanding of the behavior of solution (9.2), it is relevant to start with the special case when each soliton component will be embedded in its own CW background, that is, C1 = C2 = 0. In this special case, solution (9.2) reduces to φ j = A j exp (iϕ) (1 − 2ηW ) , j = 1, 2,   1 a cosh θ1 + sin ϕ1 b sinh θ1 + c cos ϕ1 , W ≡ −i A cosh θ1 + a sin ϕ1 cosh θ1 + a sin ϕ1

(9.12a) (9.12b)

with       a = −i A L − L /D, b = A L + L /D, and c = A2 − |L|2 /D. It is well seen that θ2 and ϕ2 are absent from expressions (9.12a) and (9.12b), and the two components share a common shape. Solution (9.2) in the present special case (C1 = C2 = 0) is then similar to that for the one-component BEC. If M I = 0, equation (9.12a) leads to the total norm of the soliton component of the solution, which coincides with the norm above the CW level:   +∞  2   η b2 + c2 |φ1 |2 − A1 + |φ2 |2 − A22 d x = I, g |M I |

(9.13)

−∞

+∞ I ≡ −∞

where

η + A (cosh x) sin (Bx + ) d x, cosh x + a sin (Bx + )

(9.14)

   = −(1/2)ηM I 1 + B 2 t + Bθ10 + ϕ10 and B = M R /M I .

In obtaining the above expression, we have used the following relations |φ1 (±∞, t)|2 = A21 and |φ2 (±∞, t)|2 = A22 for asymptotic values of the fields. One can easily verify that integral (9.14) is free of , meaning that the total norm of the soliton is a conserved quantity. If now M I = 0, it is obvious that the soliton speed vanishes and the CW background completely traps the soliton [2]. In particular, setting ξ = −k and A2 > η2 yields M I = 0, and W in solution (9.12a) turns to W =

1 η cosh θ1 + A sin ϕ1 − i M R sinh θ1 , A A cosh θ1 + η sin ϕ1

(9.15)

9.1 Soliton Management in a Binary Bose-Einstein Condensate

333

with θ1 ≡ (1/2)ηM R t − θ10 and ϕ1 ≡ M R (x − kt) + ϕ10 ,  where M R = A2 − η2 , and θ10 and ϕ10 are free real constants. It is evident that expression (9.15) is periodic in variable x with period = 2π/M R , and aperiodic in time t. For a better understanding of the modulational instability development provided by the above solution, we assume that ε = exp (−θ10 )  1

(9.16)

is a small amplitude of the initial perturbation that triggers the onset of the MI and linearize solution (9.12a) at the initial time t = 0 with respect to ε, taking Eq. (9.15) into account. We then obtain the following CW with a small modulational perturbation added to it φ j (x, 0) ≈ A j exp (iϕ) [ρ − εχ sin (M R x + ϕ10 )] ,

(9.17)

where   ρ ≡ A2 − 2η2 − 2iηM R /A2 with |ρ|2 = 1, χ ≡ 4ηM R (M R − iη) /A3 . Carrying out direct numerical simulations of the underlying equations (9.1) demonstrates that the evolution of initial configuration (9.17) closely follows the exact solution provided by Eqs. (9.12a) and (9.15) [2].

9.1.2 Results It is important to remember that the above results have been obtained under the Manakov integrable conditions a11 = a22 = a12 which, in reality, may not be exactly satisfied. Since the difference of the scattering lengths in a BEC mixture of two different hyperfine states of the same atomic species is very small, we can still use the Manakov’s system as a good approximation. As an illustration,we have set a11 = −1.03, a12 = −1, a22 = −0.97,

(9.18)

and solve numerical the coupled GP system (9.1). The obtained results yield the picture of the modulational instability development displayed in Fig. 9.1. Comparing the exact solution when a11 = a22 = a12 demonstrates that the two solutions are virtually indistinguishable. Now, we consider solution (9.2) in a more general situation when (|A1 | + |A2 |) (|C1 | + |C2 |) > 0. Under condition (9.10) of the XPM interaction, it is possible

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9 Dynamics of Matter-Wave Solitons in Multi-component …

Fig. 9.1 Plots of the spatial profile of the exact solution ( 9.12a) [dotted lines] and plots of the numerical results obtained under nearly integrable condition (9.18) [solid lines] (exact and numerical plots completely overlap). Different plots are generated for the parameters taken as η = 0.8, k = −0.1, g = 1, A1 = 1, A2 = 1.2, θ10 = 6, the latter one representing amplitude (9.16) of the initial perturbation which triggers the onset of modulational instability, with ε = 2.4788 × 10−3 and ϕ10 = 0. Reprint from Ref. [2], Copyright 2022, with permission from American Physical Society

to explicitly study the effects of the XPM interaction between the plane waves in the two-component BEC. In this general case, for condition (9.10) to be satisfied, √ √ we set C1 = 4 g A2 C and C2 = −4 g A1 C, where C = 0 is an arbitrary complex constant. To analyze the following two representative situations in detail, we further fixe ξ = −k in solution (9.2). (i) Case A2 = η2   In the present case, we have 4 A21 + A22 = A2s1 + A2s2 and solution (9.2) can be written as follows   √ φ j = − exp (iϕ) A j tanh (θ2 /2) + (−1) j 2 A3− j cosh−1 (θ2 /2) exp (iϕ3 ) , (9.19) where   θ2 ≡ η (x − kt) + θ10 , ϕ ≡ kx + η2 /4 − k 2 /2 t, ϕ3 ≡ η2 t/8 + ϕ10 , and θ10 and ϕ10 are free real constants. For solution (9.19), the density distribution is found to be   |φ1 |2 + |φ2 |2 = A22 + A21 1 + sech2 (θ2 /2) . As it is clearly seen from Fig. 9.2. solution (9.19) appears as a superposition of bright and dark solitons, produced by the action of XPM. In this figure, we can see that the solution with A1 A2 = 0 is a complex consisting of bright and dark solitons in

9.1 Soliton Management in a Binary Bose-Einstein Condensate

335

Fig. 9.2 Spatiotemporal distribution of densities of the two components in solution (9.19), with phase difference π between the components. Different plots are generated for parameters taken as k = −0.1, g = 1, A1 = 1, A2 = 0.8, θ10 = −2, and ϕ10 = 0. Reprint from Ref. [2], Copyright 2022, with permission from American Physical Society

the first or vice versa. It follows from condition A2 = η2 that

and second species,  |η| = 4g A21 + A22 ; this means that the soliton width can be controlled by the amplitude of the CW background. Also, we can observe from plots of Fig. 9.2 a shift of the soliton’s peak due to the action of XPM. (ii) Case A2 > η2 In the present case, we have   4 A21 + A22 > A2s1 + A2s2 ,

(9.20)

and solution (9.2) can be written as φ j = A j exp (iϕ) (1 − 2ηW1 ) − (−1) j 2η A3− j W2 ,

(9.21)

where 1 η cosh θ1 + A sin ϕ1 − i M R sinh θ1 , A A cosh θ1 + η sin ϕ1 + A exp (−θ2 ) 1 (M R + iη) exp [(−iϕ1 + θ1 ) /2] + A exp [(iϕ1 − θ1 )] W2 = A A cosh θ1 + η sin ϕ1 + A exp (−θ2 ) W1 =

 −θ2 + i (ϕ − ϕ2 ) , × exp 2

(9.22a)



(9.22b)

with θ1 ≡ (η/2) M R t − θ10 , θ2 ≡ x − kt + θ20 , ϕ1 ≡ M R (x − kt) + ϕ10 ,   1 2 ϕ2 ≡ −kx + k − η2 t − ϕ20 , and M R = A2 − η2 ; 2

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9 Dynamics of Matter-Wave Solitons in Multi-component …

Fig. 9.3 Soliton profile at different times generated with the use of the exact solution (9.21) for η = −1.5, k = −0.1, g = 1, A1 = 1, A2 = 1.2, θ10 = −6, ϕ10 = 0, and θ20 = −5, ϕ20 = 0. Reprint from Ref. [2], Copyright 2022, with permission from American Physical Society

Fig. 9.4 Soliton profile at different times generated with the numerical solution of Eqs. (9.1) with the nonlinearity constants taken as per Eq. (9.18). Other parameters are identical to those in Fig. 9.3. Reprint from Ref. [2], Copyright 2022, with permission from American Physical Society

θ10 , θ20 , ϕ10 , and ϕ20 being free real constants. It follows from Eqs (9.22a) and (9.22b) that W2 → 0 in the limit θ2 → ±∞, and W1 → W in the limit θ2 → +∞, and W1 → 0 in the limit θ2 → −∞. From the form of solution (9.12a) and expression (9.15), we can then conclude that solution (9.21) describes partial modulational instability, since the MI growth rate is restrained in the limit of θ2 → −∞, as illustrated by Fig. 9.3. However, when the integrable condition a11 = a12 = a22 is not satisfied, the numerical solution of the model equation (9.1) will conspicuously differ from the exact solution (9.21), as we can see from Fig. 9.4.

9.1 Soliton Management in a Binary Bose-Einstein Condensate

337

Fig. 9.5 Soliton profile at different times generated with the general-form analytical solution ( 9.2). Parameters are η = 1.5, ξ = −0.4, k = −0.5, g = 1, A1 = 1, A2 = 1.2, 10 = 5, 10 = 0, C1 = 1, and C2 is determined by constraint (9.10). Reprint from Ref. [2], Copyright 2022, with permission from American Physical Society

Finally, let us consider solution (9.2) in the general case. It then follows from Eqs. (9.4) and (9.5) that the solution contains terms with different velocities, V1 = −(1/2) [(ξ − k) + ηM R /M I ] and V2 = −ξ, leading to splitting of the soliton part of the solution on top of the continuous wave background in two wave packets, high-frequency one and low-frequency one, as we can see from plots of Fig. 9.5 where exact analytical solution is obtained under the integrable condition with ai j = 1. Also, we can observe from plots of Fig. 9.6 that plots generated with the use of exact solution are compared to those obtained with the numerical solution of Eqs. (9.1), with the normalized scattering lengths chosen as in Eq. (9.18) when the integrable condition is not satisfied. It is clearly seen from plots of Fig. 9.6 that the soliton part of the numerical solution, built on top of the continuous wave background, also splits in two wave packets, stable high-frequency and unstable low-frequency ones.

9.1.3 Conclusion and Discussions We have considered a coupled GP equation which describes binary BECs. The condition of the XPM interaction is presented in an analytical form. Under the integrable condition that leads to a Manakov system, exact analytical soliton solutions are presented, in the form of a soliton embedded on a non-vanishing continuous wave background. We found that under the integrable condition, the exact soliton solution can be used to describe the MI development which in physical applications can be

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9 Dynamics of Matter-Wave Solitons in Multi-component …

Fig. 9.6 Soliton profile at different times generated with the numerically found solution of Eqs. (9.1) with a11 = −1.03, a12 = −1, a22 = −0.97. Other parameters are identical to those in Fig. 9.5. Reprint from Ref. [2], Copyright 2022, with permission from American Physical Society

used to generate a soliton train. We also found that even if the integrable condition is not satisfied, the wave complex can be generated as a robust one. We have shown that interaction between two continuous wave amplitudes gives rise to a phase difference between the components and can cause splitting of the soliton and formation of a complex pattern. Although our study was carried out for positive g, it is important to note that the model system (9.1) is also integrable in the case of negative g when either a11 = a22 = a12 , corresponding to a binary self-repulsive condensate, or a11 = a22 = −a12 , corresponding to a mixture of two self-repulsive condensates that attract each other [2–5].

9.2 Soliton Stability in Binary Bose-Einstein Condensate Under Temporal Modulation In this Section, the mathematical model to be considered is a coupled one-dimensional GP equations whose nonlinearities exponentially decay with time. Such a coupled GP system may be used to describe the dynamics of two-component BECs with time-varying intrinsic attractive interactions trapped in an expulsive time-depending HO potential. The integrable condition of the model is presented in an explicit form. Although solitons associated with solutions of this mathematical model are subject to decay, we show that the robustness of bright solitons can be enhanced, making their lifetime longer. We show that a combination of the expulsive time-modulated HO potential with the modulated nonlinearity may sustain stable BECs, while it quickly decays in the time-independent potential. To confirm the analytical results, numerical simulations on the model equation are considered. We also show that the stability of BEC soliton is not destroyed by an addition of noise [6].

9.2 Soliton Stability in Binary Bose-Einstein Condensate Under Temporal Modulation

339

9.2.1 The Physical Model and the Lax Pair The model to be studied here pertains to a two-component BEC, having equal atomic masses and attractive interactions in both components [7, 8], trapped in an external HO potential. The evolution of the mean-field wave functions of the setting is thus described by a coupled 1D GP equation which, in the scaled form, can be written as   1 ∂2 1 2 ∂ψ1 2 2 2 = − + b11 |ψ1 | + b12 |ψ2 | + 1 (t)x ψ1 = 0, i ∂t 2 ∂x2 2   ∂ψ2 1 2 1 ∂2 2 2 2 i + b22 |ψ2 | + b21 |ψ1 | + 2 (t)x ψ2 = 0, = − ∂t 2 ∂x2 2

(9.23)

where ψ j denotes the mean-field wave function of the j-th component normalized as follows +∞ +∞ 2 |ψ1 | d x = 1 and |ψ2 |2 d x = N2 /N1 . −∞

−∞

In Eq. (9.23), b j j = 4a j j Ni /r⊥ and b jk = 4a jk Ni /r⊥ are respectively the self-phase modulation (SPM) and the cross-phase modulation (XPM) coefficients in which a j j and a jk are the respective scattering lengths, while r⊥ is the transverse-component radius. For the integrability purpose, the symmetric system is considered, with 2 , b11 = b22 = b21 = b12 ≡ −g, and 2j (t) = ω2j (t)/ω⊥

where ω j and ω⊥ stand for respectively the frequencies of the trapping potential in the longitudinal and transverse directions. Throughout this Section,√time t and coordinate x are assumed to be measured in units of 2/ω⊥ and r⊥ = / (mω⊥ ), respectively. Next, we assume that the confining and expulsive signs of the potential correspond to λ2 < 0 and λ2 > 0, respectively, and set

21 (t) = 22 (t) ≡ −λ2 (t); moreover, we assume time-varying parameter g(t) of interaction and strength λ2 (t) of the external potential. Under all the above assumptions, replacing t by 2t, the model equation (9.23) takes the following form: i



∂ 2ψ j ∂ψ j

ψ j 2 + ψ3− j 2 ψ j + λ2 (t)x 2 ψ j = 0, j = 1, 2. (9.24) + + 2g(t) ∂t ∂x2

It is clearly seen that the nonlinearity in the coupled GP equations (9.24) takes the Manakov’s form [1], and therefore is an integrable system. In the following, we

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9 Dynamics of Matter-Wave Solitons in Multi-component …

focus our attention to the case of positive time-varying parameter g(t) of interaction; this corresponds to the attractive sign of the SPM and XPM interactions. Under the above special integrable condition imposed on g(t) and λ(t), system (9.24) admits a representation in the form of the Lax pair, x + U  = 0, t + V  = 0,

(9.25)

where  is the three-component Jost function with components φ1 , φ2 , φ3 , that is,  = (φ1 , φ2 , φ3 )T , and U and V are two operators defined as ⎛

⎞ iζ (t) Q 1 Q2 U = ⎝ −Q ∗1 −iζ (t) 0 ⎠ , 0 −iζ (t) −Q ∗2 ⎛ ⎞ v11 v12 v13 V = ⎝ v21 v22 v23 ⎠ , v31 v32 v33 )

(9.26)

(9.27)

with i i i ∂ Q1 Q 1 Q ∗1 + Q 2 Q ∗2 , v12 = (t)x Q 1 − ζ (t)Q 1 + , 2 2 2 ∂x ∗ i ∂ Q2 i ∂ Q1 (t)x Q 2 − ζ (t)Q 2 + , v21 = (t)x Q ∗1 + ζ (t)Q ∗1 + , 2 ∂x 2 ∂x i i iζ 2 (t) − i(t)xζ (t) − Q 1 Q ∗1 , v23 = − Q 2 Q ∗1 , 2 2 i ∂ Q ∗2 i −(t)x Q ∗2 + ζ (t)Q ∗2 + , v32 = − Q 1 Q ∗2 , 2 ∂x 2   i i 1 ψ1 (x, t) exp (t)x 2 , iζ 2 (t) − i(t)xζ (t) − Q 2 Q ∗2 , Q 1 = √ 2 2 g(t)   1 i 2 ψ2 (x, t) exp (t)x . √ 2 g(t)

v11 = −iζ 2 (t) + i(t)xζ (t) + v13 = v22 = v31 = v33 = Q2 =

∂  Using the compatibility condition ∂∂ = ∂t∂ , we arrive to the following zerox∂t x curvature equation Ut − Vx + [U, V ] = 0, 2

which shows that the spectral parameter ζ (t) must be taken from the following nonisospectral condition    ζ (t) = μ exp − (t)dt ;

(9.28)

here, μ is a hidden complex constant and (t) is a free function of time, which can be used to define the trap strength λ(t) as follows

9.2 Soliton Stability in Binary Bose-Einstein Condensate Under Temporal Modulation

λ2 (t) =  2 (t) −

341

d , dt

(9.29)

under the condition that the time-varying strength λ(t) of the trap and the timedependent interaction strength g(t) must satisfy the following “integrable condition”: 

dg λ (t)g (t) = 2 dt 2

2

2 − g(t)

d2g . dt 2

(9.30)

Thus, the coupled 1D GP system (9.24) under the integrable condition (9.30) will be integrable. In the special case of a time-independent trap, λ(t) = const ≡ c1 , Eq. (9.30) admits the special solution g(t) = exp [c1 t]. In what follows, we assume the integrable condition (9.30) to be satisfied. It is important to note that if the integrable condition (9.30) is slightly violated, the result will depend on the accumulation of the deviation from the integrability over a characteristic time scale, T , of the dynamical regime. In other words, if the deviation from the integrability is characterized by difference λ(t) from the value given in Eq. (9.30), the system will remain close to the integrability under the following “nearly integrable condition”

T



λ(t)dt  1.



(9.31)

0

9.2.2 Analytical and Numerical Results for Two-Component Bright Solitons in the Integrable System Under the integrable condition (9.30), we use the gauge-transformation approach to look for bright soliton solutions of the coupled GP equations (9.24) as    β1 (t) 2 exp i −ξ1 + (t)x 2 /2 , ε(1) g(t) 1 cosh θ1    β2 (t) 2 = √ exp i −ξ1 + (t)x 2 /2 , ε2(1) cosh θ1 g(t)

ψ1(1) = √

(9.32a)

ψ2(1)

(9.32b)

where  θ1 = 2β1 x + 4

 α1 β1 dt − 2δ1 , ξ1 = 2α1 x + 2 

with α1 = α10 exp

(α12 − β12 )dt − 2χ1 ,

   (t)dt , β1 = β10 exp (t)dt ,

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9 Dynamics of Matter-Wave Solitons in Multi-component …

Fig. 9.7 a, b Spatiotemporal evolution of bright solitons obtained with the use of the exact analytical soliton solution of Eqs. (9.24) under the integrability condition (9.30), with g(t) taken as per Eq. (9.33) (and λ2 = 1/16). c, d Spatiotemporal evolution of bright solitons generated with the use of numerical solution for the same parameters. Reprint from Ref. [6], Copyright 2022, with permission from Elsevier

while δ1 and χ1 are free constants, and ε1(1) and ε1(1) are coupling coefficients satisfying the restriction

2 2

(1)

(1)

ε1 + ε2 = 1. As our first example, we consider the case of BEC systems with time-decaying strength g(t) of the interaction, and adopt the dependence of the nonlinearity coefficient having the form g(t) = 0.5 exp [−0.25t] . (9.33) For the nonlinearity parameter given in Eq. (9.33), Eq. (9.30) renders the external HO potential time-independent and expulsive, with λ2 = 1/16 > 0. The corresponding density profile obtained with the use of the exact analytical solution, given by Eqs. (9.29) and (9.32a), (9.32b), is depicted in Figs. 9.7a, b. Its counterpart produced with the use of the numerical solution of Eq. (9.24) is displayed in Figs. 9.7c, d. From plots of these figures, we observe perfect agreement between the analytical and numerical solutions. This agreement implies the stability of the analytical solution. Replacing the strength g(t) of the time-modulated interaction given in Eq. (9.33) by the following one g(t) = 0.5 exp [−0.5t] (9.34)

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343

Fig. 9.8 The same as in Fig. 9.7, but obtained with g(t) given by Eq. (9.34), with λ2 = 1/4. Reprint from Ref. [6], Copyright 2022, with permission from Elsevier

Fig. 9.9 The same as in Fig. 9.7, but generated with g(t) given by Eq. (9.35), with λ2 = 0.81. Reprint from Ref. [6], Copyright 2022, with permission from Elsevier

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9 Dynamics of Matter-Wave Solitons in Multi-component …

Fig. 9.10 a, b The analytical soliton solution of Eqs. (9.24), given by Eqs. (9.32a) and (9.32b), with g(t) and λ2 (t) taken as per Eqs. (9.36) and (9.37). c, d The numerically generated counterpart of the same solution. Reprint from Ref. [6], Copyright 2022, with permission from Elsevier

in Eq. (9.30), we obtain a stronger expulsive potential with λ2 = 1/4, and the condensate will quickly spread out, as we can see from Fig. 9.8a, b generated with the exact analytical solution, and its numerical counterpart in Fig. 9.8c, d. As seen in Fig. 9.9, this trend continues if the time dependence (9.34 is replaced by g(t) = 0.5 exp [−0.9t] ,

(9.35)

for which Eq. (9.30) yields with λ2 = 0.81. In order to enhance the stability of the condensates, we can switch on the time dependence of the HO strength, taking   g(t) = 0.5 exp −0.125t 2 ;

(9.36)

solving Eq. (9.30) for this g(t) yields the following time-dependent strength of the expulsive potential t2 1 (9.37) λ2 = + . 4 16 Using the corresponding exact analytical solution generated by Eqs. (9.32a) and (9.32b), we show in Fig. 9.10a, b the spatiotemporal evolution of bright solitons. Its numerical counterpart is displayed in Fig. 9.10c, d. From these figures, we conclude that the correctness and stability of the exact analytical solution is corroborated by its numerical counterpart. Next, for the much steeper modulation of parameter g(t) of interaction taken as

9.2 Soliton Stability in Binary Bose-Einstein Condensate Under Temporal Modulation

345

Fig. 9.11 The same as in Fig. 9.9, but for g(t) and λ2 (t) taken as per Eqs. (9.38) and (9.39). Note the essential difference of the spatiotemporal shape of the solution in comparison with that displayed in Fig. 9.9. Reprint from Ref. [6], Copyright 2022, with permission from Elsevier

  g(t) = 0.5 exp −2.5t 2 ,

(9.38)

it follows from the integrable condition (9.30) that λ2 = 5 + 25t 2 .

(9.39)

Using the corresponding exact analytical solution given by Eqs. (9.32a) and (9.32b) and its numerical counterpart, we show in Fig. 9.11 the spatiotemporal evolution of bright solitons which demonstrates non-monotonous evolution in time. Indeed, for both the analytical and numerical cases, the soliton’s wave fields shrink and then expand. Considering now the modulation of the interaction coefficient with the form   g(t) = 0.5 exp −12.5t 2 ,

(9.40)

the integrable condition (9.30) gives λ2 = 5 + 25t 2 .

(9.41)

The corresponding exact analytical solution is depiceted in Fig. 9.12. Comparing plots of Fig. 9.11 with the corresponding plots of Fig. 9.12, we can see that the shape of bright soliton remains qualitatively similar to that in Fig. 9.11. Comparing the results obtained for BEC with attractive interactions trapped in a time-varying expulsive trap with those resulting from BEC with attractive interactions

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9 Dynamics of Matter-Wave Solitons in Multi-component …

Fig. 9.12 The same as in Fig. 9.10, but for g(t) and λ2 (t) given by Eqs. (9.40) and ( 9.41). Reprint from Ref. [6], Copyright 2022, with permission from Elsevier

Fig. 9.13 The same as in Fig. 9.9c, d, but in the case when strong white random noise, with a standard spectral width, is added to the simulations. Reprint from Ref. [6], Copyright 2022, with permission from Elsevier

Fig. 9.14 The same as in Fig. 9.10c, d, but in the case when strong random noise is added to the simulations. Reprint from Ref. [6], Copyright 2022, with permission from Elsevier

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347

Fig. 9.15 The same as in Fig. 9.11c, d, but in the case when strong random noise is added to the simulations. Reprint from Ref. [6], Copyright 2022, with permission from Elsevier

trapped in a time-independent expulsive trap, we conclude that the first one, adjusted to the integrable system based on Eqs. (9.24) and (9.30), is more long-lived. The stability of the exact soliton solution produced by Eqs. (9.24) and (9.30) can also be confirmed by adding random noise to the simulations. Example of obtained results are showed in Figs. 9.13, 9.14 and 9.15. As we can see from these figures, the stability of the evolving condensates is not broken by the white noise. These figures also reveal that the stability of the evolving condensates depend neither on the particular correlation structure of the noise nor on its spectral width. This conclusion confirms the robustness of a two-component condensate in the time-dependent expulsive HO potential in comparison with its counterpart in the time-independent trap. These last results indicate that the life span of the two-component BEC with the time-modulated attractive interactions trapped in the time-dependent HO potential can be increased. The findings of this Section may be realized experimentally in condensates composed of 39 K [9], 85 Rb [8], and 7 Li [10] atoms.

9.2.3 Conclusion In this Section, we have considered a coupled one-dimensional GP equation for studying the dynamics of a two-component BEC with time-varying intrinsic attractive interactions trapped in an expulsive time-depending HO potential. The integrable condition is presented explicitly in terms of the nonlinearity parameter and the strength of the external potential. Under the found integrable condition, exact analytical bright soliton solutions are built and used for studying the dynamics of various binary BECs with time decaying interaction. We have showed that under the integrable condition and for BEC in time-dependent expulsive HO potential, bright solitons stay stable for a reasonably large interval of time, compared to the condensate in the time-independent expulsive HO potential. Analytical results and their stability are confirmed by numerical simulations carried out on the model equation in both the case when the noise is absent and the case when strong random noise is added to the simulations.

348

9 Dynamics of Matter-Wave Solitons in Multi-component …

References 1. S.V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Zh. Eksp. Teor. Fiz. 65, 505 (1973); Sov. Phys. JETP 38, 248 (1974) 2. L. Li, B.A. Malomed, D. Mihalache, W.M. Liu, Exact soliton-on-plane-wave solutions for two-component Bose-Einstein condensates. Phys. Rev. E 73, 066610 (2006) 3. V.G. Makhankov, N.V. Makhaldiani, O.K. Pashaev, On the integrability and isotopic structure of the one-dimensional Hubbard model in the long wave approximation. Phys. Lett. A 81, 161 (1981) 4. V.M. Pérez-García, J.B. Beitia, Symbiotic solitons in heteronuclear multicomponent BoseEinstein condensates. Phys. Rev. A 72, 033620 (2005) 5. S.K. Adhikari, Bright solitons in coupled defocusing NLS equation supported by coupling: application to Bose-Einstein condensation. Phys. Lett. A 346, 179–185 (2005) 6. R. Radha, P.S. Vinayagam, J.B. Sudharsan, W.-M. Liu, B.A. Malomed, Engineering bright solitons to enhance the stability of two-component Bose-Einstein condensates. Phys. Lett. A 379, 2977–2983 (2015) 7. E.R.I. Abraham, W.I. McAlexander, J.M. Gerton, R.G. Hulet, R. Côté, A. Dalgarno, Singlet s-wave scattering lengths of 6 Li and 7 Li. Phys. Rev. A 53, R3713 (1996) 8. S.L. Cornish, N.R. Claussen, J.L. Roberts, E.A. Cornell, C.E. Wieman, Stable 85 Rb BoseEinstein condensates with widely tunable interactions. Phys. Rev. Lett. 85, 1795 (2000) 9. G. Roati, M. Zaccanti, C. D’Errico, J. Catani, M. Modugno, A. Simoni, M. Inguscio, G. Modugno, 39 K Bose-Einstein condensate with tunable interactions. Phys. Rev. Lett. 99, 010403 (2007) 10. K.E. Strecker, G.B. Partridge, A.G. Truscott, R.G. Hulet, Bright matter wave solitons in BoseEinstein condensates. New J. Phys. 5, 73 (2003)

Chapter 10

Dynamics of Higher-Dimensional Condensates with Time Modulated Nonlinearity

Abstract The present Chapter deals generally with more general nonintegrable models, and particularly, with higher-order GP equations, that may model the dynamics of higher-dimensional Bose-Einstein condensates. These models are treated by means of both the variational approximation (VA) and direct computational simulations. We mainly focus our attention on two- and three-dimensional condensates with time modulated nonlinearity. Effects of various parameters of the model equations on the dynamics of matter wave solitons are investigated.

10.1 Dynamics of Two- and Three-Dimensional Bose-Einstein Condensates with Time Modulated Nonlinearities In this Section, we study the dynamics of two- and three-dimensional condensates with time modulated inter-atomic interactions that consist of one constant and one time-periodic oscillating terms. Analytical investigations are carried out by means of either the semi-analytical variational approximation or an averaging method; these theoretical results are confirmed with systematic direct simulations of the GP equation [1–4, 6]. All these methods used in the two-dimensional case reveal the existence of stable self-confined states when the external trap is absent; this results is in agreement with similar results obtained early for (2+1)D spatial solitons in nonlinear optics [7]. When the external trap potential is absent, the variational approximation in the case of 3D condensates also predicts the existence of self-confined state [8]. With the use of direct computatiuonal simulations, we demonstrate that the stability in the 3D free space is limited in time, eventually switching into collapse.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 E. Kengne and W. Liu, Nonlinear Waves, https://doi.org/10.1007/978-981-19-6744-3_10

349

350

10 Dynamics of Higher-Dimensional Condensates with Time Modulated …

10.1.1 The Model and Variational Approximation (VA) The mathematical model used in this Section is the following mean-field GP equation without an external trapping potential, that describes single-particle wave function in its usual form [9]:

i

  4π 2 as 2 2 ∂ψ(r,t) = − ∇ + g |ψ(r,t)|2 ψ(r,t) with g = , ∂t 2m m

(10.1)

where as and m are respectively the atomic scattering length and atomic mass, and g is the strength of the two-body inter-atomic interactions which is assumed to be time modulated has the form g = g0 + g1 sin (χ t) .

(10.2)

In Eq. (10.2), g0 and g1 are respectively the amplitude of the dc and ac parts, while χ is the ac-modulation frequency. In the following, we demonstrate in some detail that the trapping potential may be replaced, in some special situation, by a combination of the temporal modulation of the nonlinearity coefficient (10.2) with the dc and ac parts as in Eq. (10.4). Next, we introduce the following transformation  (10.3)  = 2gn 0 /, t  = t, and r = r 2m/, where n 0 is the largest value of the condensate density. It is evident that employing the transformation (10.3) with the model equation (10.1) and omitting the primes lead to the following scaled equation for isotropic states in which only the radial coordinate is kept:  2  ∂ ∂ψ(r,t) D−1 ∂ =− ψ(r, t) − [λ0 + λ1 sin (ωt)] |ψ(r, t)|2 ψ(r, t), + ∂t ∂r 2 r ∂r (10.4) where D is the spatial dimension (D = 2 or 3), λ0 ≡ −g0 / (), λ1 ≡ −g1 / (), and ω ≡ χ / . It is important to notice that positive [negative] λ0 in Eq. (10.4) correspond to the self-focusing [self-defocusing] nonlinearity. In what follows, we set, without loss of generality, |λ0 | = 1 so that λ0 remains a sign-defining parameter. We start our investigation by applying the VA to Eq. (10.4) [1, 7, 10–14]. For this purpose, we should use the following Lagrangian density generating Eq. (10.4): i

L (ψ) = where

i 2



    ∂ψ 2 1 ∂ψ ∗ ∂ψ ∗  + λ(t) |ψ|4 , ψ − ψ −  ∂t ∂t ∂r  2

(10.5)

10.1 Dynamics of Two- and Three-Dimensional Bose-Einstein …

351

λ(t) = λ0 + λ1 sin (ωt) , and the (∗ ) stands for the complex conjugation. Next, we choose the variational ansatz for the wave function as the Gaussian [10] 

1 r2 + ib(t)r 2 + iδ(t), ψG (r, t) = A(t) exp − 2 2a (t) 2

 (10.6)

where A, a, b, and δ are the time-varying amplitude, width, chirp, and overall phase, respectively. Following the approach used in Ref. [12], we insert the ansatz (10.6) in the Lagrangian density (10.5) and calculate the respective effective Lagrangian. We then arrive to ∞

(10.7) L eff = C D L ψg r D−1 dr, 0

where C D = 2π or 4π in respectively the two-dimensional or three-dimensional cases. Finally, we derive the evolution equations for various functional parameters A(t), a(t), b(t), and δ(t) of ansatz (10.6) from the expression of L eff given by Eq. (10.7) when using the corresponding Euler-Lagrange equations. In what follows, we present the analytical and numerical results separately for the two-dimensional and three-dimensional cases.

10.1.2 Two-Dimensional Case In the two-dimensional case, theoretical studies are done by means of both the VA and the averaging method, applied to the GP equation; theoretical results are confirmed by direct numerical simulations. First of all, we note that the application of the averaging method to the 2D equation (10.4) without using VA in the case of the high-frequency modulation is possible [1, 6, 15, 16].

10.1.2.1

The Variational Approximation

For the two-dimensional GP equation, we calculate from Eq. (10.7) the effective Lagrangian and arrive to   1 4 2 db 1 2 2 dδ 2 4 2 2 2 4 a − a − A λ(t)a . = π − A A − a A b + A L (2D) eff 2 dt dt 4

(10.8)

Using the Euler-Lagrange equations obtained from Eq. (10.8), we find that the total number of atoms N in the condensate is conserved:

352

10 Dynamics of Higher-Dimensional Condensates with Time Modulated …

πa 2 A2 ≡ N = const;

(10.9)

the corresponding width a and chirp b are found to satisfy the differential system da λ(t)N db 2 , = 2ab, = 4 − 2b2 − dt dt a 2πa 4 and the following closed-form evolution equation for the width: d 2a 2 (2 − λ(t)N /2π ) = , dt 2 a3

(10.10)

≡ 2 [λ0 N /(2π ) − 2] ,  = −λ1 N /π,

(10.11)

− +  sin (ωt) d 2a = . dt 2 a3

(10.12)

which by setting

can be rewritten as

In the absence of the ac component, Eq. (10.12) conserves the energy,  E 2D = (1/2) (da/dt)2 − a −2 . Evidently, E 2D → −∞ [+∞] at the limit a → 0, if > 0 [ < 0], meaning that in the absence of the ac component, the two-dimensional pulse is expected to collapse [spread out] at > 0 [ < 0]. The situation = 0 corresponds to the critical norm [17] whose numerically exact value is N = 1.862 [18]; note that the variational equation (10.11) yields N = 2 (if λ0 = +1) [11]. Let us consider the situation when the ac component of g(t) oscillates at a high frequency; in this situation, we can set a(t) = a + δa, with |δa|  |a| ,

(10.13)

where, a is a function of a slow time scale and δa is a rapidly varying function with zero mean value. Under all these assumptions, we can integrate Eq. (10.12) using the Kapitsa averaging method [1, 6, 15]. Inserting Eq. (10.13) into Eq. (10.12), we obtain, after straightforward manipulations, the following system of ODEs for a and δa:

d 2a = − a −3 + 6a −5 δa 2 − 3 δa sin (ωt) a −4 , 2 dt d2 δa = 3δa a −4 +  sin (ωt) a −3 , dt 2

(10.14a) (10.14b)

where · · · stands for averaging over period 2π/ω. Equation (10.14b) admits the special solution

10.1 Dynamics of Two- and Three-Dimensional Bose-Einstein …

δa(t) = −

 sin (ωt)

. a ω2 + 3a −4

3

353

(10.15)

Inserting now Eq. (10.15) into Eq. (10.14a), we obtain the following evolution equation for the slow variable a:

 d 2a 2 1 3 3  2 = 3 − − . (10.16)

2 + dt 2 2 ω2 a 4 + 3

a ω2 a 4 + 3

In the limit a → 0, Eq. (10.16) reduces to the following one 1 d 2a = 3 dt 2 a

  2  2 − 6 2 − + = . 6

6 a 3

(10.17)

It follows from Eq. (10.17) that if  2 > 6 2 (meaning that the amplitude of the highfrequency ac component is large enough), then the behavior of the condensate in the limit a → 0 is exactly opposite to that which would be expected in the presence of the dc component only; this means that in the case of positive , rebound occurs rather than the collapse, and vice versa in the case of negative . Now, in the limit a → ∞ , Eq. (10.16) takes the asymptotic form d 2a = − /a 3 , dt 2 showing that the condensate remains self-confined if > 0 (in the case when > 0, the norm exceeds the critical value). The above asymptotic results guarantee the stability of the behavior of the condensate for any a, solution of Eq. (10.16), the collapse and spreading out being ruled out if >

√

6 > 0.

(10.18)

In the following, we illustrate the above results in terms of the experimentally relevant setting for the condensate of 7 Li with the critical number ∼ 1500 atoms. We then conclude that, for 1800 atoms, we must add a periodic modulation with amplitude  = 0.98 for the stabilization, as one can see from Eq. (10.11) where λ0 = 1 has been used. Indeed, it follows from conditions (10.18) that for small [large] a, the right-hand side (rhs) of Eq. (10.16) is positive [negative], that is, da dt increases [decreases] with the time t increases, meaning that Eq. (10.16) must give rise to a stable fixed point (FP). In fact, the rhs of Eq. (10.16) under condition (10.18) vanishes at exactly one fixed point, 3 2 + ω2 a 4 = 4

   3 4 2 3 −  − 3 . 16 2

(10.19)

354

10 Dynamics of Higher-Dimensional Condensates with Time Modulated …

The stability of the FP (10.19) can be easily checked from the calculation of an eigenfrequency of small oscillations around it. Now, we turn to the direct numerical simulations of Eq. (10.12). Carrying out direct numerical/computational simulations on Eq. (10.12), we obtain results that are in exact agreement with those provided by the averaging method. In other words, we obtain that stable states with α(t) performing small oscillations around point (10.19) [1, 15]. We ask ourselves if the same results can be obtained in the 3D situation. The answer to this question, as we will see in what follows, is no! In 3D situation, we also needs, as in the above analysis, an approximate form of Eq. (10.16) valid when the norm is closed to the critical value (that is, in the limit

→ 0), and for large enough ω. The obtained approximate form of Eq. (10.16) reads

3 2 d 2a = − + . (10.20) dt 2 2 ω2 a 7 a3 Next, we estimate the value of the amplitude of the high-frequency ac component necessary to stop the collapse. First, we note that in physical units, a characteristic trap frequency is  ∼ 100 Hz. Then, we consider a high modulation frequency ∼ 3 kHz, and the scaled modulation frequency taken as ω 30. Assuming the norm to be N /2π = 2.2, we find from Eq. (10.11) that = 0.4; these numerical values correspond to the condensate of 7 Li with 1800 atoms, the critical number being

1500, with modulation parameters taken as λ0 = 1, λ1 = 2.3, and  = 10. From Eq. √ (10.19), we find the stationary value of the condensate width to be ast = 0.8l, l = m/ being the healing length. Based on the VA and assuming that the norm slightly exceeds the critical value, the following important result is obtained: Result: The combining ac and dc components of the two-body inter-atomic interactions corresponding to the attraction in the two-dimensional Gross-Pitaevskii equation without the trapping term may replace the collapse by a stable solitonlike oscillatory state that confines itself. A similar result was reported in Ref. [7] when studying cylindrical solitons in a bulk nonlinear-optical medium.

10.1.2.2

Averaging of the Two-Dimensional Gross-Pitaevskii Equation and Hamiltonian

Here, we assume the ac frequency ω to be large, and rewrite the two-dimensional GP equation (10.4) in the simplified form i

∂ψ + ∇ 2 ψ + λ(ωt) |ψ|2 ψ = 0. ∂t

(10.21)

Next, we employ the multiscale approach for deriving the amplitude equation for slow variations of the field. The approach consists of expanding the solution in a power series of 1/ω. For this purpose, we introduce slow temporal variables Tk as

10.1 Dynamics of Two- and Three-Dimensional Bose-Einstein …

355

Tk ≡ ω−k t, k = 0, 1, 2, . . ., and the fast time is ζ = ωt. Focusing our attention to the case when the dc part of the two-body inter-atomic interactions corresponds to attraction between the atoms (λ0 = 1), we seek the solution of Eq. (10.21) in the general form as ψ(r, t) = A(r, Tk ) +

1 u 1 (ζ, A) + ω

 2 1 u 2 (ζ, A) + · · · , ω

(10.22)

with u k = 0, where · · · stands for the average over the period of the rapid modulation. Using the approach developed in Ref. [19], the first and second corrections are found to be ζ u 1 = −i (μ1 − μ1 ) |A| A, μ1 ≡

[λ(τ ) − λ1 ] dτ,

2

0

 u 2 = (μ2 − μ2 ) 2i |A|2 At + i A2 A∗t + ∇ 2 (|A|2 A)   1 4 2 − |A| A (μ1 − μ1 ) − 2M + λ (μ2 − μ2 ) , 2

(10.23)

where, ζ μ2 ≡



(μ1 − μ1 ) ds, and M = (1/2) μ21 − μ1 2 = (1/2) λ2 − 1 .

0

With the use of these results, we derive at order (1/ω)2 to the following evolution equation for A(x, T0 ), which is valid for both the 2D and 3D cases: i

  2



∂A |A|6 A − 3 |A|4 ∇ 2 A + 2 |A|2 ∇ 2 |A|2 A + ∇ 2 A + |A|2 A + 2M ∂t ω

(10.24) + A2 ∇ 2 |A|2 A∗ = 0,

 being the same amplitude of the ac component as in Eq. (10.11). In both 2D and 3D cases, the quasi-Hamiltonian can be represented in the following form     2 δ Hq 4 ∂A |A| 1 + 6M = −i ∗ , (10.25a) ω ∂t δA      2   2 

 1 ∇ |A|2 A  , |A|8 − |A|4 + 4M Hq = d V |∇ A|2 −2M ω 2 ω (10.25b)

356

10 Dynamics of Higher-Dimensional Condensates with Time Modulated …

where d V is the infinitesimal volume in the 2D or 3D space. It follows from Eqs. (10.25a) and (10.25b) that d Hq /dt = 0. For investigating the 2D case, we apply the modulation theory developed in Ref. [20], and seek the solution in the form of a modulated TS: ψ(r, t) = exp [it] RT (r ), RT (r ) being any physical solution of the boundary-value problem,  1 d RT d RT  d 2 RT 3 − RT + RT = 0, + = 0, RT (r )|r =∞ = 0. dr 2 r dr dr r =0

(10.26)

The corresponding norm N and the Hamiltonian H read ∞ 

∞ RT2 (r |)r dr

NT ≡

= Nc ≈ 1.862, HT =

0

d RT dr

2

 1 4 − RT (r ) r dr = 0, 2

0

(10.27) respectively. Equation (10.24) shows an increase of the critical norm for the collapse, as opposed to the classical value in Eq. (10.27). With the use of Eq. (10.22), we compute the critical norm and find [21] ∞ |ψ|2 r dr = N T + 2IM

Ncrit = 0

  2 ω

, I = 11.178.

Now, let us compute the phase chirp in the soliton. The mean value of the phase chirp b in the soliton is found to be ∞ b=

0

Im [(∂ψ/∂r ) ψ ∗ ] r dr ∞ . 2 |ψ|2 0 r dr

By using expression (10.23) for the first correction, we arrive to b = − (/ω) B M (μ1 − μ1 ) , B ≡

3



∞ ∞ 2  2 4 0 r dr R (R ) − (1/4) 0 dr R ∞ 2 2 0 r dr R

 ≈ 0.596.

For developing a general analysis, we assume that the solution with the norm close to the critical value can be approximated as a modulated TS. In other words, we assume that A(r, t) ≈ [a(t)]−1 RT [r/a(t)] exp (i S) , S = σ (t) +

r 2 da dσ , = a −2 (10.28) 4a dt dt

10.1 Dynamics of Two- and Three-Dimensional Bose-Einstein …

357

with some function a(t). Assuming the initial norm to be close to the critical value (|N − Nc |  Nc ) and using the same strategies used in Ref. [20], we derive the following evolution equation for a(t): a3

d 2a 2 = −β + f 1 (t), 0 dt 2 4M0 ω2

(10.29)

where 2 N − Nc 1 β0 = β(0) − f 1 (0), β(0) = , M0 ≡ 2 4M0 ω M0 4

∞ r 3 dr RT2 ≈ 0.55, 0

f 1 (t) being an auxiliary function given as  f 1 (t) = 2a(t)Re

1 2π

 

 d xd y F(A T ) exp [−i S] {RT + ρ∇ R R (ρ)} . (10.30)

For the harmonic modulation, equations of the lowest-order of approximation lead to

1 C 2 d 2a = − + , (10.31) dt 2 a3 ω2 a 7 where

1 = (N − Nc ) /M0 − C 2 /(ω2 a04 ) and C is defined as [1] 3 C≡ M0

∞

   2  3 1 8 4 2 3 dρ 2ρ RT RT − ρ RT RT − ρ RT ≈ 39. 8

(10.32)

0

It then follows from the averaged equation that the collapse can be stopped by the rapid modulations of the nonlinear term in the two-dimensional GP equation. Comparing Eq. (10.31) with its counterpart (10.20) derived by means of averaging equation (10.12), we find that when numerical coefficients in the second terms are different due to the different profiles of the Gaussian and TS, then both approaches lead to the same behavior near the collapse threshold. Further, we estimate the FP as per numerical simulations performed in work [22] where the following parameters were used, λ = 1 +  at 0 < t < T , and λ = 1 −  at T < t < 2T with the following numerical values: T =  = 0.1, N /(2π ) = 11.726/(2π ), with the critical number Nc = 11.68/(2π ). Using these numerical values, we obtain ac = 0.49, which agrees with the value ac ≈ 0.56 produced by computational numerical simulations.

358

10 Dynamics of Higher-Dimensional Condensates with Time Modulated …

It is important to notice that based on representation (10.22) for the wave function, one can also apply the averaging procedure directly to the Hamiltonian of Eq. (10.4). The averaged Hamiltonian, which can be use to explain the possibility to arrest the collapse in the presence of the rapid modulation of the nonlinearity strength [23], is then found to be       2    2

2 1 2 2 4 8   |A| . ∇ |A| A − |A| − 6M H= d xd y |∇ A| + 2M ω 2 ω (10.33) We note that, if a given field configuration has compressed itself to a spot with size ρ, where A is ∼ ℵ, the conservation of norm N yields the relationship ℵ2 ρ D ∼ N ,

(10.34)

where D is the space dimension. For the strongest collapse-driving and collapsearresting terms, the same estimate H− and H+ in the Hamiltonian yields H− ∼ −

  2 ω

ℵ8 ρ D , H+ ∼

  2 ω

ℵ6 ρ D−2 .

(10.35)

If we employ relation (10.34) to eliminate the amplitude from Eq. (10.35), we conclude that H± asymptotically scale as ρ −5 in the limit ρ → 0 (case of the catastrophic self-compression); depending on details of the configuration, the collapse may therefore be arrested. It is important to note that in the 3d situation, the collapse cannot be prevented. Indeed, in the limit ρ → 0, the collapse-driving term in the 3D case diverges as ρ −9 , while the collapse-arresting one scales ∼ ρ −8 .

10.1.2.3

Numerical Simulations

Here, we intend to check the existence of stable self-confined solitonlike oscillating states by direct numerical simulations of the 2D equation (10.4). The above analytical study predicts the existence of stable self-confined solitonlike oscillating states in region (10.18) when (i) the dc part of the nonlinearity corresponds to self-attraction and (ii) the amplitude of the ac component is small enough. Now, we intend to check this result by carrying out direct numerical simulations on the 2D equation (10.4). A typical example of the obtained results is displayed in Fig. 10.1 showing the formation of a self-confined condensate under a combination of the self-focusing dc and sufficiently strong ac components of the nonlinearity when the external trap is absent. Comparing Fig. 10.1a with Fig. 10.1b showing respectively the collapsing state in the absence of the ac modulation at t ≈ 0.3 and the radial profile of the stable state in the presence of the ac modulation formed by the same input at t ≈ 0.6, it is clearly seen that when the ac term is taken into account, the pulse is stabilized for about 40 ac-modulation periods, after which it decays [4].

10.1 Dynamics of Two- and Three-Dimensional Bose-Einstein …

359

Fig. 10.1 A typical example of the formation of a self-confined condensate in simulations of the 2D equation (10.4). a Collapsing state at t ≈ 0.3 when the ac modulation is absence; b radial profile of the stable state formed by the same input at t ≈ 0.6, in the presence of the ac term in the nonlinearity coefficient. Different plots are generated with the following numerical values of parameters λ0 = 2.4, λ1 = 0.85, ω = 100π , and N = 5. Reprint from Ref. [1], Copyright 2022, with permission from American Physical Society

10.1.3 The Three-Dimensional Gross-Pitaevskii Model Similar to the 2D situation, we separately consider results produced by the analytical approximations (VA and averaging method) and by direct numerical simulations in the 3D case.

10.1.3.1

VA and Averaging

In the case of the 3D GP equation, we compute the effective Lagrangian (10.7) and obtain [1, 5] L (3D) eff =

  3 db dδ 1 1 3 2 3 3 π 2 A a − a2 −2 + √ λ(t)A2 − 2 − 3b2 a 2 . 2 2 dt dt a 2 2

(10.36)

The corresponding Euler-Lagrange equations yields the norm conservation 3

π 2 A2 a 3 ≡ N = const, the following evolution equation for the width a of the condensate, 4 λ(t) N d 2a = 3 − √ 3 4, dt 2 a 2 2π 2 a and the following relations including the width a and the chirp b,

(10.37)

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10 Dynamics of Higher-Dimensional Condensates with Time Modulated …

db 2 da λ(t)N = 2ab, = 4 − 2b2 − √ 3 . dt dt a 2 2π 2 a 5 Equation (10.37) for the wave width in the 3D situation is different from its counterpart (10.10) corresponding to the 2D situation. Following the same procedures as in the 2D case, the amplitudes of the dc and ac components of the nonlinearity are normalized as follows √ √

≡ λ0 N / 2π 3 and  ≡ −λ1 N / 2π 3 . Under such a normalization, equation (10.37) takes the following scaled form 4 − +  sin (ωt) d 2a = 3+ . dt 2 a a4

(10.38)

Setting  = 0 in Eq. (10.38) (case of the absence of the ac term), we arrive the following equation for the energy conservation E 3D

1 = 2



da dt

2

1 + 2a −2 − a −3 . 3

(10.39)

It follows from Eq. (10.39) that in the limit a → 0, E 3D → −∞ [+∞], if > 0 [ < 0]; this corresponds respectively to the collapse or decay of the pulse. Solving numerically Eq. (10.38) without averaging, we have found, as one can see from Fig. 10.2, the existence of a region in the parameter space where the condensate, which would decay for negative , may be stabilized by the ac component, provided that its amplitude is sufficiently large. Figure 10.2 is generated for different initial conditions, a(t = 0) = 0.3, 0.2, or 0.13 and da/dt (t = 0) = 0. It is clearly seen from

Fig. 10.2 The Poincaré section in the plane of (a, da/dt ) for = −1,  = 100, ω = 104 π , generated with the help of the numerical solution of the variational equation (10.38) with different initial conditions given in the text. Reprint from Ref. [1], Copyright 2022, with permission from American Physical Society

10.1 Dynamics of Two- and Three-Dimensional Bose-Einstein …

361

Fig. 10.3 The region in the (, ω/π ) parameter plane where the numerical solution of Eq. (10.38) with = −1 predicts stable quasiperiodic solutions in the 3D case. Here, crosses mark points show different positions where stable solutions were actually obtained, while stars correspond to minimum values of the amplitude of the ac-component  eventually leading to the collapse of the solution of equation (10.4) with = −1. Reprint from Ref. [1], Copyright 2022, with permission from American Physical Society

Fig. 10.2 that independently on the initial conditions, the solution remains bounded with quasiperiodic oscillations, avoiding the collapse or decay. The corresponding stability region in the parameter plane (ω/π, ), generated with the numerical solution is small, as we can see Fig. 10.3. Here, we can also see that for maintaining the stability, the numerical values of the frequency and amplitude of the ac component must be very large. It is important to note that for frequencies above 106 π , the condensate width a(t) becomes very small in the course of the evolution, meaning that the collapse may happen in the solution of the full 3D equation (10.4). Using the VA, we found that the stability is predicted only when < 9. If ≥ 9, the VA predicts solely the collapse. Because the frequency ω is large enough in the stability region as we can see in Fig. 10.3, we also apply the averaging method to this. For this aim, we follow the above procedures done for the 2D case and find the rapidly oscillating correction δa(t) to the solution as  sin [ωt] a , (10.40) δa = − ω2 a 5 − 12a + 4

while the resulting evolution equation for the slow variable a(t) is found to be

 6a − 5

d 2a 2 2 −4 4 2 =a + 4a − +

2 . dt 2 ω2 a 5 − 12a + 4

ω2 a 5 − 12a + 4

(10.41)

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10 Dynamics of Higher-Dimensional Condensates with Time Modulated …

In the limit a → 0, Eq. (10.41) turns to   3 2 1 d 2a . = 4 − + dt 2 16

a

(10.42)

Equation (10.42) plays the same role as Eq. (10.17) in the 2D situation, but leads to only one property (result). In the √ case of negative and with large enough amplitude of the ac component ( > (4/ 3) | |), we conclude from Eq. (10.42) that collapse takes place instead of spreading out. Other results obtained from Eq. (10.17) in the 2D with the use of an equation similar to the averaged equation (10.41) are wrong (see Figs. 10.2 and 10.3). In particular, analyzing in details the rhs of Eq. (10.41), we see that it predict neither a stable FP for negative nor for positive , and this contradicts what is revealed by the numerical simulations of Eq. (10.38). This failure follows from the existence of singular points in Eqs. (10.40) and (10.41), independently of the sign of .

10.1.3.2

Direct Simulations of the GP Equation in the 3D Case

It is necessary to compare the analytical results to the numerical ones. Ignoring the ac component ( = 0), the numerical simulations under the condition

< 0 show straightforward decay. In the presence of an ac component with a large enough amplitude, transient stabilization of the condensate takes place, confirming thus the analytical results obtained by means of the VA. Meanwhile, the stabilization is not permanent since the collapse occurs after some periods. A typical example of the obtained results for the 3D case with < 0 is displayed in Fig. 10.4 for N = 1, = −1, and ω = 104 π. Plots of Fig. 10.4 show the evolution of the density radial profiles |u(r )|2 at different times. It follows from different plots of Fig. 10.4 that a basic characteristic of the system is a dependence of the minimum , which gives rise to the collapse at fixed = −1, versus ω. This dependence

Fig. 10.4 The evolution plot of the density radial profile, |u(r )|2 , in the presence of the strong and fast ac modulation (ω = 104 π ,  = 90). The profiles of |u(r )|2 are shown at different times: t = 0.007 (a), 0.01 (b), and 0.015 (c). Reprint from Ref. [1], Copyright 2022, with permission from American Physical Society

10.2 Stable Vortex Modes in Two-Dimensional Bose-Einstein Condensates

363

is showed in Fig. 10.3 by many star points. It is evident that as a function of the frequency ω, the minimum value of  necessary for the collapse increases with the increase in ω. The numerical simulations have also showed that the collapse in the case > 0 cannot be predicted.

10.1.4 Conclusion and Discussions In this Section, we have considered as the model equation, a higher-dimensional GP equation with a time-varying two-body inter-atomic interactions without an external potential. Assuming the strength of the two-body inter-atomic interactions to be formed of two parts, one constant (dc) and one time-varying (ac) parts, the model equation is used to investigate analytically and numerically the dynamics of both 2D and 3D BECs. The effects of both the dc and ac on the condensates are studied. We found that in the 2D case, with the ac component of the two-body inter-atomic interactions, it is possible to maintain the condensate without an external trap in a stable self-confined state. For the condensates without a trap to maintain a stable self-confined state in the 3D situation, we found via the VA that the dc part of the nonlinearity must correspond to repulsion between atoms; direct numerical simulations reveals that the stability of the self-confined condensate in this case is just partial, that is, limited in time. From our investigations, we conclude that the spatially uniform ac magnetic field may play the role of an effective trap that confines the condensate, and sometimes can enforce its collapse.

10.2 Stable Vortex Modes in Two-Dimensional Bose-Einstein Condensates In this Section, we consider a physical model governed by a two-dimensional GP equation with spatial-varying parameters written in the scaled form. Exact analytical vortex-soliton (VS) solutions and approximate fundamental soliton solutions for twodimensional BECs with a spatially modulated attractive and/or repulsive nonlinearity when the external trapping potential is either absent or present. By means of the discrete energy spectrum of a related linear Schrödinger equation, the number of vortex-soliton modes is presented and the stability of built VS with vorticity higher than or equal to two are reported [24, 25]. The number of VS solutions in the case of BECs with attractive nonlinearity is found to be infinite, while for BEC systems with repulsive nonlinearity, there exists no more than a finite number of VS solutions.

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10 Dynamics of Higher-Dimensional Condensates with Time Modulated …

10.2.1 Model Description and Main Transformation The mathematical model used for investigating stable vortex modes for BECs with a spatially modulated attractive and repulsive nonlinearities trapped in a harmonic potential is the two-dimensional GP equation whose scaled form reads i

 ∂ψ = −∇ 2 + g(r ) |ψ|2 + V (r ) ψ. ∂t

(10.43)

Physically, the GP equation (10.43) describes the BECs macroscopic wave function ψ = ψ(r, θ, t), ∇ 2 is the two-dimensional Laplacian, and g(r ) and V (r ) are respectively the nonlinearity coefficient (alias strength of the two-body inter-atomic interactions) and the strength of the external harmonic potential, functions of the spatial radial coordinate r . To present exact analytical solutions of Eq. (10.43), we use the following ansatz ψ(r, θ, t) = φ(r ) exp [i Sθ − iμt]

(10.44)

in which φ(r ), θ, S, and μ are respectively the stationary wave function (spatialvarying amplitude), the azimuthal angle, an integer vorticity, and the chemical potential. Imposing to Eq. (10.44) to satisfy the model equation (10.43) yields the following ODE for φ(r ) μφ = −

1 dφ d 2φ + g(r )φ 3 + − dr 2 r dr



 S2 + V (r ) φ. r2

(10.45)

Equation (10.45) is subject to the following restriction: (i) φ(r ) must vary as r [S] at the limit r → 0 when the integer vorticity S = 0; (ii) in the case when the integer vorticity S =0, φ(r ) must satisfy the following  = 0; homogeneous boundary condition at r = 0, dφ dr r =0 (iii) φ(r ) must be localized: φ(r )|r =∞ = 0. In the special situation when the strength V (r ) of the external potential V (r ) has the form g0 (10.46) g(r ) = 2 6 , |g0 | > 0, g(0)g (+∞) ∈ R, r ρ (r ) (where g0 = is a free real constant accounting for the nonlinearity strength), we seek special solution of equation (10.45) in the following form r φ(r ) = ρ(r )U [R(r )] , with R(r ) = 0

ds . sρ 2 (s)

(10.47)

It follows from Eqs. (10.46) and (10.47) that the nonlinearity functional parameter g(r ) as well as R(r ) do not change sign; therefore, ρ(r ) has a constant sign. For

10.2 Stable Vortex Modes in Two-Dimensional Bose-Einstein Condensates

365

g(0)g (+∞) to be a finite quantity, we must have ρ(r ) ∼ r −a , with a ≥ 1/3, and ρ(+∞) = 0. Under these conditions on ρ(r ), g(r ) will be bounded and the integral (10.47) that defines R(r ) will converge. Asking now that equation (10.47) satisfies the ODE (10.45), we find that ρ(r ) must be a nonzero solution of the ODE   E ρ S2 ρ + + μ − V (r ) − 2 ρ = 2 3 , r r r ρ 

(10.48)

and U (R) must be any solution of the elliptic ordinary differential equation (EODE) −

d 2U + g0 U 3 = EU, d R2

(10.49)

where E is an arbitrary real constant.

10.2.2 Exact Vortex-Soliton Solutions for the Attractive Nonlinearity (g0 < 0) when E = 0 In the special case when E = 0 and g0 < 0, Eq. (10.48) is solvable. In the case of the harmonic potential V = kr 2 , the solution ρ(r ) of Eq. (10.48) can be written in terms of the Whittaker’s M and W functions [26, 27] as ρ(r ) = r

−1

     μ √ |S| √ 2 μ √ |S| √ 2 c1 M . , kr + c2 W , kr k, k, 4 2 4 2

The above restrictions on ρ(r ) require μ < μ0 = 2 (1 + |S|) the absence of the trap (k = 0), ρ degenerates to ρ(r ) = c3 I S

√ k and c1 c2 > 0. In

√ √



−μr + c4 K S −μr ,

with μ < μ0 = 0, where I S and K S are respectively the modified Bessel and Hankel functions, and constants C3 and C4 satisfy the condition c3 c4 > 0. It is evident that at the limit r → 0, we have ρ(r ) ∼ r −[S] , leading to g(r ) ∼ r 6|S|−2 and R(r ) ∼ r 2|S| at the limit r → 0; moreover ρ(r ) → ∞ as r → ∞. Hence, the respective nonlinearity is localized and bounded, and R(r ) is also bounded. For the boundary conditions φ(0) = φ(∞) = 0 to be satisfied, we choose as an exact solution to Eq. (10.49), the following function U (R) = n √

  √  √  η cn nη R − K 1/ 2 , 1/ 2 −g0

(10.50)

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10 Dynamics of Higher-Dimensional Condensates with Time Modulated …

Fig. 10.5 a Evolution plot of the exact vortex-solitons when the external potential is absent. (Inset) The corresponding profiles of the attractive nonlinearity coefficient. c Exact vortex-solitons of different radial quantum number n with S = 1, the respective nonlinearity-coefficient profile being depicted by the solid line in the inset of (a). Parameters are c3,4 = −μ = −g0 = 1. (b, d) The same as (a) and (c) when the trapping potential is present, V = 10−2 r 2 , with c1,2 = 3. Reprint from Ref. [24], Copyright 2022, with permission from American Physical Society

√ in which n = 2, 4, 6, . . . , n/2 is the radial quantum number, η ≡ K (2/ 2)/R(r √= ∞), cn(ξ, m) is the Jacobi elliptic cosine function with modulus m, and K (1/ 2) is the complete elliptic integral of the first kind. As we can see from Eq. (10.50), U (R) ∼ R at the limit R = 0; this implies that at the limit r → 0, the amplitude of the exact VS is ρU ∼ r |S| , as it should be. We have thus showed that for given parameters μ and S and nonlinearity strength g0 (and k, in the case when the trap is taken into account), one can build an infinite number of exact vortex-soliton solutions with n/2 bright rings surrounding the vortex core, as it is clearly seen in Fig. 10.5. It is important here to note that these VS solutions share the same chemical potential, but their energies increase with the increase of even number n.

10.2 Stable Vortex Modes in Two-Dimensional Bose-Einstein Condensates

367

10.2.3 Exact Analytical Vortex-Soliton Solutions for the Repulsive Nonlinearity (g0 > 0) Now, let us we consider the case of the repulsive nonlinearity, g0 > 0, and assume that the system is confined by a harmonic trap. For the EODE (10.49) to admit ellipticfunction solutions, we must take the free constant E from the condition E > 0; under this restriction, Eq. (10.48) becomes a nonlinear equation, which in general can be solved only in a numerical form. To build the VS solutions in the case of repulsive nonlinearity, we require ρ ∼ r −|S| (for S = 0) at r → 0 so that at the limit r → 0, the nonlinear term of Eq. (10.48) may be neglected. Under these assumptions, ρ(r ) √

is similar to the Neumann function, Y S μr , at the limit r → 0 for μ > 0 (that VS solutions do not exist when μ < 0). Due to the presence of the harmonic trap, we necessary have ρ → ∞ as r → ∞. Further, under the restriction E > 0, term Er −2 ρ −3 in equation (10.48) guarantees the sign definiteness of the spatial-varying function ρ(r ). Therefore, we can integrate numerically Eq. (10.48) to obtain R(r ) and g(r ) if we take small r0 as an initial point and then use as initial conditions, the Neumann function and its derivative at r = r0 . Vortex-soliton solutions can then be built in the numerical form, using the following exact analytical solution of Eq. (10.49) 

   2 E − B2 sn B R, E/B 2 − 1 , (10.51) U (R) = g0 with



E/2 < B ≡ n K



 √ E/B 2 − 1/R(∞) < E;

(10.52)

solution (10.51)–(10.52) is subject to a constraint with even numbers n. The maximal value of n is computed from Eq. (10.52) as √ n < n max = 2R(∞) E/π. This means that in the case of repulsive nonlinearity, there exists no more than a finite number of the VS modes (if for example n max < 2, there will not be VS mode). The number of the numerical VS solutions versus μ is depicted in Fig. 10.6a. As we can see from plots of Fig. 10.6a, the cutoff value μ0 of the chemical potential in the case of repulsive nonlinearity is the same as that for the exact VS solutions built when the attractive nonlinearity was considered. It is also seen from plots of Fig. 10.6a that the number of VS solutions jumps at points μ = μ(S) j ≡ 2 (2 j + |S| − 1)

√ k, j = 1, 2, 3, . . . ;

moreover, point μ(S) j coincides with the j-th energy eigenvalue of the vortex state in the corresponding linear Schrödinger equation. We can then conclude that the (S) number of numerically VS solutions in the interval μ(S) j < μ ≤ μ j+1 is exactly j,

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10 Dynamics of Higher-Dimensional Condensates with Time Modulated …

Fig. 10.6 a The number of numerically found VS modes versus the chemical potential in the case of the repulsive nonlinearity, with the harmonic trap. b The largest instability growth rate for numerically found vortices with S = 2, E = 1, and k = 10−2 . Reprint from Ref. [24], Copyright 2022, with permission from American Physical Society

Fig. 10.7 a Modulation profiles of the repulsive nonlinearity for numerically found vortices, which are displayed in b for S = 1 and in c for S = 2, in the presence of the harmonic trap. Open circles, squares, solid circles, and triangles denote solutions with n = 2, 4, 6, and 8, respectively. Different plots are obtained with parameters E = 1, μ = 2, k = g0 = 0.01. Reprint from Ref. [24], Copyright 2022, with permission from American Physical Society

associated with expression (10.51) for U [R] in which n are taken as n = 2, 4, 6, . . . , 2 j. A similar result has been reported for the 1D GP equation with the optical-lattice potential [28]. A characteristic example of the numerically found VS solutions is depicted in Fig. 10.7 where we can see four VS solutions. All the above solutions are built when the integer vorticity S = 0. It is important to note that it is still possible to build VS solution when S = 0. This is demonstrated in the special case of a two-tier nonlinearity, with constant gr :  g(r ) =

gr for 0 ≤ r < r0 , g0 r −2 ρ −6 for r ≥ r0 .

(10.53)

10.2 Stable Vortex Modes in Two-Dimensional Bose-Einstein Condensates

369

Exact solutions for r ≥ r0 can be built in the same way as above, except that now R(r ) must be defined as r ds . R(r ) ≡ sρ 2 (s) r0

Assuming r0 to be small enough √in comparison with the spatial scale of the external potential, as for example r0  1/k in the presence of a harmonic trap V (r ) = kr 2 , we can approximate φ in the case of r < r0 by a constant as follows  φ(r ) = [μ − V (0)] /gr . Because φ(r ) and φ  (r ) must be continuous at r = r0 , one then requires φ  (r0 ) = 0 and dU (0)/d R = 0; these two conditions yield gr =

μ − V (0) . [ρ(r0 )U (0)]2

The solution to Eq. (10.49) in this special case is then given by Eqs. (10.51) and (10.52), where B R must be replaced by B R + K (E/B 2 − 1), and n = 1, 3, 5, . . . for the case of repulsive nonlinearity, so that make φ(∞) = 0. In the same way, we can build VS solutions in the case of attractive nonlinearity. Examples of found fundamental solutions in the case of repulsive two-tier nonlinearity are displayed in Fig. 10.8. Next,we carry out the linear stability analysis and direct simulations to verify the stability of the built VS solutions. By means of the azimuthal modulational instability (AMI) in the case of the attractive nonlinearity in the absence of the trap potential, we lead to the breaking up and the collapse of higher-order VS solutions. The lowestorder exact VS solutions with n = 2 are found to be stable when the harmonic trap is present, if μ → μ0 , the wave amplitude remaining large as we can see from plots of Fig. 10.9. Similar result has not yet been reported when either S = 1 or S < 2 [29–31]. Under the action of the both repulsive nonlinearity and external harmonic trap, our numerical simulations showed that VS solutions can be stable for every n at which they exist, within some region of values of μ , as we can clearly see from Fig. 10.6b. Also, as we can see from plots of Figs. 10.6b and 10.10, VS solutions with fewer rings may be less stable than their counterparts with the largest number of rings. The numerical results plotted in Fig. 10.10a reveal that unstable VS solutions with n = 2 either exhibit a quasistability or split into vortices with lower topological charges, which periodically break up and recover the axial symmetry. This finding is similar to what was already reported in the case of the attractive nonlinearity [29–32]. Nevertheless, unstable VS solutions with n = 4, 6, . . . , as one can clearly see from plots of Fig. 10.10b, c, ultimately evolve into vortices located close to zero-amplitude points.

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10 Dynamics of Higher-Dimensional Condensates with Time Modulated …

Fig. 10.8 Evolution plots of fundamental solitons supported by the repulsive two-tier nonlinearity [see Eq. (10.53)]. The constant values of gr for n = 1, 3, 5, 7, and 9 are, respectively, 0.0200, 0.0201, 0.0224, 0.0325, and 0.1093. Different plots are generated with the parameters ρ(r0 ) = 1, ρ(r0 ) = 0, r0 = E = 1, k = g0 = 0.01, μ = 2. Reprint from Ref. [24], Copyright 2022, with permission from American Physical Society

Fig. 10.9 a Plot of the largest instability growth rate for exact vortex-solitons with S = 3 and the attractive nonlinearity; b plot os a stable vortex (solid lines, with circles showing its profile at t = 200, when it was initially perturbed by random noise), and c plot of the corresponding nonlinearity coefficient when μ = 0.7. Here g0 = −1000, k = 0.01, and c1,2 = 3. Reprint from Ref. [24], Copyright 2022, with permission from American Physical Society

10.2.4 Conclusion In this Section, we have studied a physical model governed by a 2D GP with spatiallyvarying parameters. The cases of both the attractive and repulsive nonlinearities as well as the case of the presence/absence of an external trapping potential are considered. The strategy for constructing exact analytical VS solutions are presented. This strategy consists of reducing the two-dimensional GP equation to the solvable uncoupled system formed by Eqs. (10.48) and (10.49). We also showed how to build approximate fundamental soliton solutions in the case of two-dimensional axisymmetric profiles of the nonlinearity coefficient, and harmonic trapping potential. Our results showed that BEC systems with attractive/repulsive two-body inter-atomic interactions (nonlinearity) support a finite number of exact VS solutions. We have

References

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Fig. 10.10 a Quasistable evolution of a numerically found vortex with n = 2, in the case of the self repulsion, at t = 1700. b, c Evolution of unstable vortices with n = 4 and 6 at t = 60 and 70, respectively. Different plots are generated with the following numerical parameters S = 2, E = 1, g0 = k = 0.01, μ = 1.82. The vortex with n = 8 are stable. Reprint from Ref. [24], Copyright 2022, with permission from American Physical Society

presented stable VS solutions with vorticity S ≥ 2; vortex-solitons corresponding to higher-order radial states are also reported. The strategy and the methodology elaborated for finding VS solutions in this Section can be applied to other models [33, 34].

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13. V.M. Pérez-García, H. Michinel, J.I. Cirac, M. Lewenstein, P. Zoller, Dynamics of BoseEinstein condensates: variational solutions of the Gross-Pitaevskii equations. Phys. Rev. A 56, 1424 (1997) 14. M. Desaix, D. Anderson, M. Lisak, Variational approach to collapse of optical pulses. J. Opt. Soc. Am. B 8, 2082–2085 (1991) 15. FKh. Abdullaev, A. Gammal, A.M. Kamchatnov, L. Tomio, Dynamics of bright matter wave solitons in a Bose-Einstein condensate. Int. J. Mod. Phys. B 19, 3415–3473 (2005) 16. Yu. Kivshar, S. Turitsyn, Spatiotemporal pulse collapse on periodic potentials. Phys. Rev. E 49, 2536 (1994) 17. R.Y. Chiao, E. Garmire, C.H. Townes, Self-trapping of optical beams. Phys. Rev. Lett. 13, 479–482 (1964) 18. G. Fibich, The Nonlinear Schrödinger Equation: Singular Solutions and Optical Collapse (Springer, Heidelberg, 2015) 19. T.S. Yang, W.L. Kath, Analysis of enhanced-power solitons in dispersion-managed optical fibers. Opt. Lett. 22, 985 (1997) 20. G. Fibich, G.C. Papanicolaou, A modulation method for self-focusing in the perturbed critical nonlinear Schrödinger equation. Phys. Lett. A 239, 167 (1998) 21. F.Kh, Abdullaev, J.G. Caputo, Validation of the variational approach for chirped pulses in fibers with periodic dispersion. Phys. Rev. E 58, 6637 (1998) 22. L. Bergé, V.K. Mezentsev, J.J. Rasmussen, P.L. Christiansen, Yu.B. Gaididei, Self-guiding light in layered nonlinear media. Opt. Lett. 25, 1037 (2000) 23. L. Bergé, Wave collapse in physics: principles and applications to light and plasma waves. Phys. Rep. 303, 259 (1998) 24. L. Wu, L. Li, J.-F. Zhang, D. Mihalache, B.A. Malomed, W.M. Liu, Exact solutions of the GrossPitaevskii equation for stable vortex modes in two-dimensional Bose-Einstein condensates. Phys. Rev. A 81, 061805(R) (2010) 25. D.-S. Wang, S.-W. Song, B. Xiong, W.M. Liu, Quantized vortices in a rotating Bose-Einstein condensate with spatiotemporally modulated interaction. Phys. R. A 84, 053607 (2011) 26. D.-S. Wang, S.-W. Song, B. Xiong, W.M. Liu, Quantized vortices in a rotating Bose-Einstein condensate with spatiotemporally modulated interaction, Phys. Rev. A 84, 053607 (2011) 27. E.T. Whittaker, G.N. Watson, A Course in Modern Analysis, 4th edn. (Cambridge University Press, Cambridge, 1990) 28. Y. Zhang, B. Wu, Composition relation between gap solitons and Bloch waves in nonlinear periodic systems. Phys. Rev. Lett. 102, 093905 (2009) 29. T.J. Alexander, L. Bergé, Ground states and vortices of matter-wave condensates and optical guided waves. Phys. Rev. E 65, 026611 (2002) 30. D. Mihalache, D. Mazilu, B.A. Malomed, F. Lederer, Vortex stability in nearly-two-dimensional Bose-Einstein condensates with attraction. Phys. Rev. A 73, 043615 (2006) 31. L.D. Carr, C.W. Clark, Vortices in attractive Bose-Einstein condensates in two dimensions. Phys. Rev. Lett. 97, 010403 (2006) 32. B. LeMesurier, P. Christiansen, Regularization and control of self-focusing in the 2D cubic Schrodinger equation by attractive linear potentials. Physica D 184, 226 (2003) 33. M. Vengalattore, R.S. Conroy, W. Rooijakkers, M. Prentiss, Ferromagnets for integrated atom optics. J. Appl. Phys. 95, 4404 (2004) 34. M. Vengalattore, M. Prentiss, A reciprocal magnetic trap for neutral atoms. Eur. Phys. J. D 35, 69 (2005)

Chapter 11

Engineering Matter-Wave Solitons in Spinor Bose-Einstein Condensates

Abstract We consider in this chapter a (non)integrable system of three nonlinearly coupled GP equation, which describes the dynamics of matter-wave solitons in a three-component spinor Bose-Einstein condensate. One-, two-, and three-component soliton solutions of the polar and ferromagnetic (FM) types are presented. Applying the Darboux transform (DT) in the case of an integrable system, exact analytical soliton solutions that display full nonlinear evolution of MI of CW states are also obtained. Using the Bogoliubov-de Gennes (BdG) equations for small perturbations, the multistability of the solitons is investigated analytically and numerically. The global stability of the multi-component solitons leading to the ground-state and metastable soliton states of the FM and polar types is considered too. Also, the effects of the nonlinearity parameter on the structural stability of the solitons are analyzed (Li et al. in Phys Rev A 72:033611 [1]).

11.1 Formulation of the Model The physical system to be investigated in this Chapter is an effectively onedimensional BEC loaded in a cigar-shaped trap, elongated in x and tightly confined in the transverse plane (y, z) [2, 3]. In this physical system, we denote by T  − →  (x, t) = +1 (x, t), 0 (x, t), −1 (x, t) , the one-dimensional three-component wave function describing the atoms in the hyperfine state with atomic spin F = 1. Here, components +1 , 0 , and −1 correspond to the three values of the vertical spin projection, m F = +1, 0, −1 [4]. The mathematical model associated with the physical system under consideration is the following three-component system of GP equations [4–7]

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 E. Kengne and W. Liu, Nonlinear Waves, https://doi.org/10.1007/978-981-19-6744-3_11

373

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11 Engineering Matter-Wave Solitons in Spinor …

Fig. 11.1 The evolution plot of the single-component polar soliton generated with the use of the exact analytical single-component soliton solution (11.13) with a small random perturbation added, at t = 0, to the φ+1 and φ−1 components. Parameters used in different plots are ν = 1, a = −0.5, and μ = −1. Reprint from Ref. [1], Copyright 2022, with permission from American Physical Society

  ∂±1 2 ∂ 2 ±1 + (c0 + c2 ) |±1 |2 + |0 |2 ±1 + (c0 − c2 ) =− 2 ∂t 2m ∂ x × |∓1 |2 ±1 + c2 ∗∓1 20 ,    2 ∂ 2 0 ∂0 =− i + (c0 + c2 ) |+1 |2 + |−1 |2 0 (11.1) ∂t 2m ∂ x 2 +c0 |0 |2 0 + 2c2 +1 −1 ∗0 ,

i

in which c0 = (g0 + 2g2 ) /3 and c2 = (g2 − g0 ) /3

(11.2)

denote effective constants of respectively the spin-independent and spin-exchange interactions [1, 8]. In the context of fiber optics, c2 stands for the coefficient of the four-wave mixing, while c0 + c2 and c0 − c2 account for the SPM and XPM interactions. Denoting respectively by m, a f , and a⊥ (with f = 0, 2) the atomic mass, the s-wave scattering length in the channel with total hyperfine spin f, and the size of the transverse ground state, parameters g0 and g2 in Eq. (11.2) are defined as follows 42 a f   , f = 0, 2, gf = 2 ma⊥ 1 − ca f /a⊥

11.1 Formulation of the Model

375

where c = −ζ (1/2) ≈1.46. Assuming time and length to be measured respectively in units of / |c0 | and 2 /2m |c0 | and using the following transformation for the wave function T  √ − →  → φ+1 , 2φ0 , φ−1 , the model system (11.1) takes the following scaled form: i

  ∂ 2 φ±1 ∂φ±1 =− − (ν + a) |φ±1 |2 + 2 |φ0 |2 φ±1 − (ν − a) |φ∓1 |2±1 φ±1 2 ∂t ∂x ∗ −2aφ∓1 φ02 , (11.3) 2   ∂ φ0 ∂φ0 = − 2 − 2ν |φ0 |2 φ0 − (ν + a) |φ+1 |2 + |φ−1 |2 φ0 − 2aφ+1 φ−1 φ0∗ , i ∂t ∂x

where a ≡ −c2 /c0 and ν ≡ −sgn(c0 ). The Hamiltonian that gives birth to system (11.3) is a dynamical invariant of the model (d H/dt = 0) and reads    +∞   ∂φ+1 2  ∂φ−1 2 1    +  − (ν + a) |φ+1 |4 + |φ−1 |4 d x  H =    ∂x ∂x 2 −∞    ∂φ0 2 2 2   − ν |φ0 |4 − (ν + a) − (ν − a) |φ+1 | |φ−1 | + 2  (11.4) ∂x     2    ∗ ∗ . φ−1 φ02 + φ+1 φ−1 φ0∗ × |φ+1 |2 + |φ−1 |2 |φ0 |2 − a φ+1 One can easily verify that system (11.3) conserves the momentum, the solution’s norm, and the total magnetization. In other words, the following three equations hold: (i) the conservation of momentum:  +∞ ∗ ∂φ+1 ∗ ∂φ−1 2 ∂φ0 φ+1 + φ−1 + 2φ0 d x, P =i ∂x ∂x ∂x −∞

(ii) the conservation of the solution’s norm, proportional to total number of atoms: +∞   |φ+1 (x, t)|2 + |φ−1 (x, t)|2 + 2 |φ0 (x, t)|2 d x, N= −∞

and (iii) the conservation of the total magnetization:

(11.5)

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11 Engineering Matter-Wave Solitons in Spinor …

Fig. 11.2 The evolution plots of norms N0 (solid line), N± (dashed line) of respectively the φ0 and φ±1 components, and total norm N (dotted line), as per Eqs. ( 11.17) and (11.5), in the twocomponent polar soliton solution given by Eq. (11.18) perturbed by a small random perturbation introduced in the φ0 component. Different plots are generated with the use of parameters ν = 1 and ∗ < 0; b a = 1.5 for φ φ ∗ > 0; c a = −1.5 for φ φ ∗ < 0; d μ = −1 and a a = 1.5 for φ+1 φ−1 +1 −1 +1 −1 ∗ a = −1.5 for φ+1 φ−1 > 0. Reprint from Ref. [1], Copyright 2022, with permission from American Physical Society

+∞   |φ+1 (x, t)|2 − |φ−1 (x, t)|2 d x. M=

(11.6)

−∞

Setting φ0 = 0 in system (11.3) yields the following system of two nonlinear partial differential equations, having the same behavior (form) as a system which describes light transmission in bimodal nonlinear optical fibers [1]: i

∂ 2 φ±1 ∂φ±1 =− − (ν + a) |φ±1 |2 φ±1 − (ν − a) |φ∓1 |2 φ±1 . ∂t ∂x2

(11.7)

In the context of nonlinear optical fibers, the two modes φ−1 and φ+1 represent either different wavelengths or two orthogonal polarizations. a = ν/5 and a = −ν/3 are associated with the particular cases of respectively two linear polarizations and two circular polarizations. In system (11.7), the two nonlinear terms, proportional to

11.2 Exact Analytical One-, Two-, and Three-Component Soliton Solutions

377

(ν + a) and (ν − a), account for respectively the SPM and XPM interactions of the two waves. The modulation instability of continuous wave states in system (11.7) under the condition ν + a = 0 (XPM-coupled system) has been studied in detail [9]. In the special case when ν = −1 and |a| < 1, the nonlinearity in system (11.7) will be self-defocusing, and the single NLS equation would show no MI. The XPM-coupled system (11.7) in the special case when ν = −1 and |a| < 1 of course will give rise to MI; this means that the XPM interaction is stronger than SPM interaction, that is, 0 < a < 1 [10].

11.2 Exact Analytical One-, Two-, and Three-Component Soliton Solutions Following Ieda et al. [11], we here focus our attention, for simplicity, to the special case of a physically possible integrable model (with a = ν = 1), which corresponds to c2 = c0 leading to 2g0 = −g2 > 0. For condition 2g0 = −g2 > 0 to be satisfied, we impose the following conditions on the scattering lengths a⊥ =

3ca0 a2 , a0 a2 (a2 − a0 ) > 0. 2a0 + a2

(11.8)

It is important to note that it is also possible to derive and study the stability of exact analytical soliton solutions of both polar and FM types in the non-integrable case a = ν. Following the approach elaborated by Zaidong et al. [1], we present in the following several soliton species and results for their stability.

11.2.1 Single-Component FM Solitons A one-component FM soliton can be found by means of the direct method by seeking, under the conditions a + ν > 0 and μ < 0, the single-component solution in the form  (φ−1 , φ0 , φ+1 ) = T

2μ − 1+ν



exp [−iμt] √  0, 0, cosh −μx

T .

(11.9)

In solution (11.9), the chemical potential μ is a parameter of the soliton family. Solution (11.9) corresponds to the zero-velocity soliton. Solutions leading to moving solitons can be generated from Eq. (11.9) by means of the Galilean transformation. In the special case when ν = ±1, condition a + ν > 0 means that a > 1 [a > −1] in the cases of repulsive (ν = −1) [attractive (ν = +1)] spin-independent interaction. Using Eqs. (11.5) and (11.4), the norm N and energy H of the above soliton are found to be

378

11 Engineering Matter-Wave Solitons in Spinor …

√ 4 −μ (ν + a)2 3 N= ,H =− N . ν+a 48

(11.10)

Linearizing system (11.3) about the solution (11.9) leads to the following BdG equations for small perturbations φ−1 and φ0 : ∂ 2 φ−1 ∂φ−1 =− − (ν − a) |φ+1 |2 φ−1 , ∂t ∂x2 ∂φ0 ∂ 2 φ0 = − 2 − (ν + a) |φ+1 |2 φ0 . ∂t ∂x

(11.11) (11.12)

For the soliton solution (11.9), a = 1 so that system (11.11)–(11.12) is a system of decoupled BdG equations. Therefore, solution (11.9) is stable against small perturbations. As we well know, envelop soliton solution of the single standard NLS equation is always stable, so that solution (11.9) cannot be unstable against small perturbations of φ+1 .

11.2.2 Single-Component Polar Solitons By setting φ±1 (x, t) = 0, we obtain the following simplest polar soliton for ν = +1 and μ < 0  T √ exp [−iμt] T √ , 0 . (11.13) (φ−1 , φ0 , φ+1 ) = −μ 0, cosh −μx Linearizing system (11.3) about the soliton solution (11.13) gives rise to a coupled system of BdG equations for small perturbations φ±1 of the other fields:     ∂ ∂ 2 χ±1 2μ ∗  (1 + a) χ±1 + aχ∓1 √ i + μ χ±1 = − , (11.14) + 2 2 ∂t ∂x −μx cosh where (∗ ) stands for the complex conjugation and χ±1 (x, t) ≡ φ±1 (x, t) exp [iμt] .

(11.15)

The parametric gain that comes from the x-dependent term (last term) of Eq. (11.14) may be a source of instability. To understand this instability, we can investigate system (11.14) at x = 0; we then arrive to the following system: i

  dχ±1 ∗ = μ (1 + 2a) χ±1 + 2aχ∓1 dt

(11.16)

11.2 Exact Analytical One-, Two-, and Three-Component Soliton Solutions

379

Fig. 11.3 The evolution of the three-component polar soliton, generated with the use of the exact analytical soliton solution (11.23) under the action of a small random perturbation initially added √ to the φ0 component. Parameters use to obtain different plots are ν = 1, μ = −0.8, and ε = 2/ 5 and a a = 0.5; b a = 1.5; c a = −0.5; d a = −1.5. The meaning of the solid, dashed, and dotted curves is the same as in Fig. 11.2. Reprint from Ref. [1], Copyright 2022, with permission from American Physical Society

which illustrates the qualitative mechanism of the parametric instability. One can easily show that the zero equilibrium point (χ−1 , χ+1 ) = (0, 0) of the linear equations (11.16) is unstable through a double eigenvalue in the region of a < −1/4. By means of direct numerical simulations of Eqs. (11.3), we have checked the stability of the single-component polar soliton (11.13). In these numerical simulations, we have added at time t = 0 a small random perturbation with values distributed uniformly between 0 and 0.03 in components φ+1 and φ−1 . The result of this numerical investigation showed that the polar soliton (11.13) is always unstable, as shown in Fig. 11.1. Figure 11.1d depicts the time evolution of the following total number N0 and N±1 of atoms of different components of the condensate under consideration: +∞ +∞ 2 |φ0 (x, t)| d x, N±1 = |φ±1 (x, t)|2 d x. N0 = −∞

−∞

(11.17)

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11 Engineering Matter-Wave Solitons in Spinor …

Fig. 11.4 The same as in Fig. 11.2, but generated with the three-component polar soliton solution given by Eq. (11.24). Different √ plots are generated with the following numerical values of parameters ν = 1, μ = −0.8, ε = 2/ 5, and a = 0.5 in (a), a = 1.5 in (b), a = −0.5 in (c), and a = −1.5 in (d). The meaning of the solid, dashed, and dotted curves is the same as in Figs. 9.2 and 9.3. Reprint from Ref. [1], Copyright 2022, with permission from American Physical Society

11.2.3 Two-Component Polar Solitons We now turn our attention to two-component polar solitons. By setting ν = +1 and considering a zero φ0 (x, t), we arrive to the following two-component polar solitons: φ0 (x, t) = 0, φ+1 (x, t) = ±φ−1 (x, t) =

√

−μ

exp [−iμt] √ . cosh −μx

(11.18)

In the case of the two-component polar solitons (11.18), the corresponding BdG equations for small perturbations in the φ0 component reads     ∂ ∂ 2 χ0 2μ  (1 + a) χ0 + aχ0∗ . √ i + μ χ0 = − 2 + 2 ∂t ∂x −μx cosh

(11.19)

As in the case of one-component polar solitons, the instability of solution (11.18) is possible due to the parametric gain produced by the last term of Eq. (11.18). Similar

11.2 Exact Analytical One-, Two-, and Three-Component Soliton Solutions

381

to the case of Eq. (11.14), this instability can be well understood by studying Eq. (11.19) at x = 0; setting x = 0, Eq. (11.19) turns to the following equation which illustrates the qualitative mechanism of the parametric instability: i

  dχ0 = μ (1 + 2a) χ0 ± 2aχ0∗ . dt

(11.20)

It is obvious that the zero equilibrium point χ0 = 0 of Eq. (11.20) for either sign ± is unstable in the region a < −1/4. Following the same way as for the one-component polar soliton (11.13), we can test the stability of the soliton (11.18) by means of direct numerical simulations of Eqs. (11.3) with a small uniformly distributed random perturbation added to the φ0 component. This numerical study shows that the twocomponent polar soliton (11.18) is unstable in the region of |a| ≥ 1, as shown in Fig. 11.2, and it is stable at |a| < 1 (the corresponding figure is not displayed here). It is important to notice here that the above results found numerically on the soliton stability can be explained analytically in the special case when a = +1. Indeed, writing the perturbation in its canonical form χ0 (x, t) ≡ χ1 (x, t) + iχ2 (x, t) , and looking for a χ1,2 (x, t) as χ1,2 (x, t) = U1,2 (x) exp [σ t] where σ stands for the instability growth rate, we arrive at the following differential system for the eigenfunctions U1 and U2 

 d2 2μ (1 + 2a) √  U1 , − σ U2 = − 2 − μ + dx cosh2 −μx  −σ U1 = −

2



(11.21)

d 2μ √  U2 . −μ+ dx2 cosh2 −μx

At the limit σ = 0, system (11.21) decouples, and each equation of the system becomes an explicitly solvable and produces a series of zero eigenvalue at a = an ≡ n (n + 3) /4, n = 0, 1, 2, . . . . Note that system (11.21) becomes symmetric at point a0 = 0 when σ = 0. The zero crossings at other points an = n (n + 3) /4 with vanishing σ implies stability changes. In particular, at the critical point a1 = 1, the two-component polar soliton (11.18) is destabilized as observed in the simulations; this destabilization corresponds to eigenfunctions  √ −μx 1 √  , U2 (x) = √ . U1 (x) = cosh −μx cosh −μx tanh

(11.22)

382

11 Engineering Matter-Wave Solitons in Spinor …

Fig. 11.5 The evolution of the three-component polar soliton generated with the use of the exact analytical soliton solution given by Eq. (11.25) with initially added random perturbations. Parameters used here are ν = 1, μ+1 = −1.21, μ−1 = −0.25, and a = 0.5 in (a, c) or a = −0.5 in (b, d). In cases (a) and (b), the random perturbation was added to the φ0 component, and in cases (c) and (d) it was added to components φ±1 . The curves labelled by 1, 2, 3, and 4 represent, respectively, the evolution of norms N+1 , N−1 , N0 , and N defined by Eqs. (11.17) and (11.5). Reprint from Ref. [1], Copyright 2022, with permission from American Physical Society

Each of critical points an with n > 1 implies additional destabilizations through the emergence of new unstable eigenmodes of the already unstable soliton. It is important to notice here that the destabilization of the two-component polar soliton at a = −1, observed in the simulations, can be explained by means of the bifurcation theory.

11.2.4 Three-Component Polar Solitons Now, we turn to the search of three-component polar solitons when φ0 φ±1 = 0. Setting ν = +1, one family of the three-component solitons of the polar type is found to be (11.23)

11.2 Exact Analytical One-, Two-, and Three-Component Soliton Solutions

(φ−1 , φ0 , φ+1 )T =

T,  √ exp [−iμt]  √  ∓ε, 1 − 2 , ±ε −μ cosh −μx

383

(11.23)

where ε is a free real parameter to be taken as −1 < ε < +1. Exactly as the above found one- and two-component polar solitons, the three-component polar solitons (11.23) contains one parameter, the chemistry parameter μ < 0 and does not explicitly depend on parameter a. Another three-component polar soliton solution, independent of a and containing one parameter, μ is found to be (φ−1 , φ0 , φ+1 )T =

T  √ exp [−iμt]  √  ±ε, i 1 − ε2 , ±ε . −μ cosh −μx

(11.24)

Comparing the three-component polar soliton solutions (11.23) and (11.24), it is seen that the two solutions differ by the sign of the φ±1 components and a phase shift of π/2 in the φ0 component. It is possible to build another family of one parameter three-component polar soliton solutions which explicitly depend on parameter a. An example of such species of three-component polar solitons is found to be ⎞T ⎛  √ √ μ μ exp [−iμt] +1 −1 √  ⎝1, i − , 1⎠ , (φ−1 , φ0 , φ+1 )T = −μ±1 μ±1 cosh −μx

(11.25)

where μ±1 are two free negative parameters to be taken from the condition 2 √ √ 2μ = 0. −μ+1 + −μ−1 + ν+a

(11.26)

It follows from Eq. (11.26) and the negativity of the chemical potential μ that ν + a > 0, for example, ν = −1, corresponding to repulsive spin-independent interaction. Solving Eq. (11.26) in the chemical potential μ, and inserting the result in Eq. solution (11.25), we find that the three-component polar soliton solution (11.25) contains exactly two free parameters, μ−1 and μ+1 . On the other hand, each of the two three-component polar soliton solutions (11.23) and (11.24) also contains exactly two free parameters, μ and ε. Thus, we have built three families of three-component polar soliton solutions each of which contains exactly two free parameters; two of these three families of three-component polar soliton solutions do not contain explicitly parameter a, while one of them contains a explicitly. The stability of all the above found three-component polar soliton solutions was tested in direct simulations and reported in Ref. [1]. For the direct numerical simulations for testing the stability of solutions (11.23) and (11.24), a small random perturbation was performed only in the φ0 component, and it was found that both these types are unstable, as shown in Figs. 11.3 and 11.4.

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11 Engineering Matter-Wave Solitons in Spinor …

For testing the stability of the three-component polar soliton solution given by Eq. (11.25), small random perturbations were performed in all the three components; it has been found that this three-component polar soliton (11.25) is completely stable. An example of the obtained numerical simulations is reported in Fig. 11.5. It is seen from plots of Fig. 11.5 that the added small perturbation on each soliton component induces only small oscillations of the amplitudes of the corresponding soliton component.

11.2.5 Multistability of Solitons As we have seen in the above analysis, the one-component ferromagnetic soliton (11.9), the two-component polar solitons (11.18) in the regions of −1 < a < +1 , as well as the three-component polar soliton (11.25) may all be stable in the same parameter region. In the situation when these three solitons are stable in the same parameter region, one may identify which solitons are “more stable” and “less stable.” For this aim, one may fix the soliton’s norm (11.5) and compare respective values of the Hamiltonian (11.4) for these three solutions. In this comparison, the ground-state solution is assumed to correspond to a minimum of H at given N . Substituting each derived solutions in Eq. (11.4) reveals that all the one-, two-, and two first three-component polar solitons (11.13), (11.18), (11.23), and (11.24) which do not explicitly contain a produce identical relations between N , μ, and H : √ 1 N (μ) = 4 −μ, H (N ) = − N 3 . 48

(11.27)

The situation is not the same for the one-component ferromagnetic soliton (11.9) for which N (μ) and H (N ) are given in Eq. (11.10) and differ from those given in Eq. (11.27). Computing N (μ) and H (N ) in the case of the stable three-component polar soliton (11.25) which depends explicitly on a, we found that μ, N , and H are related by exactly the same form as Eq. (11.10) for the FM soliton. Comparing expression (11.27) to expression (11.10), we find that the onecomponent FM soliton (11.9) and the stable three-component polar soliton given by (11.25) simultaneously provide the minimum of energy when ν = +1 and a > 0, which corresponds to attractive spin-independent and attractive spin-exchange interactions between atoms. Each of the two species (11.9) and (11.25) in the cases v = +1 and a > 0 plays the role of the ground state in its own class of the solitons. In the above studies, we found that the two-component soliton (11.18) is also stable in the region of 0 < a < 1, but it corresponds to higher energy; this means that the two-component soliton (11.18) in region 0 < a < 1 represents a metastable state. Also, the FM soliton (11.9) and the three-component polar soliton (11.25) exist simultaneously when ν = −1 and a > 1 under the condition that ν + a is positive. Because other soliton solutions (11.13) , (11.18) , (11.23) , and (11.24) do not exist

11.2 Exact Analytical One-, Two-, and Three-Component Soliton Solutions

385

Fig. 11.6 The evolution of the two-component polar soliton on the CW background, obtained with the help of the exact analytical soliton solution given by Eq. (11.29), under the action of initial random perturbations added to the φ0 and φ±1 components. Different parameters used in various plots are ν = +1, q = −1/2, and μ = −1. Reprint from Ref. [1], Copyright 2022, with permission from American Physical Society

when ν = −1, the two soliton species (11.9) and (11.25) provide for the energy minimum. The situation ν = +1 and −1 < a < 0 correspond to attractive spin-independent and repulsive spin-exchange interactions in Eq. (11.7); in this situation, the Hamiltonian (11.27) is smaller than the competing one (11.10) for the single-component FM soliton. In this situation, we then conclude that the two-component polar soliton (11.18) plays the role of the ground state. It has been showed that also the single-component soliton (11.9) and the three-component polar soliton (11.25) are also stable in the region ν = +1 and −1 < a < 0, but correspond to greater energy; these two solitons (11.9) and (11.25) then represent metastable states in this region. The above studies have showed that there are no stable solitons when ν = +1 and a < −1. Using the dependence N (μ) for each solution family of found solitons and employing the well-known known Vakhitov-Kolokolov criterion which gives the necessary stability condition for a soliton family supported by a self-attractive nonlinearity (this criterion reads d M/dμ < 0) [12, 13], one can state an additional characteristic of the soliton stability. Although this criterion guarantees that the soliton is always stable against perturbations with real eigenvalues, it does not say anything on the oscillatory perturbations modes that appear due to the complex eigenvalues. Computing d M/dμ from Eqs. (11.10) and (11.27), we find that d M/dμ < 0. This means that the above solitons may be unstable only against temporal oscillating perturbations that grow in time. This features/behaviors of unstable solitons are illustrated Figs. 11.1, 11.2, 11.3 and 11.4 generated with direct simulations.

11.2.6 Finite-Background Solitons Let us now study the evolution of solitons embedded on a non-vanishing continuous wave (CW) background. For simplicity, we focus our attention to the special case when ν = 1 and a = −1/2. In this special case, we can build exact analytical solutions for solitons sitting on a non-vanishing CW background. First of all, we note

386

11 Engineering Matter-Wave Solitons in Spinor …

Fig. 11.7 The same as in Fig. 11.6, but for the finite-background soliton (11.30). Parameters are ν = a = 1 and μ = −0.36. Reprint from Ref. [1], Copyright 2022, with permission from American Physical Society

that in this special case, Eq. (11.3) admits the CW solution  μ (φ−1 , φ0 , φ+1 ) = − (1, 0, 1)T exp [−iμt] 2 T

(11.28)

Using the CW solution (11.28) as the seed solution and letting φ0 = 0 yield the following two-component polar soliton solutions with a non-vanishing CW background attached on it: (φ−1 , φ0 , φ+1 )T  T √ 1 1 1 1 √  , 0, √ ± √  = −μ √ ∓ exp [−iμt] . 2 cosh −μx 2 cosh −μx (11.29) When ν = a = 1, it is also possible to use the seed solution  1 μ − (1, 1, 1)T exp [−iμt] (φ−1 , φ0 , φ+1 ) = 2 2 T

to build the following family of three-component polar soliton solution with a CW background attached on it:  T φ−1 , φ0 , φ+1 T  √ 1 −μ 1 1 1 1 1 √ , √ ∓ √ , √ ± √  = √ ± 2 2 cosh −μx 2 cosh −μx 2 cosh −μx exp [−iμt] .

(11.30)

Performing direct simulations to Eq. (11.3), we have studied the stability of the two two- and three-component polar solitons (11.29) and (11.30) by simultaneously perturbing the φ0 and φ±1 components by adding at the initial time t = 0 small

11.3 The Darboux Transform and Nonlinear Development of Modulational Instability

387

Fig. 11.8 The nonlinear development of the modulation instability, generated with the use of exact analytical soliton solution (11.41), under conditions k = 2η and αc2 + βc2 > ξ 2 . Parameters are k = 0.6, η = 0.3, ξ = 1, α = β = γ = exp (−8), αc = 1.2, and βc = 0. Reprint from Ref. [1], Copyright 2022, with permission from American Physical Society

uniformly distributed random perturbations. Examples of obtained results are illustrated in Figs. 11.6 and 11.7 showing the evolution of respectively two- and threecomponent polar solitons embedded on a non-vanishing CW background. These two figures show that both the two- and three-component polar soliton solutions are unstable, although the character of the instability is different. It is seen from plots of Fig. 11.6 [11.7] that the soliton’s core seems stable [features oscillatory instability], but the CW background appears to be modulationally unstable [stable under the modulation]. Because the soliton’s core is attached to its CW background, we conclude that both the two- and three-component solitons are unstable.

11.3 The Darboux Transform and Nonlinear Development of Modulational Instability In this Section, we focus our attention on the integrable case of Eq. (11.1) with c2 + c0 = 0, when c0 < 0, which in terms of Eq. (11.3) corresponds to ν = a = 1. This special case which corresponds to the attractive interactions, makes it possible to develop deeper analysis of MI. We will also assumed that the physically constraint (11.8) imposed on the scattering lengths of collisions between atoms is satisfied. Under all these assumptions, Eqs. (11.3) can be reduced to the following completely integrable 2 × 2 matrix NLS equation [11]   ∂Q ∂ 2 Q φ+1 φ0 † + . + 2QQ Q = 0, Q ≡ i φ0 φ−1 ∂t ∂x2

(11.31)

Darboux transform (DT) for the NLS equation (11.31) can be derived from the respective Lax pair [11, 14] − → − → ∂ − → ∂ − → = U , = V ; ∂x ∂t in Eq. (11.32), U and V are defined as

(11.32)

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11 Engineering Matter-Wave Solitons in Spinor …

U = λJ + P, V = 2iλ2 J + 2iλP + iW     I 0 0 Q , J= ,P = 0 −I −Q† 0   QQ† ∂ Q W = ∂ † ∂x † , Q −Q Q ∂x

(11.33) (11.34) (11.35)

− → where  = (1 , 2 )T is the matrix eigenfunction corresponding to λ, 1 and 2 are 2 × 2 matrices, λ is the spectral parameter, and I and 0 are respectively the unit and zero matrices. It is evident that the Lax pair (11.32) is an over-determined linear system and Eq. (11.31) is tantamount to the compatibility condition system (11.32), ∂U ∂V − + UV − VU = 0. ∂r ∂x By virtue of the Lax pair (11.32), we introduce a transformation in the form of   − → − → λ1 I 0  −1  = (λ − S)  , S = DD , = , 0 λ1 I

(11.36)

D being a nonsingular matrix satisfying the first order partial differential equation ∂D = JD + PD. ∂x Next, letting

  − →  ∂ − →  0 Q1  = U  , U = λJ + P1 , P1 ≡ , −Q†1 0 ∂x

(11.37)

P1 = P + JS − SJ.

(11.38)

yields

Moreover, the following involution property of the above linear equations is satisfied. − → Involution property: If  = (1 , 2 )T is an eigenfunction with the eigenvalue λ,   T then −2∗ , 1∗ will be an eigenfunction with the eigenvalue −λ∗ . Taking then D as follows     λI 0 1 −2∗ , = , D= 2 1∗ 0 −λ∗ I we obtain



I0 S=λ 0I



   −S11 S12  ∗ , + λ+λ S21 −S22

the matrix elements of S being given by

11.3 The Darboux Transform and Nonlinear Development of Modulational Instability

389

    −1 −1 −1 −1 S11 = I + 1 2−1 1∗ 2∗ , S12 = 2 1−1 + 1∗ 2∗ ,     −1 −1 −1 −1 S21 = 2∗ 1∗ + 1 2−1 , S22 = I + 2 1−1 2∗ 1∗ . The Darboux transform for Eq. (11.31) finally follows from Eq. (11.38), taking the form of   (11.39) Q1 = Q+2 λ + λ∗ S12 . Combining Eqs. (11.39) and (11.32), we deduce that Q1 defined by Eq. (11.39) is a new solution for Eq. (11.42), as soon as the seed solution Q of the NLS equation (11.31) is known. It is important to note that if we take the trivial zero state as the seed solution, solution (11.39) will be just a one-soliton solution of the NLS equation (11.31). Further, using the solution Q1 as the new seed solution of Eq. (11.31), we derive from Eq. (11.39) another new solution of Eq. (11.31), which now will be a two-soliton solution. Following the same procedure, we can use the (n − 1)-soliton solution to build the next solution, the n-soliton solution of the NLS equation (11.31). Thus, we have shown how to apply the DT to construct the multisoliton solution of the NLS equation (11.31). In the following, we associate to the NLS equation (11.31) nonzero boundary conditions (BCs) and consider the corresponding solution. First of all, we note that Eq. (11.31) admits the following CW solution with constant densities, appearing as its simplest solution with nonzero BCs:  Qc = −Ac exp [iϕc ] , Ac ≡

βc αc αc −βc



    , ϕc ≡ kx + 2 αc2 + βc2 − k 2 t,

(11.40) where Ac with real constant elements αc and βc (with |αc | + |βc | > 0) is the matrix amplitude (carrier amplitude) and k is the wavenumber (carrier wavenumber). In solution (11.40), we can easily verify that the constant densities of components φ±1 are equal, while their signs are opposite. Taking the CW solution with constant densities Qc as the seed solution and applying the above DT, we solve the linear system (11.32) and employ Eq. (11.39) to obtain the following new family of solutions of the NLS equation (11.31):  −1  A exp (iϕc ) , Q1 = Ac + 4ξ I + AA∗

(11.41)

where  −1  1 1 Ac  exp [θ − iϕ] + I , A =  exp [θ − iϕ] + Ac κ κ θ = M I x + [2ξ M R − (k + 2η) M I ] t, ϕ = M R x − [2ξ M I + (k + 2η) M R ] ,    M = (k + 2iλ)2 + 4 αc2 + βc2 = M R + i M I ,

(11.42) (11.43) (11.44) (11.45)

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11 Engineering Matter-Wave Solitons in Spinor …

κ being the spectral parameter given as κ≡

1 (ik − 2λ + i M) with λ = ξ + iη, 2

and  being an arbitrary 2 × 2 symmetric matrix defined as  =

βα αγ

 .

When ξ = 0, solution (11.41) reduces to the CW (11.40). On the other hand, if Ac = 0, solution (11.41) will be, as we have described in the previous Section, onesoliton solution of the NLS equation (11.31): Q1 = where

    4ξ 1 exp (−θ1 ) + σ y ∗1 σ y exp (θs ) det 1 exp (−2θs ) + 1 + exp (2θs ) |det 1 |2

exp (iϕs ) ,

(11.46)

  θs = 2ξ (x − 4ηt) − θ0 , ϕs = 2ηx + 4 ξ 2 − η2 t,

θ0 being a free real constant which determines the initial position of the soliton, σ y is the Pauli matrix, and 1 is the polarization matrix [15] defined as    −1/2 β1 α1 . ≡ 1 = 2 |α|2 + |β|2 + |γ |2 α1 γ1 We can then conclude that solution (11.41) describes one-soliton embedded on a non-vanishing CW background. It is important to note that the three-component polar soliton solution (11.30) found in the previous Section is not a special case of solution (11.41); indeed, the background fields in components φ±1 in solution (11.30) have the same sign, which is not the case for solution (11.41). Let us note that the exact solutions (11.41) can be used to describe the onset and nonlinear development of MI of the continuous wave states. A necessary condition for this situation to happen is that different parameters are taken from the condition M I = 0, which is possible only when Im(k + 2iλ) = 0. This leads to k = 2η. To observe the instability of the CW background, we must have αc2 + βc2 − ξ 2 > 0. It is evident that when k = 2η and αc2 + βc2 − ξ 2 > 0, solution (11.41) will be periodic in the spatial coordinate and aperiodic in the temporal one. Since under these condition θ will not depend on x, we conclude that the solution features no localization. It is important to note that under the above conditions, we will have M I = 0 and   M R2 = 4 αc2 + βc2 − 4ξ 2

(11.47)

11.3 The Darboux Transform and Nonlinear Development of Modulational Instability

391

Fig. 11.9 Evolution plots of the density distribution in the soliton, generated with the use of exact analytical solution (11.41) by adding a random time-varying perturbation of the nonlinear coupling constant a, with the perturbation amplitude ±5%. Different plots are generated with the following numerical values of parameters: ν = a = 1 (as concerns the unperturbed value of a), ξ = 1, η = −0.03, and θ0 = 0. a The ferromagnetic soliton, obtained with α1 = 0.48, β1 , and γ1 determined by conditions α12 = β1 γ1 and 2 |α1 |2 + |β1 |2 + |γ1 |2 = 1; b the polar soliton, generated with α1 = 0.53 and β1 = 0.43, γ1 being determined by condition 2 |α1 |2 + |β1 |2 + |γ1 |2 = 1. In different panels, solid and dotted curves show respectively the exact solutions and the perturbed ones, generated by numerical simulations. Reprint from Ref. [1], Copyright 2022, with permission from American Physical Society

will not vanish. To show in details that solution (11.41) under the above conditions will develop a particular mode of the nonlinear development of MI, we can take a special case with  = εE, where ε is a small amplitude of the initial perturbation added to the CW background and E is a matrix with all elements equal to 1. This choice of  will help us to initiate the onset of MI. Linearizing the initial value of solution (11.41) with respect to the small parameter , we arrive to

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11 Engineering Matter-Wave Solitons in Spinor …

  Q1 (x, 0) ≈ Ac ρ − χ1 E cos [M R x] − εχ2 σ z σ x Ac exp [i M R x] exp [ikx] , (11.48) where   ρ ≡ 1 − ξ (2ξ + i M R ) / αc2 + βc2 with |ρ| = 1,   χ1 ≡ ξ M R (2iξ − M R ) / αc2 + βc2 ,    χ2 ≡ βc χ1 / αc2 + βc2 , M R = 2 αc2 + βc2 − ξ 2 , and σ x,z are the Pauli matrices. It is obvious that the approximation (11.48) consists of a non-vanishing CW background, namely, its first term, two spatially modulated terms (perturbations). Equation (11.48) therefore means that a small periodic perturbation in the CW background may lead to modulational instability. A comparison of the exact solution given by Eq. (11.41) in the special case when k = 2η and αc2 + βc2 − ξ 2 > 0 with the results obtained from direct numerical simulations of Eq. (11.38) under a given initial condition, we obtain that the two solutions are very close each to another, both showing the development of the MI. For a better understanding, we have used the exact solution (11.41) when k = 2η and αc2 + βc2 − ξ 2 > 0 to depict in Fig. 11.8 the nonlinear development of the modulational instability. The nonlinear MI shown in Fig. 11.8 means that the condensate periodically transfers atoms from the spin state m F = 0 into the spin states m F = ±1 and vice versa. In the general case when a = 1 (that is, 2g0 = −g2 ), one can directly analyze the onset of the MI from Eqs. (11.3) using the linear stability analysis [16]. In this way, one should consider the CW solution (11.40) with k = 0 and perturb it as follows      = (Ac + B) exp 2iν αc2 + βc2 t , B = Q



 b+1 (x, t) b0 (x, t) , b0 (x, t) b−1 (x, t)

where matrix B with elements b±1 (x, t) and b0 (x, t) is a weak matrix of perturbations. Using the linear stability method and denoting by K and ω the perturbation wavenumber and frequency, respectively, yield the following two branches of the dispersion relation:   ω2 = K 2 K 2 − 4aαc2 − 4aβc2 ,   ω2 = K 2 K 2 − 4ναc2 − 4νβc2 .

(11.49) (11.50)

For the MI to occur, at least one of the two ω2 must be a negative quantity, which is possible as soon as at least one of the following two condition     K 2 < 4a αc2 + βc2 , K 2 < 4ν αc2 + βc2

(11.51)

holds. Note that if aν < 0, then one and only one of conditions (11.51) is satisfied. This means that, for example, the MI may occur in the case of the repulsive spin-independent interactions in Eqs. (11.3) when a is positive. Also the MI may

References

393

occur if a > 0 and ν > 0, as for example, when a = v = 1 (case of attractive spinindependent interactions in Eqs. (11.3)) with positive a. For exploring the stability of the exact analytical soliton solutions (11.46) of both the ferromagnetic and polar types rise up the question of the soliton to survive random deviations of the coupling constant a from the value a = 1 corresponding to the integrability of the system. Figure 11.9 shows the evolution of the density distribution in the soliton generated from the exact soliton solution (11.46) by adding a random time-dependent perturbation of the coupling constant a. As we can see from plots of Fig. 11.9, solitons of both types are robust against random changes of the nonlinear coupling constant a.

11.4 Conclusion Based on a system of (non)integrable three nonlinearly coupled GP equations, we have presented in this chapter soliton states of spinor BEC with atomic spin F = 1. In the case when the model equations are nonintegrable, various types of exact analytical soliton solutions are reported, including single-component ferromagnetic soliton solutions, one-, two-, and three-component polar soliton solutions. Their linear stability are checked by direct numerical simulations and, in some special cases, in an analytical form by means of BdG equations. The global stability of the solitons is analyzed too. In the special case when the model system is integrable, we employed the DT to present methodically the construction of multisoliton solutions that describe multisoliton embedded on a non-vanishing CW background. Their stability against a random time-dependent perturbation of the nonlinear coupling parameter a is investigated numerically and analytically. We found that the onesoliton solutions of both the FM and polar types are robust against random changes parameter a.

References 1. L. Li, Z. Li, B.A. Malomed, D. Mihalache, W.M. Liu, Exact soliton solutions and nonlinear modulation instability in spinor Bose-Einstein condensates. Phys. Rev. A 72, 033611 (2005) 2. W.M. Liu, E. Kengne, Schrödinger Equations in Nonlinear Systems (Springer Nature Singapore Pte Ltd., 2019) 3. S.W. Song, L. Wen, C.F. Liu, S.-C. Guo, W.M. Liu, Ground states, solitons and spin textures in spin-1 Bose-Einstein condensates. Front. Phys. 8(3), 302–318 (2013) 4. T.-L. Ho, Spinor Bose condensates in optical traps. Phys. Rev. Lett. 81, 742–745 (1998) 5. Y. Kawaguchi, M. Ueda, Spinor Bose-Einstein condensates. Phys. Rep. 520, 253–381 (2012) 6. D.M. Stamper-Kurn, M. Ueda, Spinor Bose gases: symmetries, magnetism, and quantum dynamics. Rev. Mod. Phys. 85, 1191 (2013) 7. S.W. Song, L. Wen, C.F. Liu, S.C. Guo, W.M. Liu, Ground states, solitons and spin textures in spin-1 Bose-Einstein condensates. Front. Phys. 8(3), 302–318 (2013)

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8. H. Pu, C.K. Law, S. Raghavan, J.H. Eberly, N.P. Bigelow, Spin-mixing dynamics of a spinor Bose-Einstein condensate. Phys. Rev. A 60, 1463 (1999) 9. G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995) 10. G.P. Agrawal, Modulation instability induced by cross-phase modulation. Phys. Rev. Lett. 59, 880–883 (1987) 11. J. Ieda, T. Miyakawa, M. Wadati, Exact analysis of soliton dynamics in spinor Bose-Einstein condensates. Phys. Rev. Lett. 93, 194102 (2004) 12. N.G. Vakhitov, A.A. Kolokolov, Stationary solutions of the wave equation in a medium with nonlinearity saturation. Radiophys. Quantum Electron. 16, 783 (1973) 13. L. Bergé, Wave collapse in physics: principles and applications to light and plasma waves. Phys. Rep. 303, 259 (1998) 14. T. Tsuchida, M. Wadati, The coupled modified Korteweg-de Vries equations. J. Phys. Soc. Jpn. 67, 1175 (1998) 15. J. Ieda, T. Miyakawa, M. Wadati, Exact analysis of soliton dynamics in Spinor Bose-Einstein condensates. Phys. Rev. Lett. 93, 194102 (2004) 16. N.P. Robins, W. Zhang, E.A. Ostrovskaya, Y.S. Kivshar, Modulational instability of spinor condensates. Phys. Rev. A 64, 021601(R) (2001) 17. L. Wen, Q. Sun, Y. Chen, D.-S. Wang, J. Hu, H. Chen, W.M. Liu, G. Juzeli¯unas, B.A. Malomed, A.-C. Ji, Motion of solitons in one-dimensional spin-orbit-coupled Bose-Einstein condensates. Phys. Rev. A 94, 061602(R) (2016) 18. Y.-J. Lin, K. Jiménez-García, I.B. Spielman, Spin-orbit-coupled Bose-Einstein condensates. Nature (London) 471, 83 (2011) 19. V. Galitski, I.B. Spielman, Spin-orbit coupling in quantum gases. Nature 494, 49–54 (2013) 20. H. Zhai, Degenerate quantum gases with spin-orbit coupling: a review. Rep. Progr. Phys. 78, 026001 (2015) 21. L.W. Cheuk, A.T. Sommer, Z. Hadzibabic, T. Yefsah, W.S. Bakr, M.W. Zwierlein, Spininjection spectroscopy of a spin-orbit-coupled Fermi gas. Phys. Rev. Lett. 109, 095302 (2012) 22. T. Busch, J.R. Anglin, Motion of dark solitons in trapped Bose-Einstein condensates. Phys. Rev. Lett. 84, 2298–2301 (2000) 23. S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Zh. Eksp. Teor. Fiz. 65, 505 (1973); Sov. Phys. JETP 38, 248 (1974) 24. S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G.V. Shlyapnikov, M. Lewenstein, Dark solitons in Bose-Einstein condensates. Phys. Rev. Lett. 83, 5198–5201 (1999)

Chapter 12

Engineering Magnetic Solitons in Nonlinear Systems

Abstract In this chapter, we consider a number of mathematical models which describe various nonlinear phenomena, including the dynamics of dissipative magnetic matter-wave solitons in a spinor polariton Bose-Einstein condensate, nonlinear magnetization dynamics (NMDs) of the classical ferromagnet with two single-ion anisotropies in an external magnetic field, and linear and nonlinear magnetization dynamics in the presence of spin-polarized current. With the use of these mathematical models, we investigate (i) the dynamics of dissipative magnetic polariton soliton in multi-component BECs, (ii) the effects of spin transport on nonlinear excitations in a ferromagnetic nanowire with non uniform magnetization, and (iii) the effects of a spin-polarized current on the nonlinear excitation of the magnetization in the ferromagnetic metal nanowire.

12.1 Dynamics of Dissipative Magnetic Matter-Wave Solitons in a Spinor Polariton Bose-Einstein Condensate In the present Section, we consider a driven-dissipative GP equation coupled to the rate equation to investigate the dynamics of dissipative magnetic matter-wave solitons in a spinor polariton Bose-Einstein condensate. We mainly discuss a scenario, different from the conventional single-channel one for dissipative solitons; in such a scenario, double balances rely on the presence of multiple collective excitation channels in open-dissipative quantum systems. With the help of this scenario, we prove, using exact analytical soliton solutions of the mathematical model under consideration, the existence of dissipative magnetic solitons (MSs) in a spinor polariton BEC [1].

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 E. Kengne and W. Liu, Nonlinear Waves, https://doi.org/10.1007/978-981-19-6744-3_12

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12.1.1 Introduction and Model Equations In most non-dissipative nonlinear systems such as Hamiltonian systems, a single balance between dispersion and nonlinearity generally gives rise to solitons [2]. Due to the presence of loss and/or gain of matter or energy in dissipative nonlinear systems, solitons usually exist within a finite lifetime [3, 4]. For dissipative nonlinear systems, dissipative solitons (solitons that do not decay with time) may be realize by a best balances of gain and dissipation, along with dispersion and nonlinearity [5]. In the realization of dissipative solitons, exact solutions of the model equation play important roles; indeed, exact analytical solitonlike solutions of the model equation can be employed for clarifying the bi-channel double balance of a dissipative MS, as shown in Fig. 12.1. Figure 12.1 is generated under the conditions that (i) the condensed polaritons with loss rate γC continuously replenish from reservoir polaritons at rate R; (ii) the reservoir polaritons decay at rate γ R while subjected to a uniform pumping P; (iii) the polaritons in the reservoir and condensate interact with each other modeled by the spin-independent interaction parameter g R ; (iv) there exist two excitation channels on top of the steady state, namely, the density and the spin polarization which are usually coupled; (v) nonlinear spin-polarization dynamics occurs in an isolated channel, leading to a soliton profile, when the gain balances loss and fixes the background density; the appearance of the soliton profile manifests the interplay of nonlinearity and dispersion. The aim of this Section is to employ the exact analytical solution of the model equation for investigating analytically the dynamics of dissipative MSs on top of a spinbalanced density background for spinor polariton Bose-Einstein condensates [6–9]. The dissipative magnetic solitons found in the present Section are differ from conventional dissipative solitons generated with the use of solutions of Gross-Pitaevski equation [10], as well as from dissipative solitons in open-dissipative polaritonic systems [11, 12]. The physical system to be studied in this Section is a spinor polariton BEC formed under uniform non-resonant pumping in a wire-shaped microcavity [13]. For the exciton-polariton BEC, we assume the order parameter ψ(x, t) to be a twocomponent complex vector with spin-up and spin-down wave functions ψ↑↓ = ψ1,2 : ψ(x, t) ≡ [ψ1 (x, t), ψ2 (x, t)]T [14, 15, 0-15]. Denoting by n R the density of the reservoir polaritons, the mathematical model that governs the evolution of the wave functions ψ1 and ψ2 is found to be the driven-dissipative GP equations coupled to the rate equation for n R [16, 17]: i

  2 ∂ 2 ∂ψ1 2 2 | |ψ | |ψ = − + g + g + V + D 1 12 2 R s ψ1 , ∂t 2m ∂ x 2

(12.1)

12.1 Dynamics of Dissipative Magnetic Matter-Wave Solitons in a Spinor …

397

Fig. 12.1 Schematics of bi-channel double-balance scenario for producing dissipative magnetic soliton in a spinor polariton BEC when the nonresonant pumping is taken into consideration. Reprint from Ref. [1], Copyright 2022, with permission from American Physical Society

i

with

  2 ∂ 2 ∂ψ2 2 2 | |ψ | |ψ = − + g + g + V + D 2 12 1 R s ψ2 , ∂t 2m ∂ x 2    ∂n R = P − γ R + R |ψ1 |2 + |ψ2 |2 n R , ∂t  VR = g R n R , Ds = i (Rn R − γC ). 2

(12.2) (12.3)

(12.4)

In system (12.1)–(12.3), m = 10−4 m e is the effective mass of polaritons, where m e is the free electron mass, g12 and g are respectively the constants of the two-body interactions for opposite-spin and same-spin polaritons which in this Section are assumed to be respectively negative and positive and satisfy the constraint |g12 |  g, g R is the spin-independent parameter of the interaction between condensates and reservoir polaritons. γC is the loss rate of the condensed polaritons, R is the rate of reservoir polaritons, whose gain and loss process is captured by Ds. P is the continuouswave pumping and γ R is the decay rate of the reservoir. In the following, the density  2 of spin-up (spin-down) components will be denote by n 1(2) = ψ1(2)  . Under the above restriction on g12 and g, the steady state condensate of system (12.1)–(12.3) is linearly polarized with a stochastic polarization direction [18]. For the realization of dissipative MSs on top of a spin-balanced density background, we assume two excitation channels corresponding to the density n = n 1 + n 2 and spin polarization n 1 − n 2 . Under all the above assumptions, we intend to build exact analytical vector T  soliton solutions ψ S ≡ ψ1S , ψ2S to Eqs. (12.1)–(12.3) that satisfies the condition D S ψ S = 0.

(12.5)

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12 Engineering Magnetic Solitons in Nonlinear Systems

12.1.2 Exact Analytical Solutions and Dissipative Magnetic Polariton Solitons in a Spinor Polariton Bose-Einstein Condensate To find exact analytical solutions of system (12.1)–(12.3) that generate dissipative magnetic polariton solitons, we propose the ansatz

ϕ  ϕg μ R n0  r 1 + δn 1 exp i + − t , 2 2 2 



 ϕ  ϕg μ R n0  r 1 + δn 2 exp i − + − t , 2 2 2 

ψ1 =

ψ2 =

(12.6)

where n 0 is a constant density, μ R = g R γC /R, δn 1,2 (η = x − υt) is defined from the system n 1 = n 0 (1 + δn 1 )/2 and n 2 = n 0 (1 − δn 2 )/2 and satisfy the boundary conditions lim δn 1(2) (η) = 0, η→±∞

ϕr and ϕg label respectively the relative and global phases of components ψ1 and ψ2 and satisfy the BC ∂ ϕr (g) (η) = 0. lim η→±∞ ∂η Under the above BCs, Eq. (12.6) gives an exact analytical soliton solution of system (12.1)–(12.3) as soon as n 0 = γPC − γRR and n R = γC /R, and √ 1 − U2

,

√ δn 1 = δn 2 = (12.7a) cosh ξηs 1 − U 2

√

⎤ ⎡ sinh ξηs 1 − U 2 ⎦+ π, (12.7b) ϕr = arctan ⎣ U 2

⎤

√ ⎡√ √  1 − U 2 tanh ξηs 1 − U 2 2 1 − U ⎦ − arctan ϕg = − arctan ⎣ , (12.7c) U U with respectively the dimensionless velocity U and the spin healing length ξs defined respectively as  U =υ

 2m , ξs = / 2mn 0 (g − g12 ). n 0 (g − g12 )

12.1 Dynamics of Dissipative Magnetic Matter-Wave Solitons in a Spinor …

399

Fig. 12.2 Properties of a moving dissipative magnetic soliton. a Spatiotemporal density distribution generated with the use of Eq. (12.7a). b Stokes parameters obtained with the help of Eq. (12.8). c Distribution of density of n 1 = |ψ1 |2 and n 2 = |ψ2 |2 , relative phase ϕr and global phase phase ϕg at dimensionless time of t/τ = 15. Panel (c) compares exact analytical solutions (solid lines) to numerical solutions (dashed lines) of Eqs. (12.1)–(12.3) (see main text). To generate all these plots, different parameters are chosen as γC = 0.01 ps−1 , R = 0.01 ps−1 mm2 , g = 0.01m e Vµm 2 , P = 0.41 ps−1 mm2 , γ R /γC = 40, g12 /g = −0.1, and U = 0.6. Reprint from Ref. [1], Copyright 2022, with permission from American Physical Society

The spatiotemporal density distribution of a moving soliton with U = 0.6 is illustrated in Fig. 12.2a. It is seen from this figure that the wave shape is preserved during the wave motion. Figure 12.2c shows the soliton profile at t/τ = 15 when τ = /(g − g12 )n 0 . We can see from curves of Fig. 12.2c that the relative phase exhibits an exact π -jump, which analytically can be obtained as lim ϕr − lim ϕr = π.

η→+∞

η→−∞

Dashed lines in Fig. 12.2c correspond to the numerical solutions of system (12.1)– (12.3) generated by taking the initial order parameter from Eqs. (12.6)–(12.7c) for t = 0 and by letting n R (0) = γC /R. Plots of Fig. 12.2c show perfect agreement between the analytical and numerical solution at time t/τ = 15 . Also, we have numerically checked the stability of the MS by keeping n 0 = n 1 (0) + n 2 (0) fixed and perturbing n 1 (0) − n 2 (0) from Eq. (12.7a). Solution (12.6)–(12.7c) is a MS; in other words, this solution describes a localized spin polarization sitting on top of a linearly polarized background state. We can characterize the polarization property of the polariton soliton associated with the soliton solution (12.6)–(12.7c) by means of the Stokes parameters [19]. Computing the distribution of respectively the linear and the circular polarization degrees of from the soliton solution (12.6)–(12.7c) , we arrive to     |ψ1 |2 − |ψ2 |2 2 ψ1∗ ψ2 , Sz (η) = . Sx (η) = n0 n0

(12.8)

As we can see from different curves in Fig. √12.2b, the moving soliton is strongly elliptically polarized for η < lω (here, lω = 1 − U 2 is the soliton width). It is also seen from Fig. 12.2b that, the maximum value of the circular polarization degree

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12 Engineering Magnetic Solitons in Nonlinear Systems

√ Sz (0) = 1 − U 2 , is reached at the soliton center. Away from the center, the soliton becomes linearly polarized. Computing the total spin polarization degrees of the soliton, we arrive to  dηSz (η) = π ξs , which does not depend on the velocity U . Following the strategy elaborated in works [9, 17, 20], we have computed the energy E of the soliton and found E=



 2 2   ∂ 12 ψ + g−g d x [n 1 − n 2 ]2 d xψ † − 2m ∂x2 4  12 d x [n − n 0 ]2 . + g+g 4

(12.9)

To show that the above magnetic polariton soliton (12.6)–(12.7c) preserves its energy, we employ Eqs. (12.1)–(12.3) along with soliton solutions (12.6)–(12.7c) and obtain dE = −2Re dt



  Ds

∂ψ1∗ ∂ψ2∗ + ∂t ∂t



 dx

= 0.

(12.10)

Since the moving magnetic polariton soliton (12.6)–(12.7c) preserves its energy, we conclude that (12.6)–(12.7c) is a dissipative soliton. In the case studied in this Section, the constant density n 0 = γPC − γRR of the dissipative magnetic soliton associated with solution (12.6)–(12.7c) is enabled by the stable balance between the gain and loss. On the other hand, we can employ Eq. (12.5) to obtain the following EODE for spin-polarization Sz : 

d Sz (y) dy

2

  + Sz4 (y) − 1 − U 2 Sz2 (y) = 0, y = η/ξs .

(12.11)

Real solution of Eq. (12.11) results directly from the competition between the nonlinear interaction and the kinetic energy. It is important to note that the dissipative MS relies on respective balances in two decoupled channels [12, 21, 22]. We also notice that exact solution of form (12.6)–(12.7c) has been reported for equilibrium atomic two-component BECs, when g − g12  g [6, 18, 23–25]. In the following, we analyze the properties of (i) linear spin polarization excitation, (ii) linear response function, and (iii) the excitation spectrum for a spinor polariton BEC. For this purpose, we consider, for the description of the spinor polariton BEC, Eq. (12.6) in which we assume that δn  1 and ϕ A(B)  1. Inserting Eq. (12.6) into the GP system (12.1)–(12.3) and following the standard BdG approach, we obtain, for the excitation spectra ωq , the following equation:



 2 3 2  ωq − (ω S )2 ωq + i (Rn 0 + γ R ) ωq    + Rn 0 γC + (ω B )2 ωq + ic(q) = 0,

(12.12)

12.1 Dynamics of Dissipative Magnetic Matter-Wave Solitons in a Spinor …

401

where ω S =

      εq0 εq0 + (g − g12 ) n 0 , ω B = εq0 εq0 + (g + g12 ) n 0 , (12.13)

c(q) = − (Rn 0 + γ R ) (ω B )2 + 2gn 0 γC εq0 ,

(12.14)

with the free-particle energy εq0 = 2mq . The quadratic equation from Eq. (12.12) yields ωq = ±ω S for the energy of the spin-polarization excitation. Evidently, the cubic equation from Eq. (12.12) reflects the coupled linear excitations in the reservoir and density channel of polariton BEC. ω S given by Eq. (12.13) is a real quantity, independently on the fact that the reservoir is fast or slow compared to the polariton BEC, as one can see from Fig. 12.3a. 2 2

Fig. 12.3 Visualization of the decoupling of spin-polarization channel from the density channel through linear excitations. a, b real part (a) and imaginary part (b) of the energy of density excitation ω D and spectrum ωs of spin polarization excitation. Solid curves depict numerical solutions of the Bogoliubov’s equations, and the curve with circles indicate analytical solutions generated with the used equations (12.15). c, d Density static structure factor S D (q) given by Eq. (12.16) and spin-density static structure factor SS (q) defined by Eq. (12.17) when the spinor polariton BEC is subjected to a perturbation in the form λ exp [i (q x − ωt)] +H.c (c) and λσz exp [i (q x − ωt)] +H.c (d) (see the main text). Different plots are generated with the following numerical values of parameters: γC = 0.01ps−1 , ps−1 µm2 ps−1 µm2 , g = 0.01 m e Vµm2 , P = 0.41ps−1 µm2 , γ R /γC = 40, and g12 /g = −0.1. Reprint from Ref. [1], Copyright 2022, with permission from American Physical Society

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12 Engineering Magnetic Solitons in Nonlinear Systems

Generally, the linear density excitation exhibits a complex energy ω D . In the case of the linear density excitation, adiabatic elimination of the reservoir enables the complex expression  (12.15) ω D = −i ± ω0 , 2 where ω0 =



  εq0 εq0 + (g + g12 ) n 0 − 2g R /R −  2 /4,

 = n 0 n 0R R 2 / (γ R + n 0 R) . Because ω0 is a real quantity for |q| ≤ qc , ω D will be purely imaginary due to the polariton loss, with   α 2 +  4 /4 − α 2 /2 . qc = m For a best understanding, we have displayed respectively in Fig. 12.3a, b the real and the imaginary parts of the linear density excitation ω D in the fast reservoir limit along with the numerical solutions (solid lines) of the BdG equations. It is clearly seen from these figures that the analytical and the numerical solutions agree well with each other. Concerning the linear response, we subject the spin polariton BEC to an external density perturbation described by the expression λ exp [i (q x − ωt)] + H.c with λ  1; next we assumed that the linear response in this BEC system is characterized by the spin density static structure factor SS (q) and the density static structure factor S D (q) and [26]. For simplicity, we set SS (q) = 0 and focus our attention on the case of a fast reservoir limit, leading to the following analytical expression for S D (q):

⎧ ε0 +2|ω0 | q ⎪ , if q < qc , log ⎪ | | π|ω −2|ω 0 0 ⎪ ⎪ ⎪ ⎪ ⎨ 0 4εq S D (q) = , if q = qc , π ⎪ ⎪ ⎪ ⎪  ε0

⎪ ⎪ ⎩ 1 + 1 tan−1 4ω02 −2 q , if q > qc . 2 π 4ω0 ω0

(12.16)

The result is shown in Fig. 12.3c where the density static structure factor S D (q) given by Eq. (12.16) is depicted. We can see from Fig. 12.3c that we have limq→∞ S D (q) = 1. Now, let us consider the situation when a perturbation λσz exp [i (q x − ωt)] + H.c is present. In this case, wee set S D (q) = 0 and find the spin density static structure factor SS (q) to be SS (q) = 

q2 . 2mω S

(12.17)

12.2 Nonlinear Magnetization Dynamics of a Classical Ferromagnet

403

The result is depicted in Fig. 12.3d. As we can see from Fig. 12.3d, SS (q) → 1 as q → ∞. These two results (limq→∞ S D (q) = limq→∞ SS (q) = 1) corroborate that a spin polarization perturbation only induces spin excitations.

12.1.3 Conclusion and Discussions Using exact analytical soliton solution of a driven dissipative two-component GP equations, we have introduced dissipative magnetic polariton soliton which manifests a bi-channel double balance. The properties of linear spin polarization excitation and of linear response function, as well as that of the excitation spectrum for a spinor polariton BEC are analyzed. The found here bi-channel double mechanism and its variants may lead to new dissipative solitons in the system under consideration. It is important to note that in a broader context, multi-component dissipative nonlinear system are widely seen, including mode-locked lasers and optical micro-resonators [27, 28].

12.2 Nonlinear Magnetization Dynamics of a Classical Ferromagnet We investigate the effect of an external magnetic field on nonlinear excitations in a classical ferromagnet with two single-ion anisotropies under the action of an external magnetic field. By means of the stereographic projection, the effect of an external magnetic field on the integrability of the system is analyzed and the integrable conditions of the equation of motion are presented. The exact analytical Jost solutions are presented and their properties as well as the scattering data are investigated in detail. Using the Binet-Cauchy formula, multisoliton solutions are investigated. Our results show that the external magnetic field can affect the motion of the center, the amplitude, and the width of the MSs, but has no effect on the soliton shape [29].

12.2.1 Introduction Introduced in the last century by Landau and Lifshitz, the Landau–Lifshitz (LL) equation is a fundamental tool in the magnetic recording industry [30]. It can be used to describe the nonlinear magnetization dynamics (NMDs) of the classical ferromagnet [30]. Its special solutions, including single-solitonlike and multi-solitonlike solutions, have been reported in a number of research works [31–45]. Also, special solutions of LL equation for the special initial condition have been considered by several researchers [46–60] .

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12 Engineering Magnetic Solitons in Nonlinear Systems

In all the enumerated works on the solutions of LL equation for NMDs, the anisotropy in an external magnetic field was neglected. When the anisotropy is added in the external magnetic field, analytical investigation of NMDs of the ferromagnet becomes difficult. In this situation the equation of motion for the NMDs of the ferromagnet differs from those of an isotropic ferromagnet and cannot be solved neither by separating variables in moving coordinate, by Darboux transform (DT), nor by an usual form of inverse scattering transformation (IST) [32]. Considering exact analytical solutions of the LL equation under various external actions, a general theory with terms of the continuous spectrum as a starting point, as we will see in the following, is necessary. It is evident that the system, under an external magnetic field, the initial condition of the Landau-Lifschitz equation of a ferromagnet with an anisotropy will be changed, and the system will become non-integrable. In this Section, the effect of the magnetic field on the integrability of the system will be discussed and single- and multisoliton solutions of the equation of motion will be built [34, 60–64].

12.2.2 Macroscopic Description and Equation of Motion The physical model to be investigated in this Section is the classical ferromagnet with two single-ion anisotropies in an external magnetic field. For the macroscopic description, the dynamics of the classical ferromagnet is assumed to be determined by giving at each point of the magnetization vector M = (Mx , M y , Mz ). For a classical ferromagnet with two single-ion anisotropies in an external magnetic field, the micro magnetism (alias energy of the ferromagnet) E, assumed to be composed of three terms, namely, an external magnetic field E ex , an anisotropic energy E an , and a Zeeman energy E Z , takes the following form [29]   ! ∂M ∂M 3 1 d x − βx Mx2 d 3 x E = E ex + E an ∂ x ∂ x 2 k k k   1 − βz Mz2 d 3 x − μ B M · Bd 3 x, (12.18) 2 α + EZ = 2

where μ B is the Bohn magneton. The integral of motion of Eq. (12.18) reads M2 ≡ M02 = constant. The quantity M0 in the ground state coincides with the spontaneous magnetization M0 =

2μ B S a3

12.2 Nonlinear Magnetization Dynamics of a Classical Ferromagnet

405

with the atomic spin S and inter-atomic spacing a. In the limit bβx = 0, the biaxial anisotropic ferromagnet will reduce into an uniaxial anisotropic ferromagnet with an anisotropy axis coinciding with the z axis: when βz > 0 [βz < 0], an anisotropy is of an easy-axis type and its magnetization vector in the ground state is directed along the z axis [lies in the easy plane in the absence of an external magnetic field and can be directed arbitrarily in this plane]. In the absence of the anisotropic energy (that is, E an = 0), a crystal is called an isotropic ferromagnet. As a function of space coordinates and time, the magnetization vector of the classical ferromagnet M(x, t), function of both the spatial coordinates x and time t, is governed by the Landau-Lifschitz equation 2μ B δE ∂M = M× . ∂t  δM

(12.19)

Assuming the spatial coordinate x and time t to be measured in unit of respectively l0 = (a/βz )1/2 and ω0 = (2μ B βz M0 ) /, we deduce from Eqs. (12.18) and (12.19) the following equation of motion   2 ∂ M ∂M + J M + μ B B , with J = diag(Jx , Jy , Jz ), =M× ∂t ∂x2

(12.20)

in which the matrix J is related to the anisotropic constants. The terms proportional to the Bohn magneton μ B of Eq. (12.20) describes various external actions such as the magnetic field (the case to be treated in this Section), the magnetic impurities, and the dissipative loses. It has been established that the motion equation (12.20) in the special case B = 0 is exactly integrable [48]; in this case, Eq. (12.20) can be reduced into a sine-Gordon equation in the limit Jx  Jy  Jz , when oscillations M are localized near an easy plane yz. In the limit B = 0 the motion equation (12.20 turns to a NLS equation in the limit Jx ≈ Jy  Jz , when oscillations of M are localized in the vicinity of the vacuum state M(x, t) = (0, 0, M0 ). Also, the motion equation for an isotropic ferromagnet in an external magnetic field is completely integrable, if βz = 0 [54]. The motion equation (12.20) under a zero magnetic field is equivalent to a NLS equation [36]. Therefore, the motion equation (12.20) is the most general motion equation describing the nonlinear magnetization dynamics of classical ferromagnet with two single-ion anisotropies in an external magnetic field [61]. We start by considering the effect of an external magnetic field on the integrability of the motion equation (12.20). For magnetic fields with either an easy-plane or an easy-axis (that is, with rotational symmetry), we can, by going over to a rotating coordinate frame, generalize the analysis of results of work [48]. Introducing a stereographic projection P(x, y) of the unit sphere of magnetization vector onto a complex plane as follows [46, 65] P (x, y) =

Mx + i M y , 1 + Mz

(12.21)

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12 Engineering Magnetic Solitons in Nonlinear Systems

it follows from the motion equation Eq. (12.20) that 

      2 1 − P ∗ x P, P ∗ − 1 − P 2 ∗x P, P ∗ = 0,        2 −i 1 + P ∗  y P, P ∗ − i 1 + P 2 ∗y P, P ∗ = 0,     P ∗ z P, P ∗ − P∗z P, P ∗ = 0,

(12.22)

where i with i = x, y, z  y and z can be written as 

" #2 ∂P  ∂2 P  ∂2 P 2 ∗ + 1 + |P| − 2P + 2Ji P 1 − |P|2 ∂t ∂x2 ∂x2   1  1  Bx 1 − P2 + i B y 1 + P2 − Bz P , (12.23) +μ B 1 + |P|2 2 2 

 i P, P ∗ = i 1 + |P|2

where Jx = Jz − Jy , Jy = Jy − Jx , Jz = Jy − Jx . For system (12.23) to be consistent, we must have     i P, P ∗ = 0 and i∗ P, P ∗ = 0.

(12.24)

Under the restrictions (12.24), the evolution equation for the stereographic projection P(x, t) given by Eq. (12.21) in the presence of the general direction of an external magnetic field is found to be  2 2    2    2 ∂P 2 ∂ P ∗ ∂ P + 1 + |P| − 2P + 2Ji P 1 − |P|2 i 1 + |P| 2 2 ∂t ∂x ∂x         1 1 2 x 2 y 2 z B 1 − P + i B 1 + P − B P = 0. (12.25) +μ B 1 + |P| 2 2 For analyzing the effect of an external magnetic field on the integrability of the motion equation, we will use the evolution equation (12.25) of the stereographic projection. When B x = B y = 0, that is, B = [0, 0, B z (t)] corresponding to an external field directed along an anisotropic axi, we can remove the magnetic field term from Eq. (12.25) by using the following gauge transformation    $ = P exp iμ B dt B z (t) ; P→P the system then becomes integrable. It is evident that in the situation when only B y (t) = 0, that is, B = [0, B y (t), 0] (the case when the magnetic field is transverse), the above gauge transformation cannot free Eq. (12.25) of the magnetic field term. In this case of B = [0, B y (t), 0], the combined Galilean and gauge invariance of

12.2 Nonlinear Magnetization Dynamics of a Classical Ferromagnet

407

the Landau Lifschitz equation will be broken, there will not exist Lax pairs, and the system will be non-integrable. Although the introduction of an angular variable $ ϕ = ϕ − ω B t in the polar coordinates (θ, ϕ) free Eq. (12.20) in terms (θ , $ ϕ ) from B, it is important to note that NMDs of the classical ferromagnet with an easy-axis plane is very sensitive to an external magnetic field, independently on the nature of the magnetic field (weak or strong). A perpendicular to an easy-axis plane external magnetic field does not alter the axial symmetry corresponding to the z-axis; in such a situation, the form of the ground state will depend on the strength of an external field. Let us denote by Bc the critical of the external magnetic field B: Bc =

[(Jx − 2Jz )M] . μB

The magnetization vector M in the ground state under the condition that Bz < Bc will deviate from an easy plane; in such a situation, M will be characterized by an inclination θ = θ0 = arccos(B z /Bc ) to the z axis (the ground state is referred to as “easy cone” when Bz < Bc ). The angle ϕ remains arbitrary. With the increase in the external magnetic field, the angular opening of the easy cone becomes smaller, and vector M in a non-excited ferromagnet with an easy plane lies along the z-axis. Let us consider, in the context of the experiments [63, 64], the situation when either B = [B x (t), 0, 0] or B = [0, B y (t), 0], corresponding to the situation when an external magnetic field lies in an easy plane. In this context, we consider as samples of a ferromagnet with an easy plane, Cs Ni F3 and (C6 H11 NH3 )CuBr3 , and assumed an external field to be applied as a rule in an easy plane. With these two samples, we obtain that the presence of an external field, which lies in an easy plane, it is not easy to find exact soliton solutions of the corresponding LL equation. It is not possible to free equation (12.25) from the magnetic-field term by using the above gauge transformation. This means that in general, a ferromagnet with a uniaxial anisotropy in a transverse magnetic field is non-integrable; it may become integrable only in the absence of either an external field or an anisotropic interaction. If we introduce the following two equations for 2 × 2 matrices (x, t; μ, λ) ∂(x, t; μ, λ) = L (μ, λ) (x, t; μ, λ), ∂x ∂(x, t; μ, λ) = A (μ, λ) (x, t; μ, λ), ∂t

(12.26)

where coefficients L (μ, λ) and A (μ, λ) in the Lax pairs, functions of the spectral parameters λ and μ, are defined as L (μ, λ) = −iρns (μ, λ) Mx σx − iρds (μ, λ) M y σ y − iρcs (μ, λ) Mz σz , A (μ, λ) = 2iρ 2 ds (μ, λ) cs (μ, λ) Mx σx + 2iρ 2 ns (μ, λ) cs (μ, λ) M y σ y +2iρ 2 ns (μ, λ) ds (μ, λ) Mz σz

408

12 Engineering Magnetic Solitons in Nonlinear Systems

  ∂ My ∂ Mz − Mz σx −iρns (μ, λ) M y ∂x ∂x   ∂ Mx ∂ Mz − Mx σy −iρds (μ, λ) Mz ∂x ∂x   ∂ My ∂ Mx − My σz , −iρcs (μ, λ) Mx ∂x ∂x

(12.27)

with σi (i = x, y, z) being the Pauli metrics, ns(μ, λ), ds(μ, λ), and cs(μ, λl) being elliptical functions, while μ and ρ being defined as  μ=

Jy M − Jx M 1 , ρ= √ , 2ρ 2 Jz M − Jx M

it is obvious that the motion equation (12.20) may be represented as a compatibility condition ∂A ∂L − + [L , A] = 0 ∂t ∂x of Eq. (12.26) with data (12.27). In the following, we restrict ourselves to the situation  when λ satisfies the conditions |Reλ| ≤ 2K and |Imλ|  ≤ 2K , K (μ) being a complete 1 − μ2 . This restriction on λ elliptic integral of the first kind and K  (μ) = K follows from the double-periodicity of coefficients L (μ, λ) and A (μ, λ) as function of λ. When the spectral parameters λ and μ satisfy the following relation % λ = 2

μ2 + 4ρ 2 , for βz < 0 (an easy plane) μ2 − 4ρ 2 , for βz > 0 (an easy axis),

(12.28)

with ρ given as ρ=

% 1√ − Jz ) M, 4 √(Jx 1 − Jx ) M, (J z 4

for βz < 0 (an easy plane), for βz > 0 (an easy plane),

(12.29)

the Lax pairs for an uniaxial anisotropic ferromagnet in an external magnetic field can be rewritten as follows: L (μ, λ) = −iμMx σx − iμM y σ y − iλMz σz ,

 ∂ My ∂ Mz − Mz σx A (μ, λ) = 2iμλMx σx + 2iμλM y σ y + 2iμ Mz σz − iμ M y ∂x ∂x     ∂ My ∂ Mx ∂ Mz ∂ Mx − Mz σ y − iλ Mz − Mz σz ,(12.30) −iμ Mz ∂x ∂x ∂x ∂x 

2

Next, we introduce a Riemann surface. For this aim, we assumed the spectral parameters λ and μ to be functions of a new introduced variable k, called the affine variable:

12.2 Nonlinear Magnetization Dynamics of a Classical Ferromagnet

& λ = λ(k) =

(2ρ (k 2 +1)) , k 2 −1 2 2

k −ρ k

,

& μ = μ(k) =

409

4ρk , for an easy plane, k 2 −1 k 2 +ρ 2 , for an easy axis. k

(12.31)

Then, we associate with the equation of motion (12.20) two types of BCs, namely, the boundary condition of the first type, applies at x → ±∞ and the BC of the second type, at y → ±∞. The BC of the first type M|x→±∞ = M0 = (0, 0, M0 )

(12.32)

corresponds to a breatherlike solution (alias magnetic soliton). Integrating Eq. (12.26) under the BC of the first type (12.32) yields the Jost solutions 0± (x, k), chosen as 0± (x, k) → E(x, k)as x → ±∞, where E(x, k) = exp[−iρcs(k)M0x σz ], while

(12.33)

  ' % 2ns(k)ds(k) t σz 0± (x, k) exp −iρcs(k)M0 x − cs(k)

(  for Im k ∈ 0, 2K  . E(x, t) given by Eq. (12.33) contains two independent solutions E 1 (x, k) and E 2 (x, k): E 1 (x, k) = (E 11 (x, k), E 21 (x, k))T , E 2 (x, k) = (E 12 (x, k), E 22 (x, k))T . Also, each of 0+ (x, k), 0 (x, k), and 0− (x, k) has two independent solutions, namely, 0+1 (x, k) and 0+2 (x, k), 01 (x, k) and 02 (x, k), and 0−1 (x, k) and 0−2 (x, k), respectively. Under an external magnetic field, vector M in the ground state of a ferromagnet with an easy plane will deviate from an easy plane, and will be characterized by an inclination θ0 [φ0 ] to the z− axis [x-axis], where the asymptotic magnetization vector M lies on the surface of an easy cone. In this situation, the simplest solution of the equation of motion (12.20) is found to satisfy the BC of the first type M|x→±∞ = M0 = (M0 sin θ0 cos φ0 , M0 sin θ0 sin φ0 , M0 cos [θ0 ]).

(12.34)

p

To the boundary condition (12.34) correspond Jost solutions 0± (x, k) of Eq. (12.26) that may be chosen as p

0± (x, k) → E p (x, k) as x → ±∞, where    2ρk 2ρk E p (x, k) = exp −i 2 M0 sin [θ0 x] cos [φ0 σx ] − i 2 M0 sin [θ0 x] sin φ0 σ y k −1 k −1

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12 Engineering Magnetic Solitons in Nonlinear Systems

−i

 ρ k2 + 1 k2 − 1

⎤   M0 cos θ0 xσz ⎦ ,

(12.35)

while     2ρ k 2 + 1 2ρk M0 sin [θ0 ] cos [φ0 ] x − t σx = exp −i 2 k −1 k2 − 1     2ρ k 2 + 1 2ρk M0 sin [θ0 ] sin [φ0 ] x − t σy −i 2 k −1 k2 − 1     ) ρ k2 + 1 8ρk 2 M0 cos [θ0 ] x − 4 t σz . −i k2 − 1 k −1 &

p 0 (x, k)

As it has been stated above, the magnetization will be far from an easy plane when an external magnetic field increases; in the special case of B z  Bc , the magnetization will lie along the z axis. When the magnetic fields vanish, magnetization will lie on an easy plane and takes the form M0 = (M0 cos [φ0 ] , M0 sin [φ0 ] , 0). As in the above analysis, E p (x, k) given by Eq. (12.35) contains two independent p p solutions E 1 (x, k) and E 2 (x, k) each of which has two components:  p T p p E 1 (x, k) = E 11 (x, k), E 21 (x, k) , p

 p T p p E 2 (x, k) = E 12 (x, k), E 22 (x, k) . p

p

The same, each of solutions 0+ (x, k), 0 (x, k), 0− (x, k) has two independent p p p p p solutions 0+1 (x, k) and 0+2 (x, k), 01 (x, k) and 02 (x, k), and 0−1 (x, k) and p 0−2 (x, k), respectively. Because the z axis is an easy axis in a ferromagnet, we choose the boundary condition of the first type as M|x→±∞ = M0 = (0, 0, M0 ).

(12.36)

a (x, k) of Eq. (12.26) To the BC of the first type (12.36) correspond Jost solutions 0± which may be chosen as a (x, k) → E a (x, k) as x → ±∞, 0±

where

while

  k2 − ρ2 E a (x, k) = exp −i M0 xσz , 2k

(12.37)

12.2 Nonlinear Magnetization Dynamics of a Classical Ferromagnet

411

&

0a (x, k)

 2  )  2 k + ρ2 k2 − ρ2  t σz . M0 x −  2 = exp −i 2k k k − ρ2

Similarly, E a (x, k) given by Eq. (12.37) has two independent solutions E 1a (x, k) and E 2a (x, k), each of which has two components  a T a E 1a (x, k) = E 11 (x, k), E 21 (x, k) ,

 a T a E 2a (x, k) = E 12 (x, k), E 22 (x, k) ,

a a (x, k), 0a (x, k), and 0− (x, k) has two independent solutions while each of 0+ a a a a a a (x, k) and 0−2 (x, k), 0+1 (x, k) and 0+2 (x, k), 01 (x, k) and 02 (x, k), and 0−1 respectively. Using the standard procedures of characteristic theory yields the following integral representation

+ (x, k) = E(x, k) + λ

∞

dy K + (x, y)E(y, k),

x

− (x, k) = E(x, k) + λ

(12.38)

x −∞

dy K − (x, y)E(y, k),

where the kernels K + (x, y) and K − (x, y) implicitly depend on magnetization M(x), but do not depend on eigenvalue λ; moreover kernels K + (x, y) and K − (x, y) satisfy the following BC of the second type K ± (x, y)| y=±∞ = 0. p

For a ferromagnet with an easy plane in an external magnetic field, ± (x, k) can be also taken from the following integral representation p + (x, k)

 ∞  ρ k2 + 1 p,d = E (x, k) + dy K + (x, y)E p (y, k) k2 − 1 p

x

+

2ρk −1

∞ p,nd

dy K +

k2

(x, y)E p (y, k),

x

 x  ρ k2 + 1 p p,d p − (x, k) = E (x, k) + dy K − (x, y)E p (y, k) k2 − 1 −∞

+

2ρk −1

k2

x p,nd

dy K − −∞

(x, y)E p (y, k),

(12.39)

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12 Engineering Magnetic Solitons in Nonlinear Systems p

where kernels K ± (x, y) satisfy the BC of the second type  p K ± (x, y) y=±∞ = 0. In the case of a ferromagnet with an easy axis in an external magnetic field, ±a (x, k) can be taken as +a (x, k)

k2 − ρ2 = E (x, k) + 2k

∞ dy K +a,d (x, y)E a (y, k)

a

x

+

k2 + ρ2 2k

∞ dy K +a,nd (x, y)E a (y, k), x

(12.40) −a (x, k) = E a (x, k) +

+

k2 + ρ2 2k

k2 − ρ2 2k

x dy K −a,d (x, y)E a (y, k) −∞

x dy K −a,nd (x, y)E a (y, k), −∞

where kernels K ±a,d (x, y) satisfy the BC of the second type  K ±a (x, 0) y=±∞ = 0. In the above Eqs. (12.39) and (12.40), the superscripts d and nd denote respectively the diagonal and nondiagonal parts of the matrix.

12.2.3 Soliton Solutions Using the results of the previous subsection, we investigate here soliton solutions. These soliton solutions can be built using the inverse scattering problem, which consists of constructing the magnetization vector M(x, t) from the time-dependent scattering data. Such an inverse scattering problem can be carried out either by means of the Gel’fand-Levitan-Marchenko (GLM) equation or a linear integral equation. The GLM equation in the reflectionless case (case where the reflectional coefficient r (k, t) = 0) reads K 11 (x, t) + K 12 (x, t) N  (x, t) = 0, (12.41) 

K 12 (x, t) − G (x, t) − K 11 (x, t) N (x, t) = 0,

12.2 Nonlinear Magnetization Dynamics of a Classical Ferromagnet

413

where K 11 (x, t) and K 12 (x, t) are defined as & K 11 (x, t) = i

)   det I + N  (x, t) M  (x, t) −1 , det [I + N  (x, t) N  (x, t)]

K 12 (x, t) = with



T



det I + N (x, t) N (x, t) + H (x) G (x, t) det [I + N  (x, t) N  (x, t)]

(12.42)

− 1,

M  (x, t) = N  (x, t) + i H (x)T G (x, t),

N  (x, t) and N  (x, t) being two N × N matrices. t), we will use the Cauchy formula to For obtaining  Binet   K i j (x,     t) N t) , det I + N t) M t) , and compute det I + N (x, (x, (x, (x,

T

det I + N  (x, t) N  (x, t) + H (x) G (x, t) . First, let us set   0 = det I + N  N  .

(12.43)

Using the Binet-Cauchy formula yields 0 = 1 +

!

!

!

γ0 (n 1 , n 2 , ..., nr ; m 1 , m 2 , ..., m r ) .

r =1 1≤n 1