Rogue Waves: Mathematical Theory and Applications in Physics 9783110470574, 9783110469424

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Boling Guo, Lixin Tian, Zhenya Yan, Liming Ling, Yu-Feng Wang Rogue Waves

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De Gruyter Series in Nonlinear Analysis and Applications Jürgen Appell et al. (Eds.) ISSN 0941-813X

Boling Guo, Lixin Tian, Zhenya Yan, Liming Ling, Yu-Feng Wang

Rogue Waves Mathematical Theory and Applications in Physics

Mathematics Subject Classification 2010 Primary: 37K10, 37J35, 37K15; Secondary: 70H06, 81R12 Authors Prof. Boling Guo Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics 6 Huayuan Road Haidian District 100088 Beijing People’s Republic of China [email protected] Prof. Lixin Tian Nanjing Normal University School of Mathematical Science Center for Energy Development 210023 Nanjing People’s Republic of China [email protected]

Prof. Zhenya Yan Chinese Academy of Sciences Institute of Systems Science Key Lab Mathematics Mechanization 55 Zhongguancun East Road Haidian District 100190 Beijing People’s Republic of China [email protected] Associate Prof. Liming Ling South China University of Technology Wushan RD., Tianhe District 510641 Guangzhou People’s Republic of China [email protected]

Dr. Yu-Feng Wang Institute of Applied Physics and Computational Mathematics 6 Huayuan Road Haidian District 100088 Beijing People’s Republic of China [email protected] ISBN 978-3-11-046942-4 e-ISBN (PDF) 978-3-11-047057-4 e-ISBN (EPUB) 978-3-11-046969-1 Set-ISBN 978-3-11-047058-1 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2017 Walter de Gruyter GmbH, Berlin/Boston Typesetting: Integra Software Services Pvt. Ltd. Printing and binding: CPI books GmbH, Leck Cover image: Chong Guo @ Printed on acid-free paper Printed in Germany www.degruyter.com

Contents 1 The Research Process for Rogue Waves 1 1.1 The Research Process for Rogue Wave Phenomenon 1 1.2 Some Famous Experiments of Rogue Waves 5 1.3 Research Method and Physical Mechanism of Rogue Waves 9 1.3.1 Methodology of Rogue Waves 9 1.3.2 Physical Mechanism of Rogue Waves 10 1.4 Mechanisms of Rogue Waves 12 1.4.1 Linear Mechanisms of Rogue Waves 12 1.4.2 Nonlinear Mechanisms of Rogue Waves 17 1.5 Rogue Wave Solutions for Nonlinear Partial Differential Equations 1.6 Optical Rogue Waves 26 1.7 Financial Rogue Waves 28 1.8 Nonautonomous Rogue Wave Solutions 28 2 2.1 2.2 2.3 2.4 2.4.1 2.4.2 2.5 2.5.1 2.5.2 2.5.3 2.6 2.6.1 2.6.2 2.6.3 2.6.4 2.6.5 2.6.6 3

Construction of Rogue Wave Solution by the Generalized Darboux Transformation 30 The Classical Darboux Transformation 30 Generalized Darboux Transformation for the Classical KdV Equation Darboux Transformation for N-Coupled Focusing NLS Equation 35 Rogue Wave Solutions for the Two-Component NLS Equation 37 Rogue Wave Solutions for the Two-Component NLS Equation 37 Bright-Dark Breather and Rogue Wave Solutions 41 Generalized Darboux Transformation for NLS Equation 45 Generalized Darboux Transformation 46 Higher-Order Rogue Waves in Determinant Forms 50 Mathematical Characters of the Rogue Wave Solutions for Standard NLS Equations 55 Generalized Darboux Transformation for DNLS Equation 59 Darboux Transformation-I 59 Darboux Transformation-II 63 Reductions 66 Generalized Darboux Transformations 67 Generalized Darboux Transformation-II 69 High-Order Solutions for DNLS Equation 72

Construction of Rogue Wave Solution by Hirota Bilinear Method, Algebro-geometric Approach and Inverse Scattering Method 81 3.1 Hirota Bilinear Method 81 3.1.1 Rogue Wave Solution for the NLS Equation 81 3.1.2 Rogue Wave Solution for the DS-I Equation 98

21

32

VI

3.2 3.3 3.3.1 3.3.2 3.3.3 3.4 3.4.1 3.4.2 3.4.3 3.4.4 3.4.5 3.4.6 3.4.7 4 4.1 4.2 4.2.1 4.2.2 4.2.3 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.4 4.4.1 4.4.2 4.4.3 4.4.4

Contents

Reduction from the KP Equation 105 Algebro-geometric Reduction Approach 109 Relationship Between Fredholm Determinant and (-Function 110 Wronskian Solutions 112 Construction of Rogue Wave Solution 116 Inverse Scattering Method and Rogue Wave Solution 118 Direct Problem 119 Scattering Matrix 120 Involution Relation 120 Jumps of the Eigenfunctions and Scattering Data Across the Branch Cut 122 Time Evolution 123 Inverse Problem 124 Darboux Transformation and Rogue Wave Solutions 127 The Rogue Wave Solution and Parameters Managing in Nonautonomous Physical Model 135 Introduction to the Rogue Wave Solution 135 Space-Time Modulation Nonlinear Schrödinger Equation 137 One-Dimensional Nonlinear Physical Model 137 Symmetry Analysis-Similarity Transformation and Similarity Solution 137 One Dimensional Self-Similarity Optical Rogue Wave Solution and Its Parameter Analysis 139 (3+1)-Dimensional Space-Time Modulation Gross–Pitaevskii/NLS Equation 144 Three-Dimensional Nonlinear Physical Model 144 Symmetry Analysis-Similarity Transformation and Reduction System 146 Similarity Variable, Constraint Condition and Velocity Field 147 Three-Dimensional Self-Similar Rogue Wave Solutions and Its Parameters Regulation 148 Generalized Inhomogeneous Higher-Order Nonlinear Schrödinger Equation with Modulating Coefficients 152 Symmetry Reductions – Transformation and Hirota Equation 154 Determining Similarity Variables and Controlled Coefficients 156 Darboux Transformation for the Hirota Equation 159 Optical Rogue Wave Solutions 160

Contents

4.5 4.5.1 4.5.2 4.5.3 4.5.4 4.5.5 4.6 4.6.1 4.6.2 4.6.3 4.7 4.7.1 4.7.2 4.7.3 4.7.4

Two-Dimensional Binary Mixtures of Bose–Einstein—Condensates Two-Component Gross–Pitaevskii Equations 165 Symmetry Reduction Analysis 166 Determining Similarity Variables and Controlled Coefficients 168 Types of Nonlinear Interaction 170 Self-Similar Vector Rogue Wave Solution 171 Two-Dimensional Nonlocal Nonlinear Schrödinger Equation 176 Two-Dimensional Nonlocal Nonlinear Model 176 Two-Dimensional Variable Separation Reduction 177 Two-Dimensional Rogue Wave-Like Solution 178 The Generalized Ablowitz–Ladik–Hirota Lattice with Variable Coefficients 184 Discrete Nonlinear Physical Model 184 Differential-Difference Similarity Reductions and Constraints 186 Determining the Similarity Transformation and Coefficients 187 Nonautonomous Discrete Rogue Wave Solutions and Interaction 188

Bibliography

193

VII

165

1 The Research Process for Rogue Waves 1.1 The Research Process for Rogue Wave Phenomenon Rogue wave is also named as freak wave, monster wave, extreme wave, killer wave, giant wave, etc. It is difficult to give a full explanation of rogue wave, due to its complex phenomenon. Figure 1.1 shows us the dramatic appearance of the rogue wave which was downloaded from the Internet. The authority statement in oceanography is that it develops from the ocean suddenly with high amplitude, which appears from nowhere and disappears without a trace. Nowadays, such nonlinear phenomenon has been observed in nonlinear optics, Bose–Einstein condensates (BEC), atmospherics, superfluid and even finance. It has aroused the attention of scientists from oceanography, physics and other nonlinear fields, since the term “freak rogue waves” was first coined by Draper [1] in 1965. For centuries, rogue wave has been seen as the sea legend and a part of the marine folk culture. In the ocean, rogue waves, like a deep-sea monster, engulf sailors and ships without any trace, which caused a lot of maritime disasters. In 1933, in the Pacific Northwest, USA warship Ramapo encountered the tallest rogue wave ever recorded, with 34 m (112 ft) height, which was visually measured by the crews on the deck [2]. Since 1952, in the Indian Ocean, at least 12 cases of ships encountering rogue wave have been recorded, near the Aga DeGeneres stream and along the coast of South Africa. One miserable case is that, on June 13, 1968, the tanker World Glory encountered a rogue wave, which broke the tanker into two parts (as seen in Figure 1.2) and caused the death of 22 crews, when the tanker was along the South African coast [3]. In 1966, the Michelangelo cruise ship, during the voyage from Italy to the United States, suddenly encountered a huge wave with 24 m height. The wave tore a hole in the superstructure, smashed the heavy glass, and made a crew member and two passengers die. In 1978, Munich, a German barge, sank in the Atlantic Ocean. The twisted wreckage indicates that it was destroyed by a huge wave. In 1980, a rogue wave with 25 m height was photographed by Philippe Lijour, the first mate of the French tanker Esso Languedoc, near the sea of the eastern port city Durban of South Africa (Azania). As seen in Figure 1.3, a huge water hole was observed clearly. In 1984, a rogue wave attacked the 2/4-A oil platform of Norwegian Ekofisk oil field, which was located above average sea level of 20 m. In this accident, the wall of the platform’s control room was destroyed, producing a standstill for 24 hours [4]. In 1986, a US warship SS Spray suffered three consecutive bursts of waves with approximately 25 m height in the sea of Charleston [5]. A highly asymmetrical rogue wave was also observed as shown in Figure 1.4, which was shot from the warship.

2

1 The Research Process for Rogue Waves

Figure 1.1: A photograph of a rogue wave (source:http://de.weather-forecast.com/photos/41).

Figure 1.2: The tanker World Glory was destroyed by a rogue wave.

Figure 1.3: The rogue wave photographed from the tanker Esso Languedoc.

1.1 The Research Process for Rogue Wave Phenomenon

3

Figure 1.4: The observation of a rogue wave from the warship SS Spray.

20

Wave height (m)

15 10 5 0 –5 –10

0

100

200

300

400

500

600

Time (s) Figure 1.5: The New Year wave in the North Sea.

During 1969–1994, 22 supercarriers have been attacked by the rogue waves in the Pacific and the Atlantic Oceans [6], resulting in 525 deaths (for more details see [6]). In addition, according to the available statistics, over the two decades from 1981–2000, destruction of more than 200 supercarriers, because of the heavy sea states and severe weather conditions [7], has been reported. On January 1, 1995, a rogue wave called “New Year wave” attacked the Draupner platform, which was located in the North Sea which had 70 m depth [8]. The New Year wave was unambiguously recorded, with a peak elevation of 18.5 m and average period of 12.5 s. The distribution of the wave height is shown in Figure 1.5.

4

1 The Research Process for Rogue Waves

Figure 1.6: Rogue wave on the South California coast.

On April 16, 2005, at the Georgia coast, there had been a number of huge waves with at least 7-story height. The waves collapsed like a bow, broke the windows of the ship, fell down into the deck with 10 m height, submerged 62 cabins and injured 4 passengers. US National Oceanic and Atmospheric Administration Web site published the image of a rogue wave with 18 m height, as seen in Figure 1.6, which was shot from a warship in a few seconds after the striking, in the stormy condition. From the analysis of the above figures and events, we observe the following characteristics of rogue waves: (i) it has a very steep peak; (ii) it is highly asymmetric; and (iii) it appears from nowhere and disappears without a trace. Humans have long suspected the existence of rogue waves, which was considered as mysterious as the emergence of mermaids and sea monsters. The existence of the rogue wave in the real world was confirmed by direct observations. In the sixties and seventies of the twentieth century, oceanographers began to believe in the existence of rogue waves. Radar satellites were launched in 1991 and 1995, by the scientists from the European Space Agency, the German Aerospace Center and several other European research institutions. The radar beam is seen as potentially the ideal tool for measuring wave height, based on the time for the radar beam to return from the sea. On January 1, 1995, the Draupner oil platform in the North Sea was hit by a wave with a maximum height of 84 ft, which was measured by an onboard laser device. It was the first scientific evidence for the existence of the rogue wave. In February 2000, the British ocean survey ship measured a giant wave with 95 ft height in the west of Scotland. Geophysical researchers say this is the biggest rogue wave which has been ever recorded by the scientific instruments. Studies have shown that the height of rogue wave rises up to at least 82 ft, about the height of 8 stories. The average height of the huge rogue wave is about 100 ft. The European MaxWave Project was conducted in Brest in 2000; its main goal was to improve the understanding of the rogue wave by using the

1.2 Some Famous Experiments of Rogue Waves

5

new oceanographic knowledge and new techniques with radar satellites [7]. In 2005, Muller et al. in the “Rogue Waves—The Fourteenth Aha Hulikoa Hawaiian Winter Workshop” reported that “Our understanding of rogue waves is greatly hampered by the lack of comprehensive observations in space and time.” [9] Undoubtedly, comprehensive, long-term measurements of the rogue wave are very necessary and urgently needed. Maritime observers have named the rogue wave as “wall of water,” “sea cave,” or consecutive high waves. This type of wave appears in a peaceful environment without any warning. The outstanding peak is surrounded by deep holes. Corresponding to the large-amplitude wave, rogue wave is seemed as an isolated huge wave with amplitude much larger than the average wave peaks in the surrounding area. Due to limitations of observing conditions, people had not yet fully understood in this phenomenon. Scientists have not reached a consensus on the rogue wave, and have neither given its definition, nor calculated the probability of occurrence. Although there is no precise definition of rogue wave, it should include the following characteristics: high waves, very asymmetrical, very steep. Its amplitude should at least be twice the amplitude of the background field (significant). The large-amplitude wave which appears in the ocean suddenly, and the huge wave whose amplitude is twice the amplitude of the surrounded average peaks, are usually called rogue waves. The current understanding of the rogue wave is preliminary and superficial. Rogue waves can be considered as a new “nonlinear scientific revolution” in nonlinear science, after soliton [10–12]. From the mathematical point of view, there are fundamental differences between rogue wave and soliton solutions. With the evolution of time, soliton can propagate with a stable envelop. Even for some nonautonomous system, the nonautonomous solitons almost have the similar shapes. However, the rogue wave is unstable and sensitive with the initial data, its shape is variable with the evolution of time.

1.2 Some Famous Experiments of Rogue Waves In 2007, Solli et al. reported the observation of “optical rogue waves” in nonlinear optics for the first time in Nature [13]. The propagation of the optical beams can be governed by the generalized nonlinear Schrödinger (NLS) equation im "m 𝜕m A 𝜕A i 𝜕 𝜕|A|2 2 2 –i (|A| ], = i𝛾 [|A| A + A) – T A g 𝜕z m! 𝜕tm 90 𝜕t 𝜕t

(1.2.1)

where "m is the optical fiber dispersion coefficient, 𝛾 is the nonlinear coefficient of the fiber, 90 is the central carrier frequency of the field, Tg is the parameter that characterizes the delayed nonlinear response of silica fiber. A = A(t, z) is the slowly varying electric field envelope, z means the variable, t denotes the time, 𝜕t𝜕 (|A|2 A) is

6

1 The Research Process for Rogue Waves

6 1,300

4 Tim

e (p 2 s)

0 900

1,200 1,100 m) 1,000 th (n g n ele Wav

Figure 1.7: The profile of the optical rogue wave.

the self-steepening, A 𝜕t𝜕 (|A|2 ) represents the vibrational Raman response of the mem

dium, |A|2 A stands for Kerr nonlinearity, 𝜕𝜕tmA means the dispersion term. Based on the supercontinuum generation in the presence of noise, the optical rogue wave was observed in the experiments, as seen in Figure 1.7. In 2010, the Peregrine soliton (rogue wave) was first observed in an experiment, which is in very good agreement with the numerical simulations and Peregrine’s analytic prediction [14]. These results were reported in Nature Physics by Kibler et al. The physical model is governed by the well-known NLS equation

i

𝜕8 1 𝜕8 + + |8|2 8 = 0, 𝜕. 2 𝜕42

(1.2.2)

where 4 and . denote the time and space, respectively. Equation (1.2.2) has the solution in the following form:

8(. , 4) = [1 –

4(1 + 2 i. ) ] ei. . 1 + 442 + 4. 2

Figure 1.8 shows us the correlation among the experiment observation, numerical simulations and analytic solution. Here we present some fresh results on models of the deep ocean, although there are lots of studies on it. Since it is very hard and dangerous to do experiments in the deep ocean, Chabchoub et al. observed rogue wave in a water wave tank [15] in 2011. In turn, in 2012, Chabchoub, et al. observed a hierarchy of up to fifth-order rogue wave in a water tank [16]. Thus, these experiments confirmed the existence of the rational rogue wave solutions. In addition, the characteristics rational rogue wave solution are similar to that of rogue wave. So in the following chapters, we define the rational function solution as the rogue wave solution.

1.2 Some Famous Experiments of Rogue Waves

7

Expt Analytic PS Sims

6

1

3

π Phase

2

9 Intensity (a.u.)

Power (W)

3

0 0

–2

–1

0 Time (ps)

1

2

0 0.5 1 Time (ps)

0

Figure 1.8: The correlation between the experiment observation, numerical simulations and analytic solution.

The deepwater wave model can be described by the following NLS equation: i(

9 𝜕2 a 9 k2 𝜕a 𝜕a + cg ) – 02 2 – 0 0 |a|2 a = 0, 𝜕t 𝜕x 2 8k0 𝜕x

(1.2.3)

which was first derived by Zakharov; a(x, t) is related to the surface elevation, where t and x denote time and spatial coordinates, respectively, k0 and 90 = 9(k0 ) are the wave number and wave frequency of the carrier wave, 90 = √gk0 , g is the gravitational 9 acceleration, cg = d9 | = 2k0 is the group velocity. Through the scale transformadk k=k0 0 9 9 tions T = – 02 t, X = x – cg t = x – 0 t and q = √2k02 a, we obtain the following 2k0

8k0

dimensionless form of NLS equation: iqT + qXX + 2|q|2 q = 0.

(1.2.4)

The first rogue wave solution for Eq. (1.2.4) reads as q(X, T) = (1 –

4(1 + 4iT) ) e2iT . 1 + 4X 2 + 16T 2

To describe the experiment results, we introduce the transformations X → √2k02 a0 (x – cg t), T → – transformed into

k02 a20 90 t. 4

q(x, t) = a0 e–

Thus, the the first rogue wave solution q(X, T) can be

2 a2 9 ik0 0 0t 2

(1 –

4(1 – ik02 a20 90 t) 1 + [2√2k02 a0 (x – cg t)]2 + k04 a40 920 t2

).

The comparison between the analytic solution and experimental data is shown in Figure 1.9, which confirmed the existence of rogue wave solution in water models.

8

1 The Research Process for Rogue Waves

0.03

Surface elevation (m)

0.02

0.01

0

–0.01

–0.02 0

2

4

6 Time t (s)

8

10

12

Figure 1.9: The comparison between the analytic solution and experimental data at X = 0 with a0 = 0.01 m, k0 = 11.63 m–1 , 90 = 10.7 s–1 .

The high-order rogue wave solution will be discussed in detail in the following chapters, due to its complex expression. Figure 1.10 displays the fifth-order rogue wave and experiment observation, respectively. Among many viewpoints and hypotheses on the rogue wave, our basic assumptions are that the rogue wave can be formed via modulation instability and the rational rogue wave solution is in accordance with the rogue wave. Graphically, the rational solution which we obtained almost coincides with the characteristics of rogue wave. But another inconsistent characteristic of rogue wave is that it is highly asymmetric, which is inconsistent with the strange symmetry of the first-order rational rogue wave solution. However, this is not the sticking point, since the rogue wave phenomenon is very complicated. All in all, the above experiments confirmed that our viewpoints are reasonable. The analysis on the rogue wave solution for some integrable models is the aim of this book. In theory, integrable model is very beautiful in mathematical expression, but with less physical characters. One advantage of integrable models is that we can find some analytical solutions for them. There are several methods to solve the integrable systems, including the inverse scatting transformation (IST), algebrogeometric approach, Darboux transformation and Hirota bilinear method. The IST is consistent with Darboux transformation method in essence. Darboux transformation is more convenient in the aspect of constructing the exact solution. Algebro-geometric approach relies on constructing the Baker analytic functions and spectral analysis.

1.3 Research Method and Physical Mechanism of Rogue Waves

11

|ψ5|

(a)

9

1 2 T

5

0

0 X

–2 –5

Surface elevation (cm)

(b) 1.1 2 mm

0 1.1

2 mm

0 0

10

20

30

40 Time (s)

50

60

70

80

Figure 1.10: (a) Fifth-order rogue wave; (b) Experimental demonstration of the fifth-order rogue wave at x = 0 with a0 = 1 mm, : = a0 k0 = 0.01 (upper curve) compared to the theoretical analytic solution at the same position (down curve).

Once an integrable model has been bilinearized through certain transformation, with the truncated parameter expansion at different levels, a series of solutions can be obtained through the bilinear method. The mentioned methods can also be applied to get the rogue wave solution with some modifications. In Chapter 2, we focus on the generalized Darboux transformation, which was first proposed by Guo, Ling and Liu. The Hirota bilinear method, algebro-geometric approach and inverse scattering method are discussed in Chapter 3. In Chapter 4, we present the rogue wave in nonautonomous nonlinear systems by using the similarity transformation and parameter analysis, which are the effective ways to deal with the models with variable coefficients. For the benefit of looking for other viewpoints of rogue wave conveniently, we list some references objectively to let the reader see the recent progresses on rogue wave.

1.3 Research Method and Physical Mechanism of Rogue Waves 1.3.1 Methodology of Rogue Waves In the past three decades, the development of rogue wave research is so fast that various physical models which can describe the rogue wave phenomenon were proposed. The main methods are described below. (1) Models built by experiments and numerical stimulations.

10

1 The Research Process for Rogue Waves

In 2001, M. Onorato et al. [17] studied the generation in a random sea state. They displayed the extensive numerical simulations of the NLS equation, how the rogue waves in a random sea state are more likely to occur for large values of the Phillips parameters ! and the enhancement coefficient 𝛾. In 2002, A. Calini and C. M. Schober [18] numerically investigated the dispersive perturbations of the NLS equation, which describes the wave dynamics in deepwater. In addition, they observed that the likelihood of the rogue wave formation can be greatly increased in a chaotic regime. In 2004, M. A. Tayfun et al. [19] described that the statistics of the largest crest in N waves are used to examine the nature of two exceptionally large waves recorded within the two hourly measurements. Although these waves possess similar characteristics with that of rogue waves, they are shown to be the simple outliers, which may be predicted well within the context of the extreme-value statistics. In 2009, experiment results made by X. Z. Zhao et al. [20] showed that the possibility occurrence of rogue wave in deepwater is much higher than in shallow water. In 2011, clear evidence of rogue wave in a multi-stable system was revealed by A. N. Pisarchik et al. [21] through experiments with an erbium-doped fiber laser driven by harmonic pump modulation. They pointed out that the formation of rogue wave depends on the interplay of stochastic processes with multi-stable deterministic dynamics. (2) Rogue wave can be described by solutions for the nonlinear partial differential equations, i.e., NLS equation, Korteweg–de Vries (KdV) equation, Kadomtsev– Petviashvili (KP) equation, Ablowitz–Ladik (AL) equation, Hirota equation, Davey–Stewartson system, Zakharov system, etc. In 2009, A. Ankiewicz et al. [22] studied the effect of various perturbations on rogue wave to test whether rogue waves are against the perturbations. In 2010, the integral relations of the rogue wave solutions for the focusing NLS equation were given by A. Ankiewicz et al. [23]. In 2011, N. Akhmediev et al. [24] studied the spectra of the Peregrine soliton and higher-order rational solutions for NLS equation, which they used as a model in optics and deep ocean. These solutions have the specific triangular spectra which can be easily measurable in optical systems and obey the characteristics of ocean environment. The possibility of early detection and possible localized warning of the appearance of rogue wave are increased, since the triangular feature of the solutions happened at the early stage of their evolution. The “early warning spectra” of rogue waves may become an important research field in the future [24].

1.3.2 Physical Mechanism of Rogue Waves Physical occurrence mechanism of rogue wave is one of the research focuses. Till now, the main mechanisms which are used to describe the occurrence of rogue wave are described below.

1.3 Research Method and Physical Mechanism of Rogue Waves

(1)

(2)

(3)

(4)

(5)

(6) (7)

(8) (9)

11

Interaction between water waves. In 2003, P. Peterson et al. [25] analyzed the interaction between two long-crested shallow water waves in the framework of two-soliton solutions for KP equation. The results showed that the extreme surface elevations are four times the amplitudes of the incoming waves. Three-dimensional (3D) directional wave focusing. In 2007, C. Fochesato et al. [26] deemed that the 3D directional wave focusing contributes to the generation of rogue wave in the ocean, which was simulated and analyzed in a 3D numerical wave tank. Nonlinear focusing of transient frequency modulation wave group. In 2008, E. Pelinovsky et al. [27] discussed the kinds of ships of rogue waves in the nonlinear focusing mechanism framework of transient frequency modulation wave group. Modulation instability. In 2010, M. Shats et al. [28] reported the first observation of extreme wave events in parametrically driven capillary waves. The evidence of strong four-wave coupling in nonlinear waves was first present. And they pointed out that modulation instability is the main mechanism of the occurrence of rogue wave. Dispersive focusing of unidirectional wave packets. In 2011, one of the possible mechanisms of the occurrence of rogue wave in deep water was analyzed by E. Pelinovsky et al. [29], based on the dispersive focusing of unidirectional wave packets. This mechanism is associated with the frequency dispersion of water waves and reflected in the interference of many spectral components which are moving with different velocities. Under typical conditions, the characteristic lifetime of an abnormal wave in the framework of this mechanism is approximately two minutes. During its lifetime, a rogue wave changes its shape from a high ridge to a deep depression. Dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth and wave-current interaction [30]. Effect of wind. On the one hand, the observation shows that rogue wave may occur in the presence of wind; on the other hand, focused and unfocused phase asymmetric behavior has been found based on the dispersive focusing mechanism of rogue wave formation under calm and windy conditions. Jeffreys sheltering theory played a significant role in the coherence of anomalous wave group. Numerical simulation verified that the lifetime of rogue wave increases with the higher wind speed [31]. Randomly united by some solitary waves. Soliton interaction with energy exchanging.

In recent years, with the developments of researches on both theory studies and experiment observations, rogue wave phenomena have been observed in optical pulses, BEC, superfluid, atmospherics and finance, besides in the ocean.

12

1 The Research Process for Rogue Waves

In 2010, G. Genty et al. [32] discussed the generation process of optical rogue wave in terms of interaction and turbulence. Rogue soliton can be generated from the thirdorder dispersion or Raman scattering independently, which was concluded from the simulations of picosecond pulse propagation in optical fibers. Simulations of rogue soliton emergence with dispersive perturbation in the long-distance limit were also presented [32]. In 2010, L. Stenflo and M. Marklund [33] showed the possibility of the existence of rogue wave in atmosphere. In 2010, Z. Y. Yan [34] performed the financial rogue waves in a Ivancevic option pricing model, which can replace the Black–Scholes model. These rogue wave solutions can depict the possible physical mechanisms for rogue wave phenomenon in financial markets and related fields. There exist some problems in the studying of the rogue wave: (1) Formation mechanism and characters of rogue wave; (2) Measurement of rogue wave; (3) New discovery of rogue wave models; (4) Applications of rogue wave; (5) Experiment excitation of rogue wave; (6) Prediction of rogue wave. With the development of science and technology, real-time dynamics monitoring carried by remote-sensing technology has provided massive reliable information of rogue wave. Large number of marine disasters prompted researchers to study the rogue wave. Applications of the rogue wave studies are the prediction of the occurrence of rogue wave based on the rogue wave frequency, marine and ship design, and prevention of economic crisis. In a word, deep understanding of rogue waves not only is good for the ship design, but also has a great significance on the marine safety once we can predict it as weather forecast.

1.4 Mechanisms of Rogue Waves In this chapter, we cite some reports of the mechanisms of rogue wave which are discussed in detail in Ref. [6].

1.4.1 Linear Mechanisms of Rogue Waves 1.4.1.1 Dispersion Enhancement of Transient Wave Groups (Spatio-Temporal Focusing) If a short wave with small group velocity is located in front of a long wave which possesses large group velocity, long wave will overtake the short wave with the time development. During the interaction process, a large-amplitude wave will appear at some fixed time owing to the nonlinear superposition. Afterward, the long wave will be in front of the short wave, and the amplitude of large-amplitude wave train will decrease. A significant focusing of the wave energy can occur when all the quasimonochromatic groups merge at a fixed location. Such an initial specific location of

1.4 Mechanisms of Rogue Waves

13

transient wave groups may lead to the formation of rogue wave with the resonant dynamics of the wave generation by the increasing wind. This evolution can explain the reason of the short lifetime of rogue wave. To illustrate the dispersive focusing of unidirectional water waves quantitatively, we take the kinematic equation for characteristic wave frequency 9 into account 𝜕9 𝜕9 + cgr (9) = 0, 𝜕t 𝜕x

(1.4.1)

where the group velocity cgr (9) = d9/dk can be calculated via the dispersion relation 9 = √gk tanh(kh),

(1.4.2)

with h being the water depth, g being the acceleration of gravity. For convenience, we dc set h as a constant. Multiply d9gr to both sides of Eq. (1.4.1), we get 𝜕 cgr 𝜕t

+ cgr

𝜕 cgr 𝜕x

= 0.

(1.4.3)

The physical explanation of Eq. (1.4.3) is that each component of spectral wave propagates with its own group velocity. The corresponding Riemann wave solution for Eq. (1.4.3) is cgr (x, t) = c0 (. ) = c0 (x – cgr t),

(1.4.4)

where, c0 (x) means the initial distribution of waves with different frequencies in space. The form of such a kinematic wave is continuously varied with distance (time), and its slope is calculated from Eq. (1.4.4) 𝜕 cgr 𝜕x dc0 d.

=

dc0 d. dc 1 + t d.0

.

(1.4.5)

dc

< 0 (or dx0 < 0 at t = 0) stands for the case of long waves behind the short ones, and the initial increase of the slope of the kinematic wave up to infinity with following decrease, which corresponds to the long waves overtaking the short ones during this 1 process. Several waves with different frequencies mixed together at t = max(–dc . 0 /dx) It is obvious that several focusing points are possible for arbitrary transient waves. The self-similar solution of Eq. (1.4.3) can describe the case of all waves meeting at the same point (x0 , Tf ), cgr =

x – x0 . t – Tf

(1.4.6)

Neglecting the capillary effects, the zone of the variable wave group compresses from (gh)1/2 Tf to zero at the fixed time Tf , once the group velocity of the water waves varies

14

1 The Research Process for Rogue Waves

from (gh)1/2 to zero. From the solution (1.4.6), we can see that the corresponding variation of the wave frequency is required for optimal focusing. For example, for the deepwater case (cgr ∼ 91 ), a wave train with a variable frequency 9 ∼ (t0 – t), which could be generated by the paddle in the laboratory tank, is necessary to provide the maximum effect (optimal focusing). The wave amplitude satisfies the energy conservation equation 2 𝜕 A2 𝜕 (cgr A ) + = 0, 𝜕t 𝜕x

(1.4.7)

whose solution can be expressed explicitly as A0 (x)

A(x, t) =

√1 + t

,

dc0 d.

where A0 (x) is an initial distribution of wave amplitude in space. The wave has infinite amplitude at each focal point, while near the focal point, the amplitude A ∼ (Tf –t)–1/2 . The probability of the occurrence of rogue wave should be very high, since the wind waves are seemed as the frequency and amplitude-modulated wave groups, and the infinite wave height at caustics point is predicted by the kinematic approach. But in fact, the actual situation is much more complex. Kinematic approach assumes slow variations of the amplitude and frequency (group velocity) along the wave group; however, the approximation is not valid in the vicinity of the focal points. It is a wellknown problem in the ray methods, not only for water waves. Various expressions of the Fourier integral for the wave field near the caustics made up the generalizations of the kinematic approach in linear theory, which was expressed through the Maslov integral representation. Here, we consider the simplified form of such a representation for conditions of optimal focusing (1.4.6) and use the standard form of the direct and inverse Fourier transformation for water wave displacement, '(x, t) = ∫ '(k) =

+∞

'(k)ei(kx–9 t) dk,

–∞ +∞

1 ∫ ' (x)e–ikx dx, 20 –∞ 0

where '0 (x) = '(x, 0) is the initial water displacement in unidirectional wave field, 9 is the wave frequency which satisfies Eq. (1.4.2). '(x, t) can be derived via the deltafunction and stationary-phase method, '(x, t) = Q√

cgr 20 x |d cgr /dk|

0

ei(kx–9 t– 4 ) ,

(1.4.8)

where cgr is calculated from the conditions for optimal focusing (1.4.6), Q is the intensity of the delta function. At k → 0, Eq. (1.4.8) should be replaced by the following formula:

15

1.4 Mechanisms of Rogue Waves

1

'(x, t) = Q (

with 9 = √gh (1 –

k2 h2 ), 6

1

2 3 2 3 ) Ai [( 2 ) (x – ct)] , 2 cth cth

(1.4.9)

Ai(z) being Airy function. As a consequence, the amplitude 1

1

of the leading wave decreases as the rate of t– 3 and the wave length increases as t 3 .

1.4.1.2 Spatial (Geometrical) Focusing of Water Waves Considering two horizontal coordinates x and y, wave frequency and wave vector should satisfy the following kinematic equations: 𝜕k + ∇ 9 = 0, 𝜕t

∇ × k = 0,

(1.4.10)

where 9=–

𝜕( , 𝜕t

k = ∇ (,

(1.4.11)

with ( being the phase of quasi-monochromatic wave ' = A(x, y, t)ei((x,y,t) . The characteristic forms of the above equations are written as dr 𝜕9 = , dt 𝜕k

dk 𝜕9 =– , dt 𝜕r

(1.4.12)

where the wave frequency 9 satisfies the dispersion relation (1.4.2), r = (x, y). Eq. (1.4.12) is a well-known equation of the ray theory written in Hamiltonian form. Wave amplitude can be given by the two-dimensional version of energy balance equation 𝜕 A2 + ∇ ⋅ (cgr A2 ) = 0, 𝜕t

(1.4.13)

which may transform into the energy flux conservation along the ray tube cgr D A2 = constant, with D being the differential width of the ray defined as the distance between the adjacent rays. Wave amplitude becomes infinite because D equals zero at any focal point. The energy balance equation will not be valid in the vicinity of the caustics. Detailed description of the wave field in the caustics vicinity can be done through the asymptotic Maslov representation or exact solutions for some test cases. For example, if h = h(x), the shallow water wave can be illustrated by the ordinary differential equation

16

1 The Research Process for Rogue Waves

g

d d' [h(x) ] + [92 – gh(x)ky2 ]' = 0, dx dx

(1.4.14)

where 9 stands for the frequency of the monochromatic wave, and ky means the wave number in y-direction. Location of caustics can be obtained with the second bracket of Eq. (1.4.14) being zero. In the neighborhood of caustics, depth can be simplified as h(x) = hc (1 + Lx ) with h = hc at x = 0. Therefore, in the neighborhood of caustics, Eq. (1.4.14) has the Airy equation form 2 d 2 ' ky x' = 0, – L dx2

whose solution can be described by the Airy function 2

'(x) = C ⋅ Ai (–

x ky3 1

),

(1.4.15)

L3 where C stands for a constant and can be determined through the amplitude of the incident wave A0 by using asymptotic expression for the Airy function far from the caustics. The wave field is bounded on the caustics. As a result, the wave amplification on the caustics is 1 Ac ∼ (Lky ) 6 , A0

(1.4.16)

and it is relatively weak for long waves. We need to point out that the spatial focusing procedure involves not only a wave amplification, but also a change of the wave shape. 1.4.1.3 Wave-Current Interaction The problem of the wave-current interaction requires a special investigation, since rogue waves were observed frequently in such strong currents like Gulf Stream and Agulhas Current. Formally, the ray pattern is described by the system (1.4.13) while the dispersion relation should be corrected. Considering the deepwater waves case, the dispersion relation for waves on a steady current becomes anisotropic, 9 = K(k) + k U(x, y),

K = ± √gk.

With Ux (x) only, the wave-current interaction is not trivial in one-dimensional case. When the current is opposite to incident monochromatic wave, it blocks the wave at the point x0 , where the group velocity is zero, cgr =

d9 1 g = √ + U(x0 ) = 0. dk 2 k

(1.4.17)

1.4 Mechanisms of Rogue Waves

17

The wave phase and group velocities have the same sign when the wave approaches the blocking point; after the reflection from the blocking point, they transform to the opposite sign. In the process of interaction, the wave number increases and an initial long wave transforms to a short wave. Wave amplitude can be found from the wave action balance equation 2

cgr A 𝜕 A2 ( )+∇ ⋅( ) = 0, 𝜕t K K

(1.4.18)

which is the generalization of the energy balance equation (1.4.13). For steady currents, Eq. (1.4.18) transforms into the wave action flux, cgr D A2 K

= constant,

with D being the same as above mentioned. For the propagation of unidirectional wave, the blocking point with zero group velocity (1.4.17) plays the role of caustics and the wave amplitude formally tends to infinity. In fact, Eq. (1.4.18) is not valid in the neighborhood of the caustics. More accurate asymptotic analysis should be calculated by the Maslov integral 1

𝜕U 3 [ 8 ] '(x) = C ⋅ Ai [( 𝜕 x ) k∗ (x – x0 )] eik∗ x–9 t , K(k∗ ) [ ]

with k∗ being the value of the wave number at the blocking point determined by Eq. (1.4.17), and 𝜕𝜕 Ux calculated at the same point. As a consequence, wave amplitude at the blocking point is bounded; comparing with Eq. (1.4.16), 1

Ac K 6 ∼ ( dU ) . A0 dx

1.4.2 Nonlinear Mechanisms of Rogue Waves Nonlinear mechanisms of the rogue wave will be analyzed here in the weakly nonlinear limit. In theory, the description of nonlinear mechanisms is more perfect than the linear mechanisms. 1.4.2.1 Weakly Nonlinear “Rogue Wave Packets in Deep and Intermediate Depths The simplified nonlinear model of 1+1 quasi-periodic deepwater wave trains in the lowest order in wave steepness and spectral width is based on NLS equation i(

9 𝜕2 A 90 k02 2 𝜕A 𝜕A + + cgr ) = 02 |A| A, 𝜕t 𝜕x 2 8k0 𝜕 x2

(1.4.19)

18

1 The Research Process for Rogue Waves

where the surface elevation '(x, t) is given by '(x, t) =

1 (A(x, t)ei(k0 x–90 t) + c.c. + ⋅ ⋅ ⋅) , 2

(1.4.20)

with k0 and 90 being the wave number and frequency of the carrier wave, respectively, c.c. being the complex conjugate, and “⋅ ⋅ ⋅” being the weak highest harmonics of the carrier wave. The complex wave amplitude A is a slowly varying function of x and t. NLS equation plays an important role in the understanding of nonlinear dynamics of water waves. It is well known that a uniform train of amplitude A0 is unstable to the modulation instability of the long disturbances of wave number B k, i.e., Bk < 2√2k0 A0 . k0 The maximum instability occurs at 2

Bk k0

(1.4.21)

= 2k0 A0 with the maximum growth rate that is

90 (k0 A0 ) 2

. The nonlinear stage of the modulation instability has been deeply equal to investigated analytically, numerically and experimentally. In fact, the rogue wave solutions for NLS equation can describe the rogue wave phenomenon in deepwater properly. For 3D wave trains, the governed (2 + 1)-D NLS equation is i(

9 𝜕2 A 90 𝜕2 A 90 k02 2 𝜕A 𝜕A + cgr ) = 02 |A| A. – + 𝜕t 𝜕x 2 8k0 𝜕 x2 4k02 𝜕 y2

(1.4.22)

The above 2 + 1-D NLS equation is principally anisotropic, and modulations of wave packets in the longitudinal and transversal directions behave differently. Specially, modulations in the transverse direction are stable. The domain of the modulation instability is found as √2B ky k0


i = –&i 6i , when N = 2l + 1; { { ! > + !1 &i 6i + ⋅ ⋅ ⋅ + !2l–2 &i2l–2 >i + !2l–1 &i2l–1 6i = –&i2l >i , when N = 2l, { 0 i

i = 1, 2, . . . , N. And "i = !i (>j ↔ 6j ), (i, j = 1, 2, . . . , N). The transformation between the fields is displayed as follows: 1. When N = 2l + 1 u[N] = –v – [

det(B) ] , det(A) x

v[N] = –u + [

det(B(>j ↔ 6j )) det(A(>j ↔ 6j ))

] , x

(2.6.16)

2.6 Generalized Darboux Transformation for DNLS Equation

63

where A = (AT1 , AT2 , . . . , ATN ), B = (BT1 , BT2 , . . . , BTN ), Ai = (>i , &i 6i , . . . , &i2l–2 >i , &i2l–1 6i , &i2l >i ), Bi = (>i , &i 6i , . . . , &i2l–2 >i , &i2l–1 6i , &i2l+1 6i ). 2.

When N = 2l u[N] = u – [

det(D) ] , det(C) x

v[N] = v + [

det(D(>j ↔ 6j )) det(C(>j ↔ 6j ))

] ,

(2.6.17)

x

where C = (C1T , C2T , . . . , CNT ), D = (DT1 , DT2 , . . . , DTN ), Ci = (>i , &i 6i , . . . , &i2l–3 6i , &i2l–2 >i , &i2l–1 6i ), Di = (>i , &i 6i , . . . , &i2l–3 6i , &i2l–2 >i , &i2l >i ). 2.6.2 Darboux Transformation-II In this section, we will present the so-called Dressing–Bäcklund transformation [147, 151], which we denoted as Darboux transformation-II. In fact, Darbouxtransformation-II is the second-iteration of Darboux transformation-I. For convenience, we rewrite D[1] and D[1]–1 as D[1] = (

–&1 6>1

&

&

–&1 >

1

61 ) ,

D[1]–1 =

1

&1 6>1 1 1 ( & 2 – &12 &

&

&1 6>1

).

(2.6.18)

1

Suppose J1 = (71 , 81 ) is a solution for the adjoint system (2.6.4) at & = .1 , then J1 [1] = J1 D[1]–1 |& =.1 is a new solution for adjoint system (J[1], U[1], V[1]) at & = .1 . It is easy to see that 31 J1 [1]T is a special solution for Lax Pair (I[1], U[1], V[1]) at & = –.1 . Therefore, the second step Darboux transformation D[2] can be constructed through the seed solution 31 J1 [1]T . Omitting the overall factor & 2 – .12 , we have the Darboux transformation-II T[1] = I +

3 A3 A – 3 3, & – .1 & + .1

A=

.12 – &12 ! 0 ) I1 J1 , ( 0 " 2

(2.6.19)

where .1 0 ) I1 , 0 &1

!–1 = J1 (

&1 0 ) I1 . 0 .1

"–1 = J1 (

Furthermore, we have T[1]–1 = I +

3 B3 B – 3 3, & – &1 & + &1

B=

&12 – .12 " 0 ). I1 J1 ( 0 ! 2

(2.6.20)

64

2 Construction of Rogue Wave Solution by the Generalized Darboux Transformation

̂ is The relationship between Q and new potential function Q ̂ = Q + (A – 3 A3 ) . Q 3 3 x

(2.6.21)

Above discussion indicates that T[1]; i.e., Darboux transformation-II, is indeed a two-fold Darboux transformation-I. In what follows, we consider the iteration of the Darboux transformation-II. Assume Ii (,i ) = (>i , 6i )T and Ji (-i ) = (7i , 8i ) are the N distinct solutions for Eq. (2.6.3) at & = ,i and Eq. (2.6.4) at & = -i , respectively. Like Darboux transformation-I, we have the following proposition of Darboux transformation-II. Proposition 2.6.2. The N-fold Darboux transformation for Darboux transformation-II could be written in the form of N

TN = T[N]T[N – 1] ⋅ ⋅ ⋅ T[1] = I + ∑ ( i=1

3 C3 Ci – 3 i 3), & – -i & + -i

(2.6.22)

and N

TN–1 = T[1]–1 T[2]–1 ⋅ ⋅ ⋅ T[N]–1 = I + ∑ ( i=1

3 D3 Di – 3 i 3). & – ,i & + ,i

(2.6.23)

Proof. Calculating the residues for the both sides of Eq. (2.6.22) 3 A 3 3 A3 AN A1 – 3 N 3 ) ⋅ ⋅ ⋅ Ai ⋅ ⋅ ⋅ (I + – 3 1 3), -i – -N - i + -N -i – -1 - i + -1 33 AN 33 3 A3 AN A1 Res|& =–-i (TN ) = – (I + – ) ⋅ ⋅ ⋅ 33 Ai 33 ⋅ ⋅ ⋅ (I + – 3 1 3 ). –-i – -N –-i + -N –-i – -1 –-i + -1 Res |& =-i (TN ) = (I +

Because of Res |& =-i (TN ) = –33 Res|& =–-i (TN )33 , equation Eq. (2.6.22) is valid. Similarly, equation Eq. (2.6.23) can also be proved. ◼ The N-fold Darboux transformation-II TN shows us the relationships between the fields u, v and u[N], v[N], which are expressed in Theorem 2.6.2. Theorem 2.6.2. The relationship between the fields are u[N] = u – 2 (

det M1 ) , det M x

where M = (Mij )N×N , N = (Nij )N×N , Mij =

v[N] = v + 2 (

Ji 33 Ij ,j +-i

–

Ji Ij , ,j –-i

det N1 ) , det N x

Nij = – [

Ji 33 Ij ,j +-i

(2.6.24)

+

Ji Ij ], ,j –-i

2.6 Generalized Darboux Transformation for DNLS Equation

M11 M21 ( . M1 = ( .. MN1 ( 61

M12 M22 .. . MN2 62

⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅ ⋅⋅⋅

M1N M2N .. . MNN 6N

81 82 .. ) . ), 8N 0)

N11 N21 ( . N1 = ( .. NN1 ( >1

N12 N22 .. . NN2 >2

⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅ ⋅⋅⋅

N1N N2N .. . NNN >N

65

71 72 .. ) . ). 7N 0)

Proof. Since TN (2.6.22) is the N-fold Darboux transformation for Eq. (2.6.3), we have TN,x + TN U = U[N]TN . Further, N

Q[N] = Q + ∑[Ci – 33 Ci 33 ]x . i=1

Thus, we need to calculate the explicit forms for Ci . From Proposition 2.6.2, we get Ci = Res|& =,i (TN ), which implies that the rank of matrix Ci is 1. Thus, we take Ci = |xi ⟩⟨yi | and Di = |wi ⟩⟨vi |. On the one hand, considering TN TN–1 = I and the residue of TN TN–1 at & = -l being zero, we have ⟨yl |TN–1 |& =-l = 0.

(2.6.25)

On the other hand, Jl TN–1 |& =-l = 0. Noting that the rank of TN–1 |& =-i is 1, we may obtain ⟨yl | = Jl . Substituting ⟨yl | into Eq. (2.6.25), we arrive at N

Il + ∑ ( i=1

|xi ⟩Ji Il 33 |xi ⟩Ji 33 Il – ) = 0 (l = 1, 2, . . . , N). ,l – -i ,l + -i

(2.6.26)

Solving Eq. (2.6.26), it follows that [|x1 ⟩, |x2 ⟩, . . . , |xN ⟩]1 = [>1 , >2 , . . . , >N ] M –1 , [|x1 ⟩, |x2 ⟩, . . . , |xN ⟩]2 = [61 , 62 , . . . , 6N ] N –1 , where subscripts 1 and 2 represent the first and second rows, respectively. Finally, the relations between the fields are displayed as

66

2 Construction of Rogue Wave Solution by the Generalized Darboux Transformation

81 [ 8 ]] [ [ 2 ]] det M1 –1 [ [ ]] u[N] = u + 2 [ [[>1 , >2 , . . . , >N ] M [ .. ]] = u – 2 ( det M ) , x [ . ]] [ 8 [ N ]]x [ 71 [ 7 ]] [ [ ] [ det N1 –1 [ 2 ] ]] v[N] = v – 2 [ [[61 , 62 , . . . , 6N ] N [ .. ]] = v + 2 ( det N ) . x [ . ]] [ [7N ]]x [ ◼

The proof is completed.

2.6.3 Reductions Till now, the Darboux transformation for the general Lax Pair (2.6.3) and certain solution formulas for Eq. (2.6.1) have been given. The solutions for DNLS equation can be obtained after the reduction. We consider the reductions v = u∗ and v = –u∗ , which are simply related [149]. For the Darboux transformation-I, we assume v = u∗ or Q† = –Q, where † means the complex conjugation and matrix transpose. To implement the reduction, we need to choose the seed solutions properly. Indeed, assuming &1 ∈ iℝ and

>1 = 6∗1 ,

(2.6.27)

then Q[1], which is defined by Eq. (2.6.8), satisfies the reduction relation Q[1]† = –Q[1]. The bright or dark soliton solutions for DNLS equations with the nonvanishing background can be obtained through the Darboux transformation (2.6.5) under the reduction condition (2.6.27). Next, we take into account the reduction of Darboux transformation-II. Assuming Q = –Q† and .1 = &1∗

and (71 , 81 ) = (>∗1 , 6∗1 ),

(2.6.28)

̂ = –Q ̂† . Eq. (2.6.21) yields Q To iterate the reduced Darboux transformations-I and -II, we must verify that they keep the reduction conditions (2.6.27) and (2.6.28). Equation (2.6.28) merely depends on the symmetry of Eq. (2.6.3); thus it holds automatically. For the former Eq. (2.6.27), we claim the following proposition. Proposition 2.6.3. Both Darboux transformation D[1] and T[1] keep the reduction condition (2.6.27) invariant. Proof. Direct calculations.

◼

2.6 Generalized Darboux Transformation for DNLS Equation

67

From the above analysis, both Darboux transformations-I and -II could be reduced to find solutions for DNLS equation. The Darboux transformation-I under Eq. (2.6.27) is conveniently used to construct the N-dark or bright soliton solutions for DNLS equation with nonvanishing boundary condition, while Darboux transformationII under Eq. (2.6.28) can be applied to obtain the N-bright soliton solutions and N-breathers for DNLS equation.

2.6.4 Generalized Darboux Transformations In this section, we construct the corresponding generalized Darboux transformation for D[1] and T[1]. We will follow the way which is proposed for the NLS equation [118]. In fact, both Darboux transformations-I and -II are reduced at & = &1 , because of D[1]|& =&1 I1 = T[1]|& =&1 I1 = 0; then we have I[1] 1 = lim

(D[1]I1 )|& =&1 +: :

:→0

,

or (T[1]I1 )|& =&1 +:

I[1] 1 = lim

:

:→0

,

which serves the seed solution for doing the next step transformation. 2.6.4.1 Generalized Darboux Transformation-I To construct the generalized Darboux transformation-I, we assume that the n distinct solutions for Lax Pair at & = &i (i = 1, . . . , n) are given. First, the elementary Darboux transformation is displayed as –&1 6>1

D[0] 1 =(

&

1

& –&1 6>1

).

1

As observed earlier, via the limit process, we find that 󵄨󵄨 󵄨 D[0] 1 󵄨󵄨& =&1 + :31 >1 (&1 + :) > (& ) >[1] d >1 (& ) 󵄨󵄨 󵄨 ) = D[0] ) + 31 ( 1 1 ) ( 1[1] ) = lim ( ( 1 󵄨󵄨& =&1 :→0 61 (&1 + :) 61 (&1 ) : d& 61 (& ) & =& 61 1

is a nonzero solution for Lax Pair (2.6.3) with u = u[1] and v = v[1] at & = &1 , which makes sure the next step iteration –&1

D[1] 1 =(

>[1] 1 6[1] 1

&

& –&1

6[1] 1

).

>[1] 1

Continuing this procedures, we have the following theorem.

68

2 Construction of Rogue Wave Solution by the Generalized Darboux Transformation

Theorem 2.6.3. Let (>i , 6i )T be the solutions of Lax Pair at & = &i (i = 1, . . . , n), and assume that Darboux transformation-I owns mi -order zeros at & = &i . Thus, the following generalized Darboux transformation-I is obtained: [m1 –1]

[0] DN = Dn[mn –1] ⋅ ⋅ ⋅ D[1] n Dn ⋅ ⋅ ⋅ D 1

[0] ⋅ ⋅ ⋅ D[1] 1 D1 ,

(2.6.29)

where n

N = ∑ mi , i=1

and [j–1]

–&i

=( D[j] i

>i

&

[j–1]

6i

&

–&i

[j–1] ) ,

6i

[j–1]

>i

Kj–1–l d >i 󵄨󵄨 >[j–1] i ( ) 󵄨󵄨 , [j–1] ) = ∑ l! d& l 6i 󵄨& =&i 6i l=1 j–1

(

(

>[0] > i ) = ( i) , 6 6[0] i i

(2.6.30)

and

Kl = ∑ Mi[j–2] ⋅ ⋅ ⋅ Mi[0] ⋅ ⋅ ⋅

[m –1] M1 1

⋅ ⋅ ⋅ M1[0] ,

Mi[k]

∑ $ki =l

3, when $ki = 1; { { 1 ={ { [k] 󵄨󵄨󵄨 , when $ki = 0. D { i 󵄨󵄨& =&i

Proof. To construct the generalized Darboux transformation-I, we start with the elementary Darboux transformation –&1 6>1 & 1 D[0] = ( ). 1 & –&1 6>1 1

By virtue of the nonzero solutions (>1 [1], 61 [1]), the next step Darboux transformaT tion D[1] 1 will be carried out. Taking the given seed solutions (>i , 6i ) into account, we perform the following limit: >[j] ( i[j] ) 6i

[m1 –1]

[D[j–1] ⋅ ⋅ ⋅ D[1] D[0] ⋅ ⋅ ⋅ D1 i i i = lim

:→0

:j

󵄨 [0] 󵄨󵄨󵄨 ⋅ ⋅ ⋅ D[1] 1 D1 ] 󵄨󵄨 󵄨& =&i +:

> (& + :) ), ( i i 6i (&i + :)

which yields the formulas presented in the above theorem. The proof is completed. ◼

2.6 Generalized Darboux Transformation for DNLS Equation

69

To have a compact determinant form for the generalized Darboux transformation, we may take the limit process directly on the N-fold Darboux transformation-I (2.6.15). From Eqs (2.6.16) and (2.6.17), we get the following transformation between fields: 1. If N = 2l + 1 u[N] = –v – [

det(B) ] , det(A) x

v[N] = –u + [

det(B(>j ↔ 6j )) det(A(>j ↔ 6j ))

] ,

(2.6.31)

x

where

2.

A = (AT1 ,

d T dm1 –1 dmn –1 T T d T A , . . . , A , , . . . , AT ) , A1 , . . . , A n d& d& n (m1 – 1)!d& m1 –1 1 (mn – 1)!d& mn –1 n

B = (BT1 ,

d T dm1 –1 dmn –1 T T d T B1 , . . . , B B , . . . , B , , . . . , BT ) . n d& d& n (m1 – 1)!d& m1 –1 1 (mn – 1)!d& mn –1 n

and Ai , Bi are the same as Eq. (2.6.16). If N = 2l, u[N] = u – [

det(D) ] , det(C) x

v[N] = v + [

det(D(>j ↔ 6j )) det(C(>j ↔ 6j ))

] ,

(2.6.32)

x

where C = (C1T ,

d T dmn –1 dmn –1 T T d T C1 , . . . , C C , . . . , C , , . . . , CT ) , n d& d& n (mn – 1)!d& mn –1 1 (mn – 1)!d& mn –1 n

D = (DT1 ,

d T dmn –1 dmn –1 T T d T D , . . . , D , , . . . , DT ) . D1 , . . . , D n d& d& n (mn – 1)!d& mn –1 1 (mn – 1)!d& mn –1 n

and Ci , Di are the same as Eq. (2.6.17). Therefore, the construction of Darboux transformation-I for Eq. (2.6.3) is completed, which can be considered as a generalization of Darboux transformation studied in Refs [148–150].

2.6.5 Generalized Darboux Transformation-II In this part, we consider the generalization of Darboux transformation-II. Ii (& = ,) = (>i , 6i )T are the n solutions for Lax Pair (2.6.3) at , = ,i , and Ji (& = -) = (7i , 8i ) are the n solutions for the adjoint Lax Pair (2.6.4) at - = -i , (i = 1, . . . , n). Theorem 2.6.4. Let Ii be the solutions for Lax Pair (2.6.3) at & = ,i , Ji be the solutions for adjoint Lax Pair (2.6.4) at & = -i (i = 1, . . . , n), r

∑ mi = N. i=1

70

2 Construction of Rogue Wave Solution by the Generalized Darboux Transformation

Assume that Darboux transformation-II possesses mi -order zeros at & = ±,i , and its reverse possesses mi -order zeros at & = ±-i . Then we have the following generalized Darboux transformation-II: [mi –1]

TN = Tn

[m1 –1]

⋅ ⋅ ⋅ Tn[0] ⋅ ⋅ ⋅ T1

(2.6.33)

[mi –1] –1

[m1 –1] –1

TN–1 = (T1[0] )–1 ⋅ ⋅ ⋅ (T1

⋅ ⋅ ⋅ T1[0] ,

⋅ ⋅ ⋅ (Tn[0] )–1 ⋅ ⋅ ⋅ (Tn

)

)

where Ti[j] = I + A[j] = i

A[j] i & – -i

–

3 33 A[j] i 3 & + -i

(Ti[j] )–1 = I +

,

-i2 – ,2i ![j] 0 ( i [j] ) I[j] J[j] , i i 2 0 "i

B[j] = i

-i 0 ) I[j] , i 0 ,i

)–1 = J[j] ( (![j] i i

B[j] i & – ,i

–

3 33 B[j] i 3 & + ,i

,

,2i – -i2 [j] [j] "[j] 0 Ii Ji ( i [j] ) , 2 0 !i ,i 0

0 ) I[j] , i -i

[m1 –1]

⋅ ⋅ ⋅ M1[0] ,

("[j] )–1 = J[j] ( i i

and

I[j] i

j

j

Kl dj–l =∑ I| , (j – l)! d,j–l i ,=,i l=0

J[j] i

Kl = ∑ Mi[j–1] ⋅ ⋅ ⋅ Mi[0] ⋅ ⋅ ⋅ M1 j

j

Dl dj–l , = ∑ j–l Ji |-=-i (j – l)! l=0 d,

Mi[j]

∑ $i =l

∑ $i =l

[m1 –1]

Dl = ∑ Ni[j–1] ⋅ ⋅ ⋅ Ni[0] ⋅ ⋅ ⋅ N1

1 , if $ji = 2; { ,2i –-i2 { { { { { { 2, +A[j] –3 A[j] 3 = { i i 2 32 i 3 , if $j = 1; , i { ,i –-i { { { { { [j] 󵄨󵄨 if $ji = 0. T 󵄨 , { i 󵄨󵄨& =,i

Ni[j]

⋅ ⋅ ⋅ N1[0] ,

1 , if $ji = 2; { -i2 –,2i { { { { { { 2- +B[j] –3 B[j] 3 = { i i 2 32 i 3 , if $j = 1; i { -i –,i { { { { { [j] –1 󵄨󵄨 (T ) 󵄨󵄨󵄨& =- , if $ji = 0. { i i

Proof. Noting that Darboux transformation-II is given by Eq. (2.6.19), and using the limit technique, we could obtain the special solutions for Lax Pair (2.6.3) (I[1], U[1], V[1]) at & = ,1 and adjoint Lax Pair (2.6.4) (J[1], U[1], V[1]) at & = -1 , i.e. I[1] 1 = lim

$→0

J[1] 1 = lim

$→0

T1[0] |& =,1 +$ I1 (,1 + $) $

= T1[0] |& =,1

J1 (-1 + $)T1[0]–1 |& =-1 +$ $

=

d + S1 I1 (,1 ), I| d, 1 ,=,1

d J | T [0]–1 |& =-1 + J1 (-1 )R1 , d- 1 -=-1 1

71

2.6 Generalized Darboux Transformation for DNLS Equation

where S1 =

2,1 + A1 – 33 A1 33 , ,21 – -12

R1 =

2-1 + B1 – 33 B1 33 . -12 – ,21

Thus, we may continue to construct the new Darboux transformation T1[1] = I +

A[1] 33 A[1] 1 1 33 – , & – -1 & + -1

(T1[1] )–1 = I +

B[1] 33 B[1] 1 1 33 – . & – ,1 & + ,1

Generally speaking, considering the given seed solutions Ii and Ji , we perform the following limit [m1 –1]

I[j] = lim i

(Ti[j–1] ⋅ ⋅ ⋅ Ti[0] ⋅ ⋅ ⋅ T1

⋅ ⋅ ⋅ T1[0] ) |& =,i +$

$j

$→0

[m1 –1] –1

J[j] = lim Ji (-i + $) i

(T1[0]–1 ⋅ ⋅ ⋅ (T1

)

Ii (,i + $),

⋅ ⋅ ⋅ (Ti[0] )–1 ⋅ ⋅ ⋅ (Ti[j–1] )–1 ) |& =-i +$ $j

$→0

,

and mathematical induction leads to the generalized Darboux transformationII (2.6.33). The proof is completed. ◼ Due to the above proposition, the transformations between fields are n mi –1

) ] , u[N] = u + 2 ∑ ∑ [(A[j] i 12 i=1 j=0

(2.6.34)

x

n mi –1

v[N] = v – 2 ∑ ∑ [(A[j] ) ] . i 21 i=1 j=0

(2.6.35)

x

As mentioned before, the formulas Eqs (2.6.34) and (2.6.35) could be rewritten in terms of determinants, i.e., u[N] = u – 2 (

det(P1 ) ) , det(P) x

v[N] = v + 2 (

det(Q1 ) ) det(Q) x

(2.6.36)

where P[11] P[21] ( P1 = ( ... P[r1] ̂1 (>

P[12] P[22] .. . P[r2] ̂2 >

⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅ ⋅⋅⋅

P[1r] P[2r] .. . P[rr] ̂r >

̂ 8 1 P[11] ̂ 8 2 [21] .. ) , P = (P ) .. . . ̂ 8 r P[r1] 0)

P[12] P[22] .. . P[r2]

⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅

P[1r] P[2r] .. ) , . P[rr]

72

2 Construction of Rogue Wave Solution by the Generalized Darboux Transformation

with 󵄨󵄨 󵄨󵄨 , 󵄨󵄨 󵄨-=-j 󵄨 mj –1 󵄨 𝜕 ̂j = (>j , 𝜕, >j , . . . , (m 1–1)! 𝜕 mj –1 >j ) 󵄨󵄨󵄨󵄨 > , j 𝜕, 󵄨,=,i

̂ = (8 , 8 i i

𝜕 8, 𝜕- i

... ,

𝜕mi –1 1 8) (mi –1)! 𝜕-mi –1 i

[ij] ) P[ij] = (Pkl [ij] Pkl =

mk ,ml

T

,

Ji (-)33 Ij (,) Ji (-)Ij (,) 󵄨󵄨󵄨 𝜕k+l–2 1 ( , – ) 󵄨󵄨󵄨 k–1 l–1 (k – 1)!(l – 1)! 𝜕- 𝜕, ,+,–󵄨,=,i ,-=-j

and Q[11] Q[12] ⋅ ⋅ ⋅ Q[1r] [21] [22] Q Q ⋅ ⋅ ⋅ Q[2r] ( .. .. .. Q1 = ( ... . . . Q[r1] Q[r2] ⋅ ⋅ ⋅ Q[rr] ̂ ̂ ̂ 6 ⋅⋅⋅ 6 2 r ( 61

7̂1 Q[11] 7̂2 [21] .. ) , Q = (Q ) .. . . 7̂r Q[r1] 0)

Q[12] Q[22] .. . Q[r2]

⋅ ⋅ ⋅ Q[1r] ⋅ ⋅ ⋅ Q[2r] .. ) , .. . . ⋅ ⋅ ⋅ Q[rr]

with 󵄨󵄨 󵄨󵄨 , 󵄨󵄨 󵄨-=-i 󵄨󵄨 mj –1 𝜕 1 𝜕 ̂ = (6 , 6 , ⋅ ⋅ ⋅ , 6 ) 󵄨󵄨 6 , j 𝜕, j j (mj –1)! 𝜕,mj –1 j 󵄨󵄨󵄨 ,=,j 7̂i = (7i ,

𝜕 7, 𝜕- i

⋅⋅⋅ ,

𝜕mi –1 1 7) (mi –1)! 𝜕-mi –1 i

[ij] ) Q[ij] = (Qkl [ij] Qkl =–

mk ,ml

T

,

Ji (-)33 Ij (,) Ji (-)Ij (,) 󵄨󵄨󵄨 𝜕k+l–2 1 ( . + ) 󵄨󵄨󵄨 k–1 l–1 (k – 1)!(l – 1)! 𝜕- 𝜕, ,+,–󵄨,=,i ,-=-j

From the above theorems, it is easy to see that the reductions (2.6.27) and (2.6.28) are still valid for generalized Darboux transformations-I and -II. For the generalized Darboux transformation-II, Lax Pair (2.6.3) can be reduced to DNLS equation under ,i = -i∗ . 2.6.6 High-Order Solutions for DNLS Equation We will derive some new solutions via Darboux transformation. Here, we only introduce the solution, without the analysis of the soliton dynamics and types. 2.6.6.1 Solutions with Vanishing Boundary Condition Via Darboux transformation-I, three kinds of solutions will be obtained based on the zero seed solutions, namely plane wave solutions, N-phase solutions (periodic

73

2.6 Generalized Darboux Transformation for DNLS Equation

solutions) and N-soliton solutions (as seen in Ref. [150]). In addition, we can get the rational solutions when we take limit of the spectral parameter [146, 150]. In this section, we consider the N-rational solution first. In the first two cases, generalized Darboux transformation-I will be used, while for case 3, it is more convenient to use the generalized Darboux transformation-II. Case 1: N-rational solutions. For the seed solution u = 0, the special solution for Lax Pair (2.6.3) with the reduction u = v∗ is written as –2

–2

e–i& (x+2& t+c) > ( ) = ( i& –2 (x+2& –2 t+c) ) , 6 e

(2.6.37)

where c is a complex constant, which will be taken as a polynomial function of & . To obtain the N-rational solutions, we introduce the vectors y = (>, &6, . . . , & 2N–2 >, & 2N >),

z = (>, &6, . . . , & 2N–2 >, & 2N–1 6),

and define the matrix T

Y = (y1 , y1(1) , . . . , yN , yN(1) ) ,

T

Z = (z1 , z1(1) , . . . , zN , zN(1) ) ,

with yi = y|& =&i ,c=ci and zi = z|& =&i ,c=ci , the superscript (1) being the first-order derivative to & . Then the N-rational solution can be presented as u[N] = – (

det (Y) ) . det (Z) x

(2.6.38)

Taking &1 = ia, we have 2i(a2 x–2t+a2 c) a4

4a3 [4i(a2 x – 4t + a2 c) – a4 ]e u[1] = [4i(a2 x – 4t + a2 c) + a4 ]2

.

The velocity is a2 /4, and the center is along the line a2 x – 4t + a2 c = 0. The amplitude of |u[1]|2 is 16/a2 . After asymptotic analysis, we can see that this two-rational soliton does not possess phase shift at t → ±∞, which is different from the two-soliton solutions for NLS equation. Case 2: High-order rational solitons. Next, we consider the high-order rational soliton solutions. Set the matrix T

Y1 = (y1 , y1(1) , ⋅ ⋅ ⋅ , y1(2N–1) ) ,

T

Z1 = (z1 , z1(1) , ⋅ ⋅ ⋅ , z1(2N–1) ) ,

where the superscript (i) stands for the i-th derivative with respect to & . Therefore, the N-order rational solutions for DNLS equation with vanishing boundary conditions can be expressed as

74

2 Construction of Rogue Wave Solution by the Generalized Darboux Transformation

u[N] = – (

det(Y1 ) ) . det(Z1 ) x

(2.6.39)

Case 3: High-order solitons. Setting u = 0 as the seed solution, we derive the solutions for Lax Pair (2.6.3) under reduction u = v∗ , –2

–2

e–i, (x+2, t+c) > ( ) = ( i,–2 (x+2,–2 t+d) ) , 6 e and the special solution for the adjoint Lax Pair (2.6.4) under the reduction u = v∗ (7, 8) = (ei-

–2

(x+2-–2 t+c∗ )

, e–i-

–2

(x+2-–2 t+d∗ ) ) ,

where c and d are constants depending on the spectral parameters. N-order soliton solutions for DNLS equation reads as u[N] = –2 (

det(M1 ) ) , det(M) x

M = (Mij )N×N

(2.6.40)

where

Mij =

di+j–2 2[-ei(d-i–1 d,j–1

–2

M YT M1 = (

),

X = (>,

X 0

–,–2 )[x+2(-–2 +,–2 )t]

+ ,e– i(-2 – ,2

d dN–1 >, . . . , N–1 >) , d, d,

–2

–,–2 )[x+2(-–2 +,–2 )t]

Y = (8,

]

|-=,∗ ,

d dN–1 8, . . . , N–1 8) . dd-

The intensities of two- and three-soliton are plotted in Fig. 2.9.

(a)

(b) 6

2

|u|

|u|

4 2

2

10 8 6 4 2 20

60 20

t

0 –20 –5 –60 –10

5

x

10

20 t –20 –10

–5

0

5

10

x

Figure 2.9: Intensities of high-order soliton with parameters , = 1 + i, c = d = 0. (a) Two soliton and (b) three soliton.

2.6 Generalized Darboux Transformation for DNLS Equation

75

2.6.6.2 Solutions with Nonvanishing Boundary Condition The plane wave solutions with nonvanishing boundary condition can be obtained by Darboux transformation-I from the zero seed solution [149]. Thus, we will consider the higher-order rational solutions based on the zero solution. Also, we may apply the Darboux transformation to the general plane wave solutions. In the following, we will consider the higher-order rational solutions corresponding to the two cases, respectively. Case 1: Higher-order rational solutions from the zero solution. To obtain the higher-order rational solutions with nonvanishing background condition, the order of determinants should be odd. Define the matrices, T ̂ = (̂ Y y1 , ŷ1 (1) , . . . , ŷ1 (2N) ) , 1

T Ẑ1 = (̂ z1 , ẑ1 (1) , . . . , ẑ1 (2N) ) ,

where ŷ1 = (>, &1 6, . . . , &12N–1 6, &12N+1 6),

ẑ1 = (>, &1 6, . . . , &12N–1 6, &12N >),

6, 8 are given by Eq. (2.6.37). Then the higher-order rational solutions with nonvanishing boundary condition can be expressed as u[N] = – (

̂) det(Y 1 ) . det(Ẑ ) 1

(2.6.41)

x

Taking the parameters &1 = ia (a is a real constant) and c = 0, we have the first-order rational solutions with nonvanishing boundary condition 2L2 L∗1 – 2i(a 4x–2t) a e , aL21 2

u[1] =

(2.6.42)

where L1 = 16. 2 + a8 + 8ia4 (. – 4t), L2 = 16. 2 – 3a8 – 8ia4 (. + 4t), and . = a2 x–4t. The norm of u[1] contains the maximum value √ (0, ± 163 a4 ).

6 a

at the origin point and

The “ridge” of the solution (2.6.42) lays approximately vanishes at (x, t) = on the line a2 x – 4t = 0, and decays to a2 slowly. Case 2 : Higher-order rational solutions from the plane wave solution. To construct these solutions, we choose the seed solutions as u = A exp(2i(2 ), where (2 = 21 [ax – (A2 a + a2 )t + c], c ∈ ℝ. The corresponding fundamental solution for Lax Pair (2.6.3) is

76

2 Construction of Rogue Wave Solution by the Generalized Darboux Transformation

I=(

ei((1 +(2 )+6 e–i((1 –(2 )–6 ), e–i((1 +(2 )+6 ei((1 –(2 )–6

(2.6.43)

where 2 + a& 2 1 arccos (– ), 2 2iA&

(1 = and

1 6 = √–(2& –2 + a)2 – 4A2 & –2 [x – (a + A2 – 2& –2 )t + d]. 2 To resolve the reduction (2.6.27), we assume –4A2 & 2 – (2 + a& 2 )2 > 0, d ∈ ℝ, & ∈ iℝ. The corresponding bright and dark solitons have been studied and analyzed in Refs. [149, 150]. Here, we consider the limit cases. Under the case –4A2 & 2 – (2 + a& 2 )2 = 0, I given by the formula (2.6.43), which does not qualify as the fundamental solution, is a constant. Thus, the generalized Darboux transformation could not find interesting solutions. To derive meaningful solutions, nonconstant solution for Eq. (2.6.3) is essential. We turn to the limit technique. For convenience, we consider the special case a = c = 0, A = 1, which leads to the genuine rational solutions. Expanding the solutions for Lax Pair at & = i with the special solution I1 =

I|& =i(1+f ) C f 1/2

,

1 C = ( ), 1

at f = 0, which satisfies the reduction (2.6.27), we have I1 = Y0 + Y1 f + ⋅ ⋅ ⋅ + Yn f n + ⋅ ⋅ ⋅ , x 1 dn I1 (f ). Yn = ( n ) = lim f =0 n! df n yn The higher-order genuine rational solution with nonvanishing background condition can be expressed as follows: 1. When N = 2l – 1, we get u[2l – 1] = –1 – [

det(B) ] , det(A) x

with min(i–1,2j–2)

Ai,2j–1 = Bi,2j–1 = i2j–1

∑

k C2j–1 xk ,

k=0 i–1

k xk , Bi,2l–1 = i2l ∑ C2l k=0

(j = 1, 2, . . . , l – 1),

(2.6.44)

77

2.6 Generalized Darboux Transformation for DNLS Equation

min(i–1,2j–1)

Ai,2j = Bi,2j = i2j

∑

k C2j yk ,

n Cm =

k=0

2.

m! . n!(m – n)!

When N = 2l, we get u[2l] = 1 – [

det(D) ] , det(C) x

(2.6.45)

with min(i–1,2j–2)

Ci,2j–1 = Di,2j–1 = i2j–1

∑

k C2j–1 xk ,

k=0 min(i–1,2j–1)

Ci,2j = i2j

∑

k C2j yk ,

(j = 1, 2, . . . , l)

k=0 i–1

Di,2k = Ci,2k ,

(k = 1, 2, . . . , l – 1),

k Di,2l = i2l+1 ∑ C2l+1 yk . k=0

Specially, taking d = ef , with e being real number, we have x0 = √2(2x – 6t – i), y0 = √2(2x – 6t + i), 2 1 x1 = √2 [ x3 – 6x2 t + 18xt2 – 18t3 – 4x + 20t + 2e + i ( – x2 + 6xt – 9t2 )] , 3 2 2 3 1 2 2 3 √ y1 = 2 [ x – 6x t + 18xt – 18t – 4x + 20t + 2e + i (x2 – – 6xt + 9t2 )] . 3 2 Via formula Eq. (2.6.44), the first-order genuine rational solutions read as u[1] = –

(–2x + 6t – i)(–2x + 6t + 3i) , (–2x + 6t + i)2

(2.6.46)

which is nothing but rational traveling wave solution with nonvanishing background. Similarly, the second-order genuine rational solution can also be obtained from Eq. (2.6.45) u[2] =

L∗1 L2 , L21

(2.6.47)

where ∗ is the complex conjugate, L1 = 8'3 + 18' + 48t + 12e + i(12'2 + 3), L2 = 8'3 – 30' + 48t + 12e + i(36'2 – 15), ' = 3t – x, e is an arbitrary real constant. The maximum value 5 for solution (2.6.47) 3

is located at (x, t) = (– 43 e, – 41 e), and the zero point is located at ( 74–448

–6e 154–443 –6e , ), 24

with 4 = ±√ 125 . The “ridge” of solution (2.6.47) approximately lays on the line x = 3t. When t → ±∞, Eq. (2.6.47) approaches to Eq. (2.6.46) along its “ridge.”

78

2 Construction of Rogue Wave Solution by the Generalized Darboux Transformation

Case 3 : Higher-order rogue wave solutions. According to the above, generalized Darboux transformation is an efficient way to obtain the higher-order solutions. We start with u[0] = exp(–ix), and the corresponding fundamental solution for Lax pair I = E(

! !–1 " 0 )( ), 0 "–1 –!–1 –!

exp (– 21 ix) 0 ) 0 exp ( 21 ix)

E=(

where 1 ! = [(2& )–1 (+ – 2i + i& 2 )]1/2 , " = exp [ +& –2 (x + 2& –2 t + F(& ))], + = (–4 – & 4 )1/2 , 2 F(& ) is a polynomial function of & . Similarly, through the limit technique, expanding the special solution at & = 1 + i, I1 =

I|& =(1+i)(1+f ) C f 1/2

,

1 C = ( ), 1

at f = 0, and find I1 = Y0 + Y1 f + ⋅ ⋅ ⋅ + Yn f n + ⋅ ⋅ ⋅ , where x 1 d I1 (f ). Yn = ( n ) = lim f =0 n! df n yn Explicitly, we have 1 x0 = exp [– ix](2x – 2it – 1 – i), 2 1 y0 = exp [ ix](2x – 2it + 1 + i), 2 1 1 1 1+ i 2 1+ i 2 x + (1 – i)xt – t x1 = exp [– ix] [– x3 + ix2 t + xt2 – it3 + 2 3 3 2 2 5 13 1 1 1 – ix – x + it – t + 2e + + 2ig] , 2 2 2 2 2 1 1 1 1+ i 2 1+ i 2 x – (1 – i)xt + t y1 = exp [ ix] [– x3 + ix2 t + xt2 – it3 – 2 3 3 2 2 5 13 1 1 1 – ix – x + it – t + 2e – + 2 ig] . 2 2 2 2 2

79

2.6 Generalized Darboux Transformation for DNLS Equation

Therefore, N-order rogue wave solutions can be written as u[N] = exp[–ix] – 2 (

det(M1 ) ) , det(M) x

(2.6.48)

with M = (Mij )N×N , M11 M21 ( M1 = ( ... MN1 ( x0

M12 M22 .. . MN2 x1

⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅ ⋅⋅⋅

M1N M2N .. . MNN xN–1

y0 y1 .. ) , . ) yN–1 0 )

i–1,j–1

Mij =

1 i+j–(k+l+1) i–k–1 Ci+j–(k+l+2) ∑ – (– ) 2 k=0,l=0 × [Yk† 33 Yl (1 – i)i–k–1 (1 + i)j–l–1 + iYk† Yl (1 + i)i–k–1 (1 – i)j–l–1 ] .

Particularly, we present the first- and second-order rogue wave solutions u[1] = –

[2t2 + 2x2 – 3 – 2i(x + 3t)][2x2 + 2t2 + 1 – 2i(x – t)] exp[–ix], [2x2 + 2t2 + 1 + 2i(x – t)]2

and u[2] =

L∗1 L2 exp[–ix], L21

where L1 = 72[e2 + g 2 ] + [48x3 – 144xt2 + 72ix2 + 144ixt – 72it2 – 144x – 72t + 36i]e + [–144x2 t + 48t3 + 72ix2 – 144ixt – 72it2 – 72x + 432t – 36i]g + 8x6 + 24x4 t2 + 24x2 t4 + 8t6 + 24ix5 – 24ix4 t + 48ix3 t2 – 48ix2 t3 + 24ixt4 – 24it5 – 12x4 + 48x3 t – 216x2 t2 + 48xt3 + 180t4 + 48ix3 – 288ixt2 – 336it3 + 90x2 – 72xt + 666t2 + 54ix – 198it + 9, L2 = 72[e2 + g 2 ] + [48x3 – 144xt2 – 72ix2 + 432ixt + 72it2 + 144x + 216t – 180i]e + [–144x2 t + 48t3 + 216ix2 + 144ixt – 216it2 + 216x + 144t + 36i]g + 8x6 + 24x4 t2 + 24x2 t4 + 8t6 – 24ix5 – 72ix4 t – 48ix3 t2 – 144ix2 t3 – 24ixt4 – 72it5 – 60x4 – 144x3 t – 504x2 t2 – 144xt3 – 60t4 + 48ix3 + 288ix2 t + 576ixt2 – 528it3 – 198x2 + 504xt – 486t2 + 90ix + 414it + 45, e and g are real numbers. Dynamics of the rogue wave are plotted in Fig. 2.10.

80

2 Construction of Rogue Wave Solution by the Generalized Darboux Transformation

(b)

(a) 10 2

|u|

8

7 5 3 1 10

|u|

5 t

0 –5 –10

–10

5 0 x –5

10

2

5 3 1 10

5 t

0

–5 –10

5 0 –5 x –10

10

Figure 2.10: Second-order rogue wave with parameters: (a) e = g = 0; (b) e = 0 and g = 100.

In fact, except the higher-order rational solutions, genuine rational solutions, rogue wave solutions, and the higher-order breathers can also be obtained with the modification of seed solution for Eq. (2.6.36). In addition, the above formula can be easily modified and applied to the so-called Fokas–Lenells equation [147], and the details are listed in Ref. [152].

3 Construction of Rogue Wave Solution by Hirota Bilinear Method, Algebro-geometric Approach and Inverse Scattering Method 3.1 Hirota Bilinear Method Hirota bilinear method is an effective way to solve the soliton equations. The method provides a procedure to obtain the soliton solutions for the nonlinear evolution equation [153]. In this chapter, we make some modifications to get the rational solution (rogue wave solution) for the NLS and DS-I equations [154, 155].

3.1.1 Rogue Wave Solution for the NLS Equation In this section, we discuss the procedure to obtain the rogue wave solutions for the NLS equation via the Hirota method. Rogue wave will tend to the nonzero background when the space and time are infinite. Thus, we need to consider the amplitude and phase, which are the two factors corresponding to space and time. Note the classical NLS equation, iut = uxx + 2|u|2 u,

(3.1.1)

keeping invariant under the scaling transformation x → !x,

t → !2 t,

u → u/!

and the Galilean transformation u(x, t) → u(x – vt, t) exp(–ivx/2 + iv2 t/4), where v stands for the velocity, and ! is real. The amplitude and phase can be measured by the scaling and Galilean transformations, respectively. Thus, we only consider the rogue wave solutions in the following form: u(x, t) → e–2it

as

x, t → ±∞.

Through u → ue–2it , NLS equation (3.1.1) can be transformed into iut = uxx + 2(|u|2 – 1)u,

(3.1.2)

where u(x, t) → 1,

x, t → ±∞.

(3.1.3)

82

3 Construction of Rogue Wave Solution by Hirota Bilinear Method

The rogue wave solutions belong to the rational solutions. To derive them, we introduce the elementary Schur polynomials Sn (x) [156], which are defined by ∞

∞

n=0

k=1

∑ Sn (x)+n = exp ( ∑ xk +k ) , with x = (x1 , x2 , . . .). For example, S0 (x) = 1,

S1 (x) = x1 ,

1 S2 (x) = x12 + x2 , 2

S3 (x) =

1 3 x + x1 x 2 + x3 , . . . . 6 1

The general Schur polynomials can give the complete set of homogeneous-weight algebraic solutions for the KP equation. (More details are listed in Ref. [156], which displayed the relationship between soliton equations and infinite-dimensional Lie algebra. It plays an important role in studying the algebra construction and exact solutions for the integrable systems.) Next, we display the procedure of the Hirota bilinear method. Under the transformation u=

g , f

(3.1.4)

we get the bilinear forms for Eq. (3.1.2) {

(D2x + 2)f ⋅ f = 2|g|2 , (D2x – iDt )g ⋅ f = 0,

(3.1.5)

where g and f are the complex and real functions, respectively. Dx and Dt are the bilinear derivative operators defined by P(Dx , Dy , Dt )F(x, y, t) ⋅ G(x, y, t) =P(𝜕x – 𝜕x󸀠 , 𝜕y – 𝜕y󸀠 , 𝜕t – 𝜕t󸀠 )F(x, y, t) ⋅ G(x󸀠 , y󸀠 , t󸀠 )|x=x󸀠 ,y=y󸀠 ,t=t󸀠 , where P is the polynomial of Dx , Dy , Dt , . . .. The (2+1)-dimensional forms of Eqs. (3.1.5) are {

(Dx Dy + 2)f ⋅ f = 2gh, (D2x – iDt )g ⋅ f = 0,

(3.1.6)

where h is the complex function. In fact, Eqs. (3.1.6) is the bilinear form of DS equation, which is also called (2+1)-dimensional NLS equation. We first construct a set of solutions for Eq. (3.1.6) in the form of Gram determinant. Under the following conditions (𝜕x – 𝜕y )f = Cf ,

f : real,

h = g∗ ,

∗

means the complex conjugation,

(3.1.7)

3.1 Hirota Bilinear Method

83

the solutions will satisfy Eq, (3.1.5). Higher-order rogue wave solutions will be obtained via the methods. In fact, the solutions are derived through the Hirota bilinear method with some new solving techniques, which will be displayed in detail in the following. The bilinear forms for NLS hierarchy have the solutions in the form of Schur polynomials, which are linked to the rectangular Young diagrams. However, due to the different degree of g and h, these solutions generally do not satisfy the requirements of complex conjugate (h = g ∗ ), unless f is linked to the square Young diagram. If f is a square-shaped Young diagram, we can derive the defocusing NLS equation with h = –g∗ . In order to construct the rational solutions for the focusing NLS equation, we need to consider the weight-inhomogeneous polynomials. To satisfy the reduction condition (3.1.7), we need to consider the following Schur polynomials. Here, we present the following theory. Theorem 3.1.1. Under condition (3.1.3), the nonsingular rational solutions for NLS equation (3.1.1) are given as u=

31 , 30

(3.1.8)

where 3n = det (m(n) 2i–1,2j–1 ), 1≤i,j≤N

min(i,j) (n) (n) m(n) ij = ∑ Ii- Jj- , -=0

I(n) i- =

1 i–(x+ (n) + -s), ∑a S 2- k=0 k i–-–k

j–-

J(n) j- =

1 (x– (n) + -s), ∑ a∗ S 2- l=0 l j–-–l

ak (k = 0, 1, . . .) is the complex constant, (x1± (n), x2± (n), . . .), s = (s1 , s2 , . . .)

∗

denotes the complex conjugate, x± (n) =

1 x ∓ 2k it – rk , (k ≥ 2), x1± (n) = x ± 2it ± n – , xk± = 2 k! ∞ ∞ + 2 + ∑ rk +k = ln (cosh ) , ∑ sk +k = ln ( tanh ) . 2 + 2 k=1 k=1 3n also can be written in the series form 1

3n = ∑

3

∑

-1 =0 -2 =-1 +1 -3 =-2 +1

Ii- = 0,

2N–1

5

∑ ...

J(n) i- = 0,

∑

(n) det (I(n) 2i–1,- ) det (J2i–1,- )

-N =-N–1 +1 1≤i,j≤N

(i < -).

j

1≤i,j≤N

j

84

3 Construction of Rogue Wave Solution by Hirota Bilinear Method

We point out some statements before the proof. The odd terms of rk and sk are zero, and the coefficients of even terms are displayed as 1 1 1 , r4 = – , r6 = ,... 8 192 2, 880 1 7 31 s2 = – , s4 = , s6 = – ,.... 12 1, 440 90, 720 r2 =

(a) Determinant Solutions for (2+1)-dimensional System In this section, we derive the determinant solutions for Eq. (3.1.5). , >(n) and 8(n) are functions of x1 , x2 and x–1 , which satisfy the Lemma 3.1.1. m(n) ij i j differential and difference relations (n) (n) 𝜕x1 m(n) ij = >i 8j ,

(3.1.9a)

(n+1) (n) (n–1) 𝜕x2 m(n) 8j + >(n) , ij = >i i 8j

(3.1.9b)

𝜕x–1 m(n) ij

=

–>(n–1) 8(n+1) , i j

(3.1.9c)

(n) (n+1) m(n+1) = m(n) , ij ij + >i 8i (n+k) 𝜕xk >(n) , i = >i

(3.1.9d)

𝜕xk 8(n–k) = –8(n–k) j j

(k = 1, 2, –1).

(3.1.9e)

Then 4n = det (m(n) ij ) 1≤i,j≤N

(3.1.10)

satisfies the bilinear forms (Dx1 Dx–1 – 2)4n ⋅ 4n = –24n+1 4n–1 , (D2x1 – Dx2 )4n+1 ⋅ 4n = 0.

(3.1.11)

Proof. Giving the following differential formula of determinant, N

𝜕x det (aij ) = ∑ Bij 𝜕x aij , 1≤i,j≤N

(3.1.12)

i,j=1

and the expansion formula of bordered determinant aij bi ) = – ∑ Bij bi cj + d det(aij ), cj d i,j

det (

with Bij being the (i, j)-cofactor of matrix aij . Using the above formulas repeatedly, the derivatives and shifts of 4 function (3.1.10) can be verified,

3.1 Hirota Bilinear Method

𝜕x1 4n = 𝜕x21 4n = 𝜕x2 4n = 𝜕x–1 4n =

(𝜕x1 𝜕x–1 – 1)4n =

4n+1 = 4n–1 = 𝜕x1 4n+1 =

𝜕x21 =

𝜕x2 4n+1 =

85

󵄨󵄨 (n) (n) 󵄨󵄨 󵄨󵄨 m 󵄨 󵄨󵄨 ij >i 󵄨󵄨󵄨 , 󵄨󵄨 (n) 󵄨 󵄨󵄨–8j 0 󵄨󵄨󵄨 󵄨 󵄨 󵄨󵄨 (n) (n+1) 󵄨󵄨 󵄨󵄨 (n) (n) 󵄨󵄨 󵄨󵄨 m 󵄨󵄨 󵄨󵄨 m 󵄨 > i 󵄨󵄨 ij 󵄨󵄨 + 󵄨󵄨 ij >i 󵄨󵄨󵄨 , 󵄨󵄨 (n) 󵄨󵄨 󵄨󵄨 (n–1) 󵄨 󵄨󵄨–8j 0 󵄨󵄨 󵄨󵄨8j 0 󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 (n) (n+1) 󵄨󵄨 󵄨󵄨 (n) (n) 󵄨󵄨 󵄨󵄨 m 󵄨 󵄨 > 󵄨 󵄨 mij >i 󵄨󵄨󵄨 󵄨󵄨󵄨 ij(n) i 󵄨󵄨󵄨 – 󵄨󵄨󵄨 (n–1) 󵄨󵄨 , 󵄨󵄨󵄨–8j 0 󵄨󵄨󵄨 󵄨󵄨󵄨8j 0 󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 (n) (n–1) 󵄨󵄨 󵄨󵄨 mij >i 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨8n+1 0 󵄨󵄨󵄨 , 󵄨󵄨 󵄨󵄨 j 󵄨󵄨󵄨 m(n) >(n–1) >(n) 󵄨󵄨󵄨 󵄨󵄨 ij i i 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 (n+1) 0 –1 󵄨󵄨󵄨 , 󵄨󵄨8j 󵄨󵄨 󵄨󵄨 (n) 󵄨󵄨 –8 –1 0 󵄨󵄨󵄨󵄨 󵄨󵄨 j 󵄨󵄨 (n) (n) 󵄨󵄨 󵄨󵄨󵄨 mij >i 󵄨󵄨󵄨 󵄨󵄨 (n+1) 󵄨󵄨 , 󵄨󵄨–8 1 󵄨󵄨󵄨 󵄨󵄨 j 󵄨 󵄨󵄨 (n) (n–1) 󵄨󵄨 󵄨󵄨m > 󵄨󵄨 i 󵄨󵄨 ij 󵄨󵄨 , 󵄨󵄨 (n) 󵄨 󵄨󵄨 8j 1 󵄨󵄨󵄨 󵄨 󵄨 󵄨󵄨 (n) (n+1) 󵄨󵄨󵄨 󵄨󵄨 m >i 󵄨󵄨 ij 󵄨󵄨󵄨 (n+1) 󵄨󵄨 , 󵄨󵄨󵄨–8j 0 󵄨󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 (n) 󵄨󵄨 󵄨 >(n) >(n+1) 󵄨󵄨 (n) (n+2) 󵄨󵄨󵄨 󵄨󵄨 mij i i 󵄨󵄨 󵄨󵄨 m 󵄨󵄨 󵄨󵄨󵄨 󵄨 (n) 󵄨󵄨 ij >i 󵄨 0 0 󵄨󵄨󵄨 , 󵄨󵄨 (n+1) 󵄨󵄨 + 󵄨󵄨󵄨 –8j 󵄨󵄨 󵄨󵄨–8j 0 󵄨󵄨 󵄨󵄨 (n+1) 󵄨 󵄨 󵄨󵄨–8 1 0 󵄨󵄨󵄨󵄨 󵄨 j 󵄨󵄨 󵄨󵄨 (n) 󵄨󵄨 󵄨 >(n) >(n+1) 󵄨󵄨 (n) (n+2) 󵄨󵄨󵄨 󵄨󵄨 mij i i 󵄨󵄨 󵄨󵄨 m >i 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 ij 󵄨󵄨󵄨 (n+1) 󵄨󵄨 + 󵄨󵄨 –8(n) 0 0 󵄨󵄨 . j 󵄨󵄨–8 󵄨󵄨 󵄨󵄨 󵄨󵄨 0 󵄨󵄨 j 󵄨󵄨 󵄨󵄨 (n+1) 󵄨󵄨 –8 1 0 󵄨󵄨 󵄨󵄨 j

On the other hand, from the Jacobi determinants, 󵄨 󵄨󵄨 󵄨󵄨aij bi ci 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 dj e f 󵄨󵄨󵄨 × |aij | = 󵄨󵄨󵄨󵄨aij ci 󵄨󵄨󵄨󵄨 × 󵄨󵄨󵄨󵄨aij bi 󵄨󵄨󵄨󵄨 – 󵄨󵄨󵄨󵄨aij bi 󵄨󵄨󵄨󵄨 × 󵄨󵄨󵄨󵄨aij ci 󵄨󵄨󵄨󵄨 . 󵄨󵄨 󵄨󵄨 󵄨󵄨 gj k 󵄨󵄨 󵄨󵄨 dj e 󵄨󵄨 󵄨󵄨 gj h 󵄨󵄨 󵄨󵄨 dj f 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 gj h k 󵄨󵄨󵄨 So we have (𝜕x1 𝜕x–1 – 1)4n × 4n = 𝜕x1 4n × 𝜕x–1 4n – (–4n–1 )(–4n+1 ) 1 2 1 (𝜕 – 𝜕x2 )4n+1 × 4n = 𝜕x1 4n+1 × 𝜕x1 4n – 4n+1 (𝜕x21 + 𝜕x2 )4n , 2 x1 2 which are just the bilinear forms (3.1.11).

◼

86

3 Construction of Rogue Wave Solution by Hirota Bilinear Method

Because the matrix m(n) can be rewritten as ij x1

(n) (n) m(n) ij = ∫ >i 8j dx1 ,

determinant (3.1.10) is called Gram determinant solutions. Denoting f = 40 ,

g = 41 ,

h = 4–1 ,

which satisfy the bilinear equations, (Dx1 Dx–1 – 2)f ⋅ f = –2gh, (D2x1 – Dx2 )g ⋅ f = 0. Letting x1 = x, x2 = –it and x–1 = –y, they are just the bilinear forms (3.1.5). (b) Algebraic Solutions for the (1+1)-dimensional System In the next procedure, we derive the algebraic solutions for the bilinear forms (3.1.5) and conditions (𝜕x – 𝜕y )f = Cf , which satisfy the (1+1)-dimensional system 2 { (Dx + 2)f ⋅ f = 2gh, { 2 { (Dx – iDt )g ⋅ f = 0.

(3.1.13)

g, f and h can be obtained from the Gram determinant of Lemma 3.1.1. as Lemma 3.1.2. Define the matrix elements m(n) ij (n) m(n) ij = Ai Bj m |p=1,q=1

(3.1.14)

and m(n) =

1 p n (– ) e. +' , p+q q

. = px1 + p2 x2 ,

' = qx1 – q2 x2 ,

where Ai and Bj are the differential operators of p and q, respectively, A0 = a0 , A1 = a0 p𝜕p + a1 , a A2 = 0 (p𝜕p )2 + a1 p𝜕p + a2 , 2 .. . i

ak (p𝜕 )i–k , (i – k)! p k=0

Ai = ∑

(3.1.15)

3.1 Hirota Bilinear Method

87

B0 = b0 , B1 = b0 q𝜕q + b1 , B2 =

b0 (q𝜕q )2 + b1 q𝜕q + b2 , 2

.. . j

bl (q𝜕q )j–l , (j – l)! l=0

Bi = ∑ with ak and bk being constants. Then

󵄨󵄨 (n) m(n) 󵄨󵄨 m11 13 󵄨󵄨 (n) (n) 󵄨󵄨 m m 󵄨 31 33 󵄨󵄨 (n) 4n = det (m2i–1,2j–1 ) = 󵄨󵄨 . . 󵄨 . . 1≤i,j≤N 󵄨󵄨 . . 󵄨󵄨 󵄨󵄨 (n) (n) 󵄨󵄨m2N–1,1 m2N–1,3

󵄨󵄨 . . . m(n) 1,2N–1 󵄨󵄨󵄨 󵄨 (n) . . . m3,2N–1 󵄨󵄨󵄨󵄨 󵄨󵄨 .. .. 󵄨󵄨 󵄨󵄨 . . 󵄨󵄨 󵄨 (n) . . . m2N–1,2N–1 󵄨󵄨󵄨

(3.1.16)

fulfills the bilinear equations 2 { { (Dx1 + 2)4n ⋅ 4n = 24n+1 4n–1 { 2 { (D – Dx2 )4n+1 ⋅ 4n = 0. { x1

Proof. Let m̃ (n) =

1 p n ̃ ̃ (– ) e. +' , p+q q

̃

>̃ (n) = pn e. ,

̃ 8̃ (n) = (–q)–n e' ,

with .̃ =

1 x + px1 + p2 x2 , p –1

'̃ =

1 x + qx1 – q2 x2 , q –1

which satisfy the following differential and difference relations: 𝜕x1 m̃ (n) = >̃ (n) 8̃ (n) , 𝜕x2 m̃ (n) = >̃ (n+1) 8̃ (n) + >̃ (n) 8̃ (n–1) , 𝜕x–1 m̃ (n) = –>̃ (n–1) 8̃ (n+1) , m̃ (n+1) = m̃ (n) + >̃ (n) 8̃ (n+1) , 𝜕 >̃ (n) = >̃ (n+k) , 𝜕 8̃ (n) = –8̃ (n–k) xk

xk

(k = 1, 2, –1).

Through the definitions ̃ (n) m̃ (n) ij = Ai Bj m ,

̃ (n) >̃ (n) i = Ai > ,

̃ (n) 8̃ (n) j = Bj 8 ,

(3.1.17)

88

3 Construction of Rogue Wave Solution by Hirota Bilinear Method

we can find that m̃ (n) , >̃ (n) and 8̃ (n) follow the differential and difference relaij i j tions (3.1.9), because of the operators Ai and Bj commuting with the differentials 𝜕xk . From Lemma 3.1.1, we find that for the arbitrary sequence of indices (i1 , i2 , . . . , iN ; j1 , j2 , . . . , jN ), the determinant 4̃ n = det (m̃ (n) i ,j ) 1≤-,,≤N

- ,

obeys Eq. (3.1.11), i.e., 4̃ n = det (m̃ (n) 2i–1,2j–1 ) 1≤i,j≤N

is the solution for Eq. (3.1.11), with p and q being arbitrary parameters. Next, we take the reduction condition into account. From the Leibniz rule, the following relation is displayed: (p𝜕p )m (p +

m m 1 1 ) = ∑ ( ) (p + (–1)l ) (p𝜕p )m–l , l p p l=0

then Ai (p +

i ak i–k i – k 1 1 ] (p + (–1)l ) (p𝜕p )i–k–l )= ∑ ∑[ l p (i – k)! p k=0 l=0 i

i–l

ak 1 (p + (–1)l ) (p𝜕p )i–k–l l!(i – k – l)! p l=0 k=0

=∑∑ i

1 1 (p + (–1)l ) Ai–l , l! p l=0

=∑

and similarly j

1 1 1 Bj (q + ) = ∑ (q + (–1)l ) Bj–l . q l! q l=0 satisfies Therefore, we discover that m̃ (n) ij ̃ (n) = Ai Bj (p + (𝜕x1 + 𝜕x–1 )m̃ (n) ij = Ai Bj (𝜕x1 + 𝜕x–1 )m i

1 1 (p + (–1)k ) Ai–k Bj m̃ (n) k! p k=0

= ∑

1 1 + q + ) m̃ (n) p q

89

3.1 Hirota Bilinear Method

j

1 1 (q + (–1)l ) Ai Bj–l m̃ (n) l! q l=0

+∑

j

i

1 1 1 l1 ̃ (n) (p + (–1)k ) m̃ (n) i–k,j + ∑ l! (q + (–1) q ) mi,j–l . k! p k=0 l=0

= ∑

| leads to Choosing p = 1 and q = 1, then m̃ (n) ij p=1,q=1 j

(𝜕x1 + 𝜕x–1 )(m̃ (n) ij |p=1,q=1 ) = 2

1 (n) 1 (n) m̃ i–k,j |p=1,q=1 + 2 ∑ m̃ i,j–l |p=1,q=1 . (3.1.18) k! l! k=0,k:even l=0,l:even ∑

Via Eq. (3.1.12) and the above relations, the differential of the determinant 4̃̃ n = det (m̃ (n) 2i–1,2j–1 |p=1,q=1 ) 1≤i,j≤N

can be calculated as (𝜕x1 + 𝜕x–1 )4̃̃ n N N

= ∑ ∑ Bij (𝜕x1 + 𝜕x–1 )(m̃ (n) 2i–1,2j–1 |p=1,q=1 ), i=1 j=1 N N

2j–1

2i–1

= ∑ ∑ Bij (2 i=1 j=1

1 (n) 1 (n) m̃ 2i–1–k,2j–1 |p=1,q=1 + 2 ∑ m̃ 2i–1,2j–1–l |p=1,q=1 ) k! l! k=0,k:even l=0,l:even ∑

2i–1

N

1 N ∑ Bij m̃ (n) 2i–1–k,2j–1 |p=1,q=1 k! i=1 k=0,k:even j=1

= 2∑

∑

2j–1

N

1 N ∑ Bij m̃ (n) 2i–1,2j–1–l |p=1,q=1 , l! j=1 l=0,l:even i=1

+2 ∑

∑

where Bij is the (i, j)-cofactor of the matrix (m̃ (n) | ). When we take k = 2, 4, . . ., 2i–1,2j–1 p=1,q=1 the summation of j is a determinant with two same lines, which lead to zero. Thus, in the first term of the right hand, only k = 0 survives. Similarly, the second term of the right hand only holds at l = 0. Therefore, the expression in the right hand goes to N N

N N

i=1 j=1

j=1 i=1

̃̃ ̃ 2 ∑ ∑ Bij m̃ (n) 2i–1,2j–1 |p=1,q=1 + 2 ∑ ∑ Bij m2i–1,2j–1 |p=1,q=1 = 4N 4n ; thus 4̃̃ n satisfies (𝜕x1 + 𝜕x–1 )4̃̃ n = 4N 4.̃̃

(3.1.19)

90

3 Construction of Rogue Wave Solution by Hirota Bilinear Method

4̃̃ n is also suitable to the bilinear equation (3.1.11), because it is the special case of 4̃ n . From Eqs (3.1.11) and (3.1.19), we obtain the equations as follows: (D2x1 + 2)4̃̃ n ⋅ 4̃̃ n = 24̃̃ n+1 4̃̃ n–1 , (D2x1 – Dx2 )4̃̃ n+1 ⋅ 4̃̃ n = 0, which are just Eq. (3.1.17). If we take x–1 = 0, m̃ (n) | and 4̃̃ can be reduced to m(n) ij p=1,q=1 ij and 4n , respectively. ◼ The fundamental idea of the above reduction is a procedure to reduce the higher dimensional system to a lower dimensional one. Via Eq. (3.1.19), the derivative of x–1 is replaced by the derivative of x1 . Then, x–1 is the arbitrary parameter in the solution. Taking x1 = x and x2 = –it in Lemma 3.1.2, we get f = 40 , g = 41 and h = 4–1 , which satisfy the (1+1)-dimensional system (3.1.13). (c) Complex Conjugate and Regularity (Nonsingularity) Considering the complex conjugate condition, f : real,

h = g∗ ,

and the regularity (nonsingularity) of the solutions, the complex conjugate condition turns to 40 : real,

4–1 = 4∗1 .

Because x1 = x is real and x2 = –it is pure imaginary in Lemma 3.1.2, the above condition can be easily satisfied when we take ak and bk being the complex conjugate to each other, bk = a∗k .

(3.1.20)

In fact, under Eq. (3.1.20), we have (–n) = m(n) m(n)∗ ij ij |ak ↔bk ,x2 ↔–x2 = mji ;

thus, 4∗n = 4–n . In the further step, we can verify that the rational solution u = g/f = 41 /40 is nonsingular, i.e., 40 is nonzero for any (x, t). Noting that f = 40 is the determinant of the Hermitian matrix, M = mat1≤i,j≤N (m(0) 2i–1,2j–1 ).

91

3.1 Hirota Bilinear Method

For any nonzero column vector v = (v1 , v2 , . . . , vN )T and v† , we get N

N

i,j=1

i,j=1

∗ v† Mv = ∑ vi∗ m(0) 2i–1,2j–1 vj = ∑ vi vj A2i–1 B2j–1 N

= ∑ vi∗ vj A2i–1 B2j–1 ∫

–∞

i,j=1

=∫

x

x

1 . +' e |p=1,q=1 p+q

e. +' dx|p=1,q=1

N

∑ vi∗ vj A2i–1 B2j–1 e. +' |p=1,q=1 dx

–∞ i,j=1

󵄨󵄨 N 󵄨󵄨2 󵄨󵄨 󵄨󵄨 ∗ . 󵄨 = ∫ 󵄨󵄨∑ vi A2i–1 e |p=1 󵄨󵄨󵄨 dx > 0, 󵄨 󵄨󵄨 –∞ 󵄨󵄨 i=1 󵄨 x

which proves the positive definition of the Hermitian matrix M. Therefore, f = det M > 0, which means u is nonsingular. Summing up the above, we present the following theory. Theorem 3.1.2. The nonsingular rational solutions for NLS equation (3.1.2) read as u=

41 , 40

where 4n = det (m(n) 2i–1,2j–1 ), 1≤i,j≤N

i

j

a∗l

2 2 ak p n 1 (p𝜕p )i–k (q𝜕q )j–l (– ) e(p+q)x–i(p –q )t |p=1,q=1 , (i – k)! (j – l)! p+q q k=0 l=0

m(n) ij = ∑ ∑

where ak is the complex constant. (d) Simplification of Rogue Wave Solution In this part, we simplify the rogue wave solutions in 3.1.2, and derive the expression of solution in 3.1.1. The generator G of the differential operators (p𝜕p )k (q𝜕q )l is displayed as ∞ ∞

*k +l (p𝜕p )k (q𝜕q )l = exp(*p𝜕p + +q𝜕q ) = exp(*𝜕ln p + +𝜕ln q ); k! l! k=0 l=0

G = ∑∑

for any function F(p, q), we have G F(p, q) = F(e* p, e+ q).

(3.1.21)

92

3 Construction of Rogue Wave Solution by Hirota Bilinear Method

This relation can also be obtained through expanding the right hand to Taylor series (*, +) near the point (0, 0). Applying this relation to m(n) =

1 p n (– ) exp((p + q)x – i(p2 – q2 )t), p+q q

we have 1 e* p (– ) exp((e* p + e+ q)x – i(e2* p2 – e2+ q2 )t); e* p + e + q e+ q n

G m(n) = therefore,

1 G m(n) |p=1,q=1 m(n) 2 = * en(*–+) exp((e* + e+ – 2) – i(e2* – e2+ )t) e + e+ 1 = 1 – [(e* – 1)(e+ – 1)]/[(e* + 1)(e+ + 1)] × exp (n(* – +) + (e* + e+ – 2)x – i(e2* – e2+ )t – ln

(e* + 1)(e+ + 1) ). 4

The exponent in the above expression can be written as ∞

∞ l * + *k + *++ (x – 2k it) + ∑ (x + 2l it) – – ln (cosh cosh ) k! l! 2 2 2 k=1 l=1

n(* – +) + ∑ ∞

∞

k=1

l=1

≡ ∑ xk+ *k + ∑ xl– +l , with xk+ and xl– being the same as in Lemma 3.1.1. At the same time, we have ∞

∑( -=0

-

∞ (e* – 1)(e+ – 1) *+ 4 * + = ( ) ) exp (- ln ( tanh tanh )) ∑ 4 *+ 2 2 (e* + 1)(e+ + 1) -=0 ∞

= ∑( -=0

∞ *+ ) exp (- ∑ sk (*k + +k )) , 4 k=1

where sk is identical with Lemma 3.1.1. Thus, ∞ ∞ ∞ 1 *+ (n) + k ) G m | = ( exp ( (x + -s )* + (xl– + -sl )+l ) ; ∑ ∑ ∑ p=1,q=1 k k 4 m(n) -=0 k=1 l=1

3.1 Hirota Bilinear Method

93

collecting the coefficients of *k +l , we find 1 1 (p𝜕p )k (q𝜕q )l m(n) |p=1,q=1 = m(n) k!l!

min(k,l)

∑ -=0

1 S (x+ + -s)Sl–- (x– + -s). 4- k–-

Therefore, the matrix element of the Gram determinant is given as j

i 1 Ai Bj m(n) |p=1,q=1 = ∑ ∑ ak a∗l (n) m k=0 l=0 min(i,j)

= ∑ -=0

min(i–k,j–l)

∑ -=0

1 S (x+ + -s)Sj–l–- (x– + -s) 4- i–k–-

j–-

1 i–(x+ + -s)Sj–l–- (x– + -s). ∑ ∑ a a∗ S 4- k=0 l=0 k l i–k–-

Noting 3n = 4n /(m(n) |p=1,q=1 )N , the determinant expression in eq. (3.1.1) is obtained. Finally, in order to derive the expression in series, we use the formula repeatedly a b det(aij + bi cj ) = det ( ij i ) ; –cj 1 thus, 3n can be written in the form of 3N × 3N determinant min(2i–1,2j–1)

3n = det ( 1≤i,j≤N

∑ -=0

2N–1

(n) (n) (n) I(n) 2i–1,- J2j–1,- ) = det ( ∑ I2i–1,- J2j–1,- ) 1≤i,j≤N

-=0

󵄨󵄨 󵄨 I 󵄨󵄨󵄨 󵄨0 󵄨󵄨 , = 󵄨󵄨󵄨󵄨 󵄨󵄨–J I2N×2N 󵄨󵄨󵄨 with I(n) . . . I(n) I(n) 10 11 1,2N–1 (n) (n) I(n) I . . . I 30 31 3,2N–1 ), I=( . .. .. .. . . I(n) I(n) . . . I(n) 2N–1,0 2N–1,1 2N–1,2N–1 J(n) J(n) . . . J(n) 10 30 2N–1,0 (n) (n) J(n) J . . . J 11 31 2N–1,1 ). J=( . .. .. .. . . J(n) J(n) . . . J(n) 1,2N–1 3,2N–1 2N–1,2N–1 0 and I2N×2N being the N ×N zero matrix and 2N ×2N unit matrix, respectively. Through Laplace expansion, we get

94

3 Construction of Rogue Wave Solution by Hirota Bilinear Method

󵄨󵄨 󵄨󵄨 󵄨󵄨 I(n) I(n) . . . I(n) 1-1 1-2 1-N 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 I(n) 󵄨 I(n) . . . I(n) 󵄨󵄨 3-1 3-2 3-N 󵄨󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 3n = ∑ .. .. .. 󵄨󵄨 󵄨󵄨 󵄨󵄨 . . . 0≤-1 1 )2 + 4(t – >2 )2 – (2√3 + 4i)(t – >2 ) + i√3 2it e , (x – >1 )2 + 4(t – >3 )2 – 2√3(t – >2 ) + 1

where >1 and >2 are real parameters. When the parameters are chosen as >1 = 0, √ >2 = – 43 , the classical Peregrine solutions are obtained.

3.3 Algebro-geometric Reduction Approach The algebro-geometric solutions are the most generalized solutions for the integrable systems. Analytic solutions can be derived through degenerating or limiting the algebro-geometric solutions [164]. Ref. [165] has reported that soliton and ellipse function solutions can be degenerated from algebro-geometric solutions. As to the rational solutions, the methods need to do some modification. In this section, we cite the results in Ref. [166]. More details are listed in Refs [164, 167]. Cao and Geng have done much work on the algebro-geometric solutions [168, 169]. Through constructing Baker–Akhezier function, there are methods to construct the algebro-geometric solutions, which are used by Jimbo and Miwa [170, 171], listed in their classical book [165], based on the Riemann–Hilbert problem [172], and used in [173]. When the genus of the associated hyperelliptic spectral curve tends to zero, the corresponding (-function solutions for NLS equation

110

3 Construction of Rogue Wave Solution by Hirota Bilinear Method

ivt + vxx + 2|v|2 v = 0

(3.3.1)

can be expressed as [163, 165]) v(x, t) =

(3 (x, t) exp(2it – i>); (1 (x, t)

(3.3.2)

more details are listed in Refs [163], [165], where (r (x, t) =

2N 𝛾- – 𝛾, 2 { 2N exp { ∑ ln ( ) *, *- + (∑ i*- (x – x0- ) 𝛾- + 𝛾, -=1 k∈{0;1}2N {,>-,,,-=1 󵄨 2N 󵄨󵄨 𝛾- + 𝛾, 󵄨󵄨󵄨 𝛾 –i 󵄨󵄨 + i0: ) * } , –2$- (t – t0- ) + (r – 1) ln + ∑ ln 󵄨󵄨󵄨󵄨 󵄨 -} 𝛾- + i ,=1,,=,̸ 󵄨󵄨 𝛾- – 𝛾, 󵄨󵄨󵄨 }

∑

*- = 2√1 – +2- ,

𝛾- = √

$- = *- +- ,

1 – +, 1 + +-

:- = {0; 1},

>, x0- , t0- , *- , $- and 𝛾- (- = 1, . . . , 2N) are real numbers and satisfy x0N+j = x0j ,

t0N+j = t0j ,

0 < +j < 1,

+N+j = –+j ,

*N+j = *j ,

$N+j = –$j ,

𝛾N+j = 1/𝛾j ,

j = 1, 2, . . . , N.

3.3.1 Relationship Between Fredholm Determinant and (-Function (r can be rewritten as (r =

󵄨󵄨 𝛾 + 𝛾 󵄨󵄨 󵄨 , 󵄨󵄨 󵄨󵄨 ∏(–1): ∏ 󵄨󵄨󵄨󵄨 󵄨 𝛾 – 𝛾 󵄨 , 󵄨󵄨 J⊂{1,...,2N} -∈J -∈J,,∈J̄ 󵄨 ∑

} { × exp {∑ i*- (x – x0- ) – 2$- (t – t0- ) + xr,- } , } {-∈J where xr,- = (r – 1) ln

𝛾- – i , 𝛾- + i

1 ≤ j ≤ 2N,

especially xr,j = (r – 1) ln

𝛾j – i 𝛾j + i

xr,N+j = –(r – 1) ln :j = j,

,

𝛾j – i 𝛾j + i

1 ≤ j ≤ N;

1 ≤ j ≤ N, – (r – 1)i0, :j = j + 1,

1 ≤ j ≤ N, N + 1 ≤ j ≤ 2N.

3.3 Algebro-geometric Reduction Approach

111

The matrix A = (ajk )1≤j,k≤2N is defined as

ajk =

󵄨󵄨 K + K 󵄨󵄨 󵄨 j l 󵄨󵄨 󵄨󵄨 exp(–2Kj x), ∏ 󵄨󵄨󵄨󵄨 Kj + Kk l=j̸ 󵄨󵄨 Kj – Kl 󵄨󵄨󵄨 2:j Kj

:j ∈ {–1; +1};

then the Fredholm determinant det(I + A) with I being the identity matrix has the following form:

det(I + A) =

󵄨󵄨 K + K 󵄨󵄨 󵄨 j k 󵄨󵄨 󵄨󵄨 exp(–2x ∑ Kj ) . ∏ :j ∏ 󵄨󵄨󵄨󵄨 󵄨 K – K 󵄨 j k 󵄨󵄨 J⊂{1,...,N} j∈J j∈J,k∈J̸ 󵄨 j∈J ∑

(3.3.3)

When we define Ar = (a-, )1≤-,,≤2N as

a-, =

󵄨󵄨 𝛾 + 𝛾 󵄨󵄨 2(–1):- 𝛾󵄨 ' 󵄨󵄨 󵄨󵄨 exp (iK- (x – x0- ) – 2$- (t – t0- ) + xr,- ), ∏ 󵄨󵄨󵄨󵄨 𝛾- + 𝛾, '=-̸ 󵄨󵄨 𝛾- – 𝛾' 󵄨󵄨󵄨

det(I + Ar ) is transformed into 󵄨󵄨 𝛾 + 𝛾 󵄨󵄨 󵄨 , 󵄨󵄨 󵄨󵄨 exp (iK- (x – x0- ) – 2$- (t – t0- ) + xr,- ) . ∏(–1): ∏ 󵄨󵄨󵄨󵄨 󵄨 𝛾 – 𝛾, 󵄨󵄨󵄨 J⊂{1,...,2N} -∈J -∈J,,∈J̸ 󵄨 (3.3.4) It is easy to find that det(I + Ar ) =

∑

(r = det(I + Ar ). The solutions for NLS equation (3.3.1) are listed as

v(x, t) =

det(I + A3 (x, t)) exp(2it – i>). det(I + A1 (x, t))

Next, we point out some simple examples. For r = 3,

(3 (x, t) =1 – exp (–2 ln – exp (–2 ln

1 + 𝛾12 𝛾1 – i + ln + iK1 (x – x01 ) + 2$1 (t – t01 )) 𝛾1 + i 1 – 𝛾12

1 + 𝛾12 𝛾1 – i + iK1 (x – x01 ) – 2$1 (t – t01 )) + ln 𝛾1 + i 1 – 𝛾12

+ exp(iK1 (x – x01 ) + 2$1 (t – t01 )).

(3.3.5)

112

3 Construction of Rogue Wave Solution by Hirota Bilinear Method

It is the exact expression of det(I + A3 ). For r = 1, the expression (1 (x, t) =1 – exp (ln – exp (ln

1 + 𝛾12 + iK1 (x – x01 ) + 2$1 (t – t01 )) 1 – 𝛾12

1 + 𝛾12 + iK1 (x – x01 ) – 2$1 (t – t01 )) 1 – 𝛾12

+ exp(iK1 (x – x01 ) + 2$1 (t – t01 )) is equal to det(I + A1 ), obviously.

3.3.2 Wronskian Solutions 3.3.2.1 Relations Between Fredholm and Wronskian Determinants Consider the following functions: 6- (y) = sin(K- (x – x0- )/2 + i$- (t – t0- ) – ixr,- /2 + 𝛾- y),

1 ≤ - ≤ N,

6- (y) = cos(K- (x – x0- )/2 + i$- (t – t0- ) – ixr,- /2 + 𝛾- y),

N + 1 ≤ - ≤ 2N,

which depend on r = 1, r = 3, and should be denoted as 6r- (y). For simplicity, we notate them as 6- (y) in this section, and C- = K- (x – x0- )/2 + i$- (t – t0- ) – ixr,- /2 + 𝛾- y, Wr (y) = Wr(61 , 62 , . . . , 62N ) ≡

1 ≤ j ≤ 2N,

det [(𝜕y,–1 6- )-,,∈[1,...,2N] ] .

Matrix Dr = (d-, )-,,∈[1,...,2N] is defined as d-, =

󵄨󵄨 𝛾 + 𝛾 󵄨󵄨 2(–1):- 𝛾󵄨 ' - 󵄨󵄨 󵄨󵄨 exp [i*- (x – x0- ) – 2$- (t – t0- ) + xr,- ] , ∏ 󵄨󵄨󵄨󵄨 𝛾- + 𝛾, '=,̸ 󵄨󵄨 𝛾' – 𝛾, 󵄨󵄨󵄨

1 ≤ - ≤ 2N,

1 ≤ , ≤ 2N,

with xr,- = (r – 1) ln

𝛾- – i . 𝛾- + i

Then we have the following theorem. Theorem 3.3.1. det(I + Dr ) = kr (0) × Wr (0),

3.3 Algebro-geometric Reduction Approach

113

where kr (y) =

22N exp(i ∑2N -=1 C- ) -

1 ∏2N -=2 ∏,=1 (𝛾- – 𝛾, )

.

Proof. Omitting (2i)–1 eiC- in each line - by the Wronskian determinant Wr (y), with 1 ≤ - ≤ 2N. Thus 2N

Wr = ∏ eiC- 2–N × Wr1 ,

(3.3.6)

-=1

with 󵄨󵄨 (1 – e–2iC1 ) i𝛾 (1 + e–2iC1 ) 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 (1 – e–2iC2 ) i𝛾2 (1 + e–2iC2 ) 󵄨󵄨 Wr1 = 󵄨󵄨󵄨 .. .. 󵄨󵄨 . . 󵄨󵄨 󵄨󵄨 󵄨󵄨(1 – e–2iC2N ) i𝛾2N (1 + e–2iC2N )

. . . (i𝛾1 )2N–1 (1 + (–1)2N e–2iC1 ) 󵄨󵄨󵄨󵄨 󵄨 . . . (i𝛾2 )2N–1 (1 + (–1)2N e–2iC2 ) 󵄨󵄨󵄨󵄨 󵄨󵄨 . 󵄨󵄨 .. .. 󵄨󵄨 . . 󵄨󵄨 2N–1 2N –2iC2N 󵄨󵄨 . . . (i𝛾2N ) (1 + (–1) e )󵄨󵄨

The determinant Wr1 can be written in the form of Wr1 = det(!jk ej + "jk ), with !jk = (–1)k (i𝛾j )k–1 , ej = e–2i(j and "jk = (i𝛾j )k–1 , if 1 ≤ j ≤ N, 1 ≤ k ≤ 2N; !jk = (–1)k–1 (i𝛾j )k–1 , ej = e–2i(j and "jk = (i𝛾j )k–1 , N + 1 ≤ j ≤ 2N, 1 ≤ k ≤ 2N. Making U = (aij )i,j∈[1,...,2N] and V = (")i,j∈[1,...,2N] , then det(U) = (i)

2N(2N–1) 2

∏ (𝛾l – 𝛾m ). 1≤m 0 denotes nonself-focusing interaction [193–195]. If "(t) is real constant, then NLSE is also called Gross–Pitaevskii equation in the BECs. If g(t) < 0 denotes attractive interaction, g(t) > 0 denotes the repulsive interaction [196–198]. Based on the Lagrangian density (4.2.1) and Euler–Lagrange equation (variation principle) $L 󵄨󵄨󵄨 𝜕L 𝜕 𝜕L 𝜕 𝜕L 󵄨󵄨 ≡ – – = 0, $8∗ 󵄨󵄨(4.2.1) 𝜕8∗ 𝜕t 𝜕8∗t 𝜕x 𝜕8∗x

(4.2.2)

we can obtain the space-time modulation NLSE i

𝜕8 "(t) 𝜕2 8 + + V(x, t)8 + g(t)|8|2 8 = i𝛾(t)8. 𝜕t 2 𝜕x2

(4.2.3)

In the nonlinear optics, group dispersion can change with the time. But in the BECs, "(t) must be chosen as the constant. For " = 1, then Eq. (4.2.3) reduces space-time modulation Gross–Pitaevskii equation i

𝜕8 1 𝜕2 8 + V(x, t)8 + g(t)|8|2 8 = i𝛾(t)8. + 𝜕t 2 𝜕x2

(4.2.4)

4.2.2 Symmetry Analysis-Similarity Transformation and Similarity Solution To study the similarity rogue wave solution of NLSE (4.2.3), we use the following transformation [199]: 8(x, t) = [JR (x, t) + iJI (x, t)] ei>(x,t) ,

(4.2.5)

where the intensity of wave function is |8(x, t)|2 = |JR (x, t)|2 + |JI (x, t)|2 , JR (x, t), JI (x, t) and phase >(x, t) are the functions of (x, t). Inserting the

138

4 The Rogue Wave Solution and Parameters Managing

transformation (4.2.5) into Eq. (4.2.3), we have the following coupled real nonlinear partial differential equations: "(t) (JI,xx + 2>x JR,x – >2x JI + >xx JR ) + [V(x, t) – >t ]JI 2 +g(t)(J2R + J2I )JI – 𝛾(t)JR = 0, "(t) –JI,t + (JR,xx – 2>x JI,x – >2x JR – >xx JI ) + [V(x, t) – >t ]JR 2 +g(t)(J2R + J2I )JR + 𝛾(t)JI = 0. JR,t +

(4.2.6)

We consider the following similarity transformation [199]: JR (x, t) = A(t) + B(t)P('(x, t), 4(t)), { { { JI (x, t) = C(t)Q('(x, t), 4(t)), { { { {>(x, t) = 7(x, t) + ,4(t),

(4.2.7)

where '(x, t) and 4(t) are new similarity variables, , is a real constant, 7(x, t), A(t), B(t), C(t), P(', 4) and Q(', 4) are undetermined functions. Inserting the transformation (4.2.7) into system (4.2.6), we can obtain the following partial differential system: 'xx = 0,

(4.2.8a)

't + "(t)7x 'x = 0,

(4.2.8b)

"(t)7x2

(4.2.8c)

27t +

– 2v(x, t) = 0,

23t + ["(t)7xx – 2𝛾(t)]3 = 0 (3 = A, B, C),

(4.2.8d)

"(t) 2 ' CQ – ,4t CQ + g(t)CQ [C2 Q2 + (A + BP)2 ] = 0, 4t BP4 + 2 x '' "(t) 2 ' BP + (A + BP){g(t)[C2 Q2 + (A + BP)2 ] – ,4t } = 0. –4t CQ4 + 2 x ''

(4.2.8e) (4.2.8f)

From Eqs (4.2.8a)–(4.2.8d), we have '(x, t) = !(t)x + $(t), A(t) = a0 √|!(t)| e

(4.2.9a)

t

∫0 𝛾(s)ds

,

B(t) = bA(t),

$t !t x2 – x + 70 (t), 2!(t)"(t) !(t)"(t) "(t) 2 V(x, t) = 7t + 7 , 2 x

7(x, t) = –

C(t) = cA(t),

(4.2.9b) (4.2.9c) (4.2.9d)

where a0 , b, and c are real constants, !(t) (the inverse of the wave width), $(t) (–$(t)/!(t) being the position of its center of mass) and 70 (t) are the differential functions with the time. From these results, we can see that frequency !(t) and 𝛾(t) control the amplitude of wave function, and !(t), "(t) and $(t) control the potential function.

4.2 Space-Time Modulation Nonlinear Schrödinger Equation

139

To reduce Eqs (4.2.8e)–(4.2.8f) as the constant coefficient partial differential equations, we require the following condition: 4t =

"(t) 2 ' , 2 x

g(t) =

G"(t)'2x , 2A2 (t)

(4.2.10)

where G is a constant. According to Eqs (4.2.9a) and (4.2.9b), it follows that 4(t) =

1 t 2 ∫ [! (s)"(s)]ds, 2 0

g(t) =

G!2 (t)"(t) t

2a20 e2 ∫0 𝛾(s)ds

.

(4.2.11)

From the above equations, we can see that frequency !(t), group dispersion "(t) and gain–loss term 𝛾(t) control the nonlinear interaction. In virtue of Eqs (4.2.7), (4.2.9c) and (4.2.11), we can see that phase function >(x, t) is >(x, t) = –

!t $t 1 t x2 – x + 70 (t) + ∫ !2 (s)"(s)ds, 2!(t)"(t) !(t)"(t) 2 0

!

(4.2.12)

$

t t where – 2!(t)"(t) denotes phase deviation, – !(t)"(t) denotes frequency shift, 70 (t) +

1 t 2 ∫ ! (s)"(s)ds 2 0

denotes the phase-front curvature. Based on the above conditions, Eqs (4.2.8e)–(4.2.8f) can be reduced as the following constant coefficient partial differential equations: bP4 + cQ'' – ,cQ + cGQ[c2 Q2 + (1 + bP)2 ] = 0, 2 2

(4.2.13a) 2

–cQ4 + bP'' – ,(1 + bP) + G(1 + bP)[c Q + (1 + bP) ] = 0.

(4.2.13b)

4.2.3 One Dimensional Self-Similarity Optical Rogue Wave Solution and Its Parameter Analysis From the results in Refs [175, 200, 201], we can obtain the rational solution of system (4.2.13). Furthermore, according to the above transformations (4.2.5) and (4.2.7), we can obtain the similarity rogue wave solution of Eq. (4.2.3). In what follows, we consider two groups of low-order rogue wave solutions. 4.2.3.1 First-Order Self-Similarity Rogue Wave Solution When , = G = 1, through the system (4.2.13), it follows that P(', 4) = –

4 , b[1 + 2'2 + 442 ]

Q(', 4) = –

84 , c[1 + 2'2 + 442 ]

(4.2.14)

140

4 The Rogue Wave Solution and Parameters Managing

Moreover, we can obtain the self-similar rogue wave solution of Eq. (4.2.3) [199] 4 + 8i4(t) ] 1 + 2[!(t)x + $(t)]2 + 442 (t)

t

81 (x, t) = a0 √|!(t)| e∫0 𝛾(s)ds [1 – × exp{i[ 7(x, t) + 4(t)]},

(4.2.15)

where 7(x, t) and 4(t) are determined by Eqs (4.2.9c) and (4.2.11). It is ready to find the self-similar rogue wave solution (4.2.15) of Eq. (4.2.3). Different from the rogue wave solution of classical NLSE [175, 183, 184, 191, 200, 201], there are much more colorful phenomena. Specially, when ! = 2, a0 = " = 1, 70 = 𝛾 = 0, g = 1, Eq. (4.2.3) reduces to classical NLSE; meanwhile the self-similar rogue wave solution (4.2.15) reduces to the ordinary rogue wave solution [175, 200]. In the following, we choose some parameters to illustrate the propagation rule of the first-order self-similar rogue wave solution (4.2.15). For the fixed parameters !0 = 1,

70 (t) = 0,

𝛾(t) = 0.2 sin(t).

(4.2.16)

For the other parameters, we classify several cases to study the physical mechanism self-similar rogue wave solution(4.2.15). (i) If !(t), "(t) and $(t) choose triangular periodic function !(t) = 2 + 0.5 sin(5t2 ), "(t) = 3 + 0.2 cos(2t), $(t) = 2 cos(4t), 5 sin(2t2 ).

(4.2.17)

Fig. 4.3 shows the nonlinear interaction g(t), GVD "(t), potential function V(x, t) and gain–loss term 𝛾(t)’s variation, where the parameters are determined by Eqs (4.2.16), (4.2.17) and $(t) = 2 cos(4t). Fig.4.4 shows the evolution of self-similar rogue wave solution (4.2.15), where the parameters are determined by Eqs (4.2.16)–(4.2.17). Different (a)

(b)

6

g V

4

400 0

2

–400

β

–3

–3 0 –8

γ t

0

x 8

0

0

t

3 3

Figure 4.3: (a): Nonlinear interaction g(t), group velocity dispersion "(t) and gain-loss term 𝛾(t)’s variation; (b) Potential function V(x, t), where the parameters are determined by Eqs (4.2.16)–(4.2.17) and $(t) = 2 cos(4t).

4.2 Space-Time Modulation Nonlinear Schrödinger Equation

(a)

141

(b) 1

3

|Ψ1| t

2

0 2

1 0.3 –3

0 –2 x

–1

0

1

t

–1

–2

(c)

–3

–2

–1

x 0

1

0

x

2

(d) 1

3

|Ψ1| t

2

0 2

1 0.3

0 –2

x0

2

–2

t –1 –2

Figure 4.4: Time evolution of amplitude |81 | of self-similar rogue wave (4.2.15) (Left: wave propagations in (x, t), Right: density plot), where the parameters are determined by Eqs (4.2.16)–(4.2.17). (a),(b) $(t) = 2 cos(4t); (c),(d): $(t) = 5 sin(2t2 ).

$(t) = 2 cos(4t) and $ = 5 sin(2t2 ) arouse the variation of amplitude of self-similar rogue wave solution (4.2.15). (ii) If !(t), "(t) and $(t) choose double periodic function !(t) = c1 dn(t, k1 ), "(t) = 1 + 0.5cn(t, k1 ), $(t) = 2cn(t, k2 ).

(4.2.18)

Figure 4.5 shows the nonlinear interaction g(t) GVD "(t) potential function V(x, t) and gain–loss term 𝛾(t)’s variation, where parameters are determined by Eqs (4.2.16) and (4.2.18). Figure 4.6 shows the evolution of amplitude of self-similar rogue wave solution (4.2.15), where parameters are determined by Eqs (4.2.16) and (4.2.18). Different parameters c1 = 1, 1.8 arouse the variation of amplitude of self-similar rogue wave solution (4.2.15). 4.2.3.2 Second Order Self-Similar Rogue Wave Solution Similarly, from the system (4.2.13), the following rational solutions follow: P(', 4) =

P2 (', 4) , bA2 (', 4)

Q(', 4) =

Q2 (', 4) , cA2 (', 4)

(4.2.19)

142

4 The Rogue Wave Solution and Parameters Managing

(a) 1.4

(b) β V 400

1.0

200 0.6

0

g

–200 –20

0.2 γ –0.2 –10

t

–20 x

0

10

0

0 20

t

20

Figure 4.5: (a): Nonlinear interaction g(t) group velocity dispersion "(t) and gain–loss term𝛾(t)’s variation; (b) Potential function V(x, t), where the parameters are determined by Eqs (4.2.16), (4.2.18) and c1 = 1.0, k1 = 0.5, k2 = 0.6.

(a)

(b) 10

Ψ1 3

t

2

0

1

10 0

0 –6 x

0

6

–10

t

–10

–6

(c)

x

0

6

x

0

6

(d) 10 Ψ1

4

t

3

0

2 10 1 –6 x

0

0t –10 6

–10 –6

Figure 4.6: The time evolution of self-similar rogue wave solution’s amplitude |81 | (4.2.15)(Left: wave propagation in (x, t), Right: density plot), where the parameters are determined by Eqs (4.2.16) and (4.2.18). (a),(b) c1 = 1.0; (c),(d) c1 = 1.8, k1 = 0.5, k2 = 0.6.

4.2 Space-Time Modulation Nonlinear Schrödinger Equation

143

where 1 3 3 P2 ('(x, t), 4(t)) = – '4 – 6'2 42 – 1044 – '2 – 942 + , 2 2 8 15 Q2 ('(x, t), 4(t)) = –4 ('4 + 4'2 42 + 444 – 3'2 + 242 – ) , 4 1 6 1 4 2 2 6 1 4 9 4 3 2 2 2 4 A2 ('(x, t), 4(t)) = ' + ' 4 + ' 4 + 4 + ' + 4 – ' 4 12 2 3 8 2 2 9 2 33 2 3 + ' + 4 + . 16 8 32 Thus, based on the transformations (4.2.5) and (4.2.7), the second-order self-similar rogue wave solution of Eq. (4.2.3) follows [199]: t

82 (x, t) = a0 √|!(t)| e∫0 𝛾(s)ds [1 +

P2 (', 4) + iQ2 (', 4) i[7(x,t)+4(t)] , ]e A2 (', 4)

(4.2.20)

where '(x, t) = !(t)x + $(t), 7(x, t) and 4(t) are determined by Eqs (4.2.9c) and (4.2.11). This second-order rogue wave solution (4.2.20) is different from the ordinary one [175, 200], since some free functions of time t are involved. Similar to the first-order selfsimilar rogue wave solution, when ! = 2, a0 = " = 1, 70 = 𝛾 = 0, g = 1, the solution (4.2.20) can also reduce to the known one of Eq. (4.2.3) in Refs [175, 200]. In what follows, we fixed some parameters to study the evolution of secondorder self-similar rogue wave solution (4.2.20). Figure 4.7 shows the evolution of (a)

(b)

1.5

|Ψ2| 4

t 0

1.5

2 0

0 –3 –2 –1 x

0

t –1.5

–1.5

1

(c)

(d)

–3

–2

–1 x 0

1

1.5

|Ψ2| 4

t 1.5

2 0 –3 –2 –1 x

0 0

1

2

–1.5

0

t –1.5 –3

–2

–1 x 0

1

2

Figure 4.7: The evolution of amplitude |82 | for the second-order rogue wave solution (Left: wave propagation in (x, t), Right: density plot), where the parameters are determined by Eqs (4.2.16) and (4.2.18). (a),(b): $(t) = 2 cos(4t), (c),(d): $(t) = 5 sin(2t2 ).

144

4 The Rogue Wave Solution and Parameters Managing

(a)

5

(b) |Ψ2|

t

3 1

0 x0

10

t –15 –10

–15

(c)

14

0

15

–10

15

(d)

x

0

10

x

0

10

15

|Ψ2| t

10

15

6 2 –10

0 x0

10

–15

0

t –15 –10

Figure 4.8: The evolution of amplitude |82 | for the second-order rogue wave solution (Left: wave propagation in (x, t), Right: Density plot), where the parameters are determined by Eqs (4.2.16) and (4.2.18). (a),(b) c1 = 1.0; (c),(d) c1 = 1.8, k1 = 0.5, k2 = 0.6.

the second-order rogue wave solutions’ amplitude, where the parameters are determined by Eqs (4.2.16)–(4.2.17). Different $(t) = 2 cos(4t) and $(t) = 5 sin(2t2 ) arouse the variation of the second-order self-similar rogue wave solution. Figure 4.8 shows the evolution of the second-order rogue wave solutions’ amplitude, where the parameters are determined by Eqs (4.2.16) and (4.2.18). Different parameters c1 = 1, 1.8 arouse the variation of amplitude for the second-order self-similar rogue wave solution (4.2.20).

4.3 (3+1)-Dimensional Space-Time Modulation Gross–Pitaevskii/NLS Equation 4.3.1 Three-Dimensional Nonlinear Physical Model We study the self-similar rogue wave solution for the one-dimensional variable coefficient NLSE Actually, three-dimensional nonlinear Schödinger/Gross–Pitaevskii equations possess important significance in the nonlinear science. It is used in nonlinear optics, condensed-matter physics, and, in particular, modeling Bose–Einstein condensate [176, 194, 196]. In what follows, we consider the Lagrangian density

4.3 (3+1)-Dimensional Space-Time Modulation Gross–Pitaevskii/NLS Equation

145

ℏ2 |∇8|2 + gnon (t)|8|4 + 2[Vext (r, t) – iA(t)]|8|2 , m

(4.3.1)

L = iℏ(88∗t – 8t 8∗ ) +

where 8 ≡ 8(r, t), r ∈ ℝ3 , ∇ ≡ (𝜕x , 𝜕y , 𝜕z ), ℏ is Planck constant, m is the atomic mass, Vext (r, t) is potential function, A(t) is gain–loss term, gnon (t) = 40ℏas (t)/m denotes nonlinear interaction, where as (t) represents the s-wave scattering length of Feshbach resonance (as (t) < 0 represents the attractive interaction, and as (t) > 0 represents the repulsion interaction). Based on the Lagrangian density (4.3.1) and Euler–Lagrange equation $L 󵄨󵄨󵄨 𝜕 𝜕L 𝜕L 𝜕 𝜕L 󵄨󵄨 ≡ – – ∑ = 0; ∗ ∗ ∗ 󵄨 $8 󵄨(4.2.1) 𝜕8 𝜕t 𝜕8t 3=x,y,z 𝜕3 𝜕8∗3

(4.3.2)

it follows that the (3+1)-dimensional space-time modulation Gross–Pitaevskii/ NLSE [189, 194, 196–198, 202] iℏ

𝜕8 ℏ2 = – ∇2 8 + Vext (r, t)8 + gnon (t)|8|2 8 + iA(t)8. 𝜕t 2m

(4.3.3)

Under the scaling transformation r → √ℏ/(m9⊥ ) r, t → 9–1 ⊥ t, Eq. (4.3.3) becomes a dimensionless form [189] i

𝜕J 1 = – ∇2 J + v(r, t)J + g(t)|J|2 J + i𝛾(t)J, 𝜕t 2

(4.3.4)

where v(r, t), g(t) and 𝛾(t) are related to the potential function Vext , nonlinear interaction gnon (t) and gain–loss term A(t) in the Eq. (4.3.3). Equation (4.3.4) also appears in nonlinear optics, plasma and oceanography; specially in the nonlinear optics, second-order term’s coefficient ∇2 J (also called the GVD) can depend on the time. Especially, the three-dimensional harmonic potential merely depends on time [196] v(r, t) = wx2 x2 + wy2 y2 + wz2 z2 .

(4.3.5)

(i) Insert Gaussian transformation 8(r, t) = I(x, t) exp [–

√2 (wy y2 + wz z2 )] e–i,t 2

into Eq. (4.3.4), and consider the obtained results in the mean of y, z direction, where , = (wy + wz )/√2; then three dimensional-equation (4.3.4) reduces to the one-dimensional Gross–Pitaevskii equation i

𝜕I 1 𝜕2 I 2 ̂ ̂ =– + v(x)I + g(t)|I| I + i𝛾(t)I, 𝜕t 2 𝜕x2

(4.3.6)

146

4 The Rogue Wave Solution and Parameters Managing

Similarly, (ii) insert Gaussian transformation 8(r, t) = I(x, y, t) exp (–

√2 w z2 ) e–i,t 2 z

into Eq. (4.3.4), and consider the obtained results in the mean of y, z direction, where , = wz /√2; then three-dimensional equation (4.3.4) reduces to the two-dimensional Gross–Pitaevskii equation i

𝜕I 1 𝜕2 I 𝜕2 I 2 ̂ ̂ I + i𝛾(t)I. = – ( 2 + 2 ) + v(x, y)I + g(t)|I| 𝜕t 2 𝜕x 𝜕y

(4.3.7)

4.3.2 Symmetry Analysis-Similarity Transformation and Reduction System We aim at looking at the proper similarity transformation to relate the solution of Eq. (4.3.4) with the one of (1+1)-dimensional classical NLSE i

𝜕I(', 4) 𝜕2 I(', 4) + G|I(', 4)|2 I(', 4), =– 𝜕4 𝜕'2

(4.3.8)

where the physical field I(', 4) is the function of ' ≡ '(r, t) and 4 ≡ 4(t), G is a real constant. Since here we consider the three-dimensional rogue wave solution, choose G as the negative constant (for instant G = –1), which corresponds to the attractive case (equivalent with the self-focusing interaction in the nonlinear optics and the negative scattering length in the BEC theory). To control the boundary condition in the infinity, we consider the very natural constraint [203] ' → 0 (r → 0),

' → ∞ (r → ∞).

(4.3.9)

Considering the physical field J(r, t) possesses the following form [189]: J(r, t) = 1(t)ei>(r,t) I['(r, t), 4(t)],

(4.3.10)

where 1(t), >(r, t), 4(t) and '(r, t) are undetermined real functions. Transformation (4.3.10) can reduce Eq. (4.3.4) with many different forms, where we reduce (4.3.4) to NLS equation (4.3.8). Insert Eq. (4.3.10) into Eq. (4.3.4) and require I['(r, t), 4(t)] to satisfy Eq. (4.3.8); the following partial differential equations follow: ∇2 ' = 0,

(4.3.11a)

't + ∇> ⋅ ∇' = 0, 2

24t – |∇'| = 0, 2

21t + 1∇ > – 2𝛾(t)1 = 0, 2

2

2g(t)1 – G|∇'| = 0, 2

2v(r, t) + |∇>| + 2>t = 0.

(4.3.11b) (4.3.11c) (4.3.11d) (4.3.11e) (4.3.11f)

4.3 (3+1)-Dimensional Space-Time Modulation Gross–Pitaevskii/NLS Equation

147

In general, for the arbitrary potential function and nonlinear term, Eq. (4.3.11) is not always consistence. But for some proper potential function v(r, t), nonlinear interaction g(t) and gain–loss term 𝛾(t) and Eq. (4.3.11) may be solvable. In the following, we analyze it by two steps: – Firstly, based on the boundary condition (4.3.9), through solving Eqs (4.3.11a)– (4.3.11c) to determine the similarity variable '(r, t), 4(t) and phase >(r, t). – Secondly, based on the determined similarity variable '(r, t), 4(t) and phase >(r, t), use Eqs (4.3.11d)–(4.3.11f) to obtain 1(t), v(r, t) and g(t).

4.3.3 Similarity Variable, Constraint Condition and Velocity Field Through Eqs (4.3.11a)–(4.3.11c), we can obtain the similarity variable '(r, t), 4(t) and phase >(r, t) t

'(r, t) = c(t) ⋅ r – ∫ c(s) ⋅ a(s)ds,

(4.3.12a)

0

1 t ∫ |c(s)|2 ds, 2 0 ̂ r + a(t) ⋅ r + 9(t), >(r, t) = r K(t) 4(t) =

(4.3.12b) (4.3.12c)

where we introduce the 3×3 diagonal matrix ̂ = diag(Kx (t), Ky (t), Kz (t)) K(t) and K3 (t) = –

ċ3 (t) , 2c3 (t)

where 3 = x, y, z, the dot denotes the derivative with the time, coefficient c(t) = (cx (t), cy (t), cz (t)), a(t) = (ax (t), ay (t), az (t)) and 9(t) are real functions that depend on the time t. Through Eqs (4.3.11d)–(4.3.11f), we can obtain that 1(t), v(r, t) and g(t) t

1(t) = 10 √|cx (t) cy (t) cz (t)| e∫0 𝛾(s)ds , G |c(t)|2

, t 2120 |cx (t) cy (t) cz (t)| e2 ∫0 𝛾(s)ds 1 ̂ r + b(t) ⋅ r – 9(t) ̇ – |a(t)|2 , v(r, t) = r A(t) 2

g(t) =

(4.3.13a) (4.3.13b) (4.3.13c)

where 10 is integration constant. Gain–loss term 𝛾(t) is determined by the initial state, which is used to control the amplitude 1(t) and nonlinear term g(t). In the potential ̂ = diagonal time-dependent function v(r, t), we introduce the 3×3 diagonal matrix A(t) 3 × 3 matrix (Ax (t), Ay (t), Az (t)) with the entries

148

4 The Rogue Wave Solution and Parameters Managing

A3 (t) =

ċ2 (t) c̈3 (t) – 32 2c3 (t) c3 (t)

(4.3.14)

and the vector function b(t) = (bx (t), by (t), bz (t)) and b3 (t) =

ċ3 (t) a3 (t) – ȧ 3 (t). c3 (t)

(4.3.15)

It is ready to see that the velocity field v(r, t) = ∇>(r, t) corresponding to the abovementioned phase >(r, t) is given by v(r, t) = 2(Kx (t)x, Ky (t)y, Kz (t)z) + a(t)

(4.3.16)

such that we have the divergence of the vector field v(r, t) in the form div v(r, t) = 2[Kx (t) + Ky (t) + Kz (t)] = –𝜕t ln |cx (t)cy (t)cz (t)|,

(4.3.17)

which measures the fluid per unit volume. These time-dependent functions c3 (t) are used to control the density. By Eq. (4.3.17), we can obtain the following: (i) When [ ln |cx (t) cy (t) cz (t)|]t < 0, the density is decreased (fluid outflux); (ii) When [ ln |cx (t) cy (t) cz (t)|]t > 0, the density is increased (fluid influx); (iii) When ln |cx (t) cy (t) cz (t)| = const, the density is invariant, and vj (r, t) is incompressible. In BEC theory, the potential function usually chooses the harmonic function. From Eqs (4.3.13c) and (4.3.14), we can see that if we require the potential function v(r, t) is a second-degree polynomial for every space x, y, z, then we have A3 ≠ 0, i.e., c3 c̈3 – 2ċ23 ≠ 0, which denotes that c3 are not equivalent to constants, but are some functions of time. These time-dependent functions c3 (t) will affect the other variables (see Eqs (4.3.13a)–(4.3.15)) such that self-similar solutions of Eq. (4.3.10) in form (4.3.4) exhibit abundant structures. Therefore, we have established the relation between three-dimensional variable coefficient GP/NLS equation (4.3.4) and classical NLS equation (4.3.8). With this way, we can obtain much more self-similar rogue wave solutions for three-dimensional variable coefficient GP/NLS equation (4.3.4).

4.3.4 Three-Dimensional Self-Similar Rogue Wave Solutions and Its Parameters Regulation As two representative examples, we consider the lowest-order rational solutions of the classical NLS Eq. (4.3.8) which serve as prototypes of rogue waves. First, we use

149

4.3 (3+1)-Dimensional Space-Time Modulation Gross–Pitaevskii/NLS Equation

(a)

(b) 0.4

(c) 0.4

–200

γ(t) 0.2

0.2

g(t)

A(t) 0

–400

0 –0.2

–0.2

–0.4

–600 –0.4 0

10

t

20

30

0

6

t

12

18

0

10 t

20

Figure 4.9: (a) Nonlinear interaction g(t) (4.3.13b); (b) A3 (t) second-degree term of the linear potential v(r, t) (4.3.14); (c) the gain or loss term 𝛾(t) vs time for the parameters are given by (4.3.19), and kx = 0.9, ky = 0.6, kz = 0.1 and k = 0.6. (b): Ax (t) solid line, Ay (t) dashed line, Az (t) dashed-dotted line.

the first-order rational solution of Eq. (4.3.8). As a result, we obtain the first-order nonstationary rogue wave solutions of Eq. (4.3.4) in the form [189] t

J1 (r, t) = 10 √|cx (t) cy (t) cz (t)| e∫0 𝛾(s)ds × [1 –

4 + 8i4(t) ] ei[>(r,t)+4(t)] , 1 + 2'2 (r, t) + 442 (t)

(4.3.18)

where the variables '(r, t), 4(t) and the phase >(r, t) are given by Eqs (4.3.12a)– (4.3.12c). For the illustrative purposes, we can choose these free parameters in the form c3 (t) = a3 (t) = dn(t, k3 ), 𝛾(t) = sn(t, k)cn(t, k), 10 = 1, 9(t) = 0,

(4.3.19)

where sn(t, k), cn(t, k) and dn(t, k3 ) stand for the respective Jacobi elliptical functions and k3 , k ∈ (0, 1) are their moduli. Figure 4.9 depicts the profiles of nonlinearity g(t), given by Eq. (4.3.13b), the coefficients of second-degree terms of the linear potential v(r, t) given by Eq. (4.3.14), and the gain or loss term 𝛾(t) vs time for the chosen parameters given by Eq. (4.3.19). The evolution of intensity distribution of the 3D field (4.3.18) is shown in Fig. 4.10. The solution is localized in space and keeps the localization infinitely in time, which differs from the usual rogue wave solutions (see [175]). On the other hand, if we choose the free parameters in the form cx (t) = ax (t) = 1 + c0 sin(t), cy (t) = ay (t) = 1.2 + c0 cos(t), cz (t) = az (t) = 0.8 + c0 sin(t), c0 = 0.01,

(4.3.20)

150

4 The Rogue Wave Solution and Parameters Managing

(a)

(b)

(c)

40

t

–40 –8

8

x

–8

8

y

–8

8

z

Figure 4.10: Color coded plot of wave intensity of the first-order self-similar rogue wave solution (4.3.18). (a) |J1 |2 (x, 0, 0, t) with max{x,0,0,t} |J1 |2 = 0.06; (b) |J1 |2 (0, y, 0, t) with max{0,y,0,t} |J1 |2 = 0.068; (c) |J1 |2 (0, 0, z, t) with max{0,0,z,t} |J1 |2 = 0.068. The parameters given by Eq. (4.3.19) with kx = 0.9, ky = 0.6, kz = 0.1, k = 0.6.

(a)

(b)

–150

0.006

–200 0.002

–250 g(t) –300

A(t) –0.002

–350 –400

–0.006 0

5

10

t 15

20

25

0

5

10

t 15

20

25

Figure 4.11: (a) Nonlinearity g(t) (4.3.13b); (b) The coefficients A3 (t) of second-degree term of the linear potential v(r, t) given by Eqs (4.3.14); The parameters are determined by Eq. (4.3.20). (b): Ax (t) solid line, Ay (t) dashed line, Az (t) dashed-dotted line.

and 10 , 9(t) and 𝛾(t) are determined by Eq. (4.4.26), then the evolution of intensity distribution of the 3D rogue wave solutions (4.3.18) will be changed. For the fixed parameters (4.3.20), Fig. 4.11 displays the profiles of nonlinearity g(t) given by Eq. (4.3.13b), the coefficients of second-degree terms of the linear potential v(r, t) given by Eq. (4.3.14) and the gain–loss term 𝛾(t). Figure 4.12 shows the evolution of intensity of the first-order self-similar rogue wave solution (4.3.18). The solution is localized both in time and in space, thus revealing the usual rogue wave features [175]. Generally speaking, we have large degree of freedom in choosing the coefficients of transformation. As a result, we can describe infinitely large class of solutions of the three-dimensional GP/NLS equation (4.3.4) with every exact solution of the onedimensional NLS equation. Additional possibility of choosing the solution of the latter

151

4.3 (3+1)-Dimensional Space-Time Modulation Gross–Pitaevskii/NLS Equation

(a)

(b)

(c)

10

t

–10 –15

x

15

–15

y

15

–15

z

15

Figure 4.12: Color coded plot of wave intensity of the first-order self-similar rogue wave solution (4.3.18). (a) |J1 |2 (x, 0, 0, t) with max{x,0,0,t} |J1 |2 = 0.027; (b) |J1 |2 (0, y, 0, t) with max{0,y,0,t} |J1 |2 = 0.03; (c) |J1 |2 (0, 0, z, t) with max{0,0,z,t} |J1 |2 = 0.029, the parameters given by Eq. (4.3.20).

one increases tremendously the variety of solutions that we can obtain, especially, the self-similar rogue wave solution. When a higher-order rational solution of the NLS equation (4.3.8) (see Refs [175, 200]) is applied to transformation (4.3.10), we obtain the second-order nonstationary rogue wave solutions of Eq. (4.3.4) in the form [189] t

J2 (r, t) = 10 √|cx (t) cy (t) cz (t)| e∫0 𝛾(s)ds × [1 +

P(', 4) – i4(t) Q(', 4) i[>(r,t)+4(t)] ]e , H(', 4)

(4.3.21)

where functions P(', 4), Q(', 4) and H(', 4) are given by [189] 3 3 1 P(', 4) = – '4 – 6'2 42 – 1044 – '2 – 942 + , 2 2 8 15 Q(', 4) = '4 + 4'2 42 + 444 – 3'2 + 242 – , 4 1 2 1 9 3 1 H(', 4) = '6 + '4 42 + '2 44 + 46 + '4 + 44 – '2 42 12 2 3 8 2 2 9 33 3 + ' 2 + 42 + . 16 8 32

(4.3.22)

The variables '(r, t), 4(t) and and the phase >(r, t) here are given by Eqs. (4.3.12a)(4.3.12c). As in the previous cases, we choose the parameters given by Eqs (4.3.19) and (4.3.20) except for a3 (t) = 0. The intensity distributions of the second-order rogue wave solutions (4.3.21) are depicted in Figs 4.13 and 4.14. Clearly, the field evolution in this case is more complicated. Figure 4.13 shows that the solution is localized in all three dimensions in space. Figure 4.14 shows that the solution is localized both in space and in time, thus displaying the basic feature of a rogue wave. It follows from the above-mentioned two cases for the parameters that the parameters c3 (t), a3 (t) and gain–loss term 𝛾(t) can be used to control the wave propagations related to rogue waves, which may raise the possibility of relative experiments

152

4 The Rogue Wave Solution and Parameters Managing

(a)

(b)

(c)

60

t

–60 –8

x

8

–8

y

8

–8

z

8

Figure 4.13: Color coded plot of wave intensity of the second-order self-similar rogue wave solution (4.3.21) (a) |J2 |2 (x, 0, 0, t) with max{x,0,0,t} |J2 |2 = 0.135; (b) |J2 |2 (0, y, 0, t) with max{0,y,0,t} |J2 |2 = 0.13; (c) |J2 |2 (0, 0, z, t) with max{0,0,z,t} |J2 |2 = 0.125. The parameters given by Eq. (4.3.19) with a3 (t) = 0, kx = 0.9, ky = 0.6, kz = 0.1, k = 0.6.

(a)

(b)

(c)

10

t

–10 –10

x

10

–10

y

10

–10

z

10

Figure 4.14: Color coded plot of wave intensity of the second-order self-similar rogue wave solution (4.3.21) (a) |J2 |2 (x, 0, 0, t) with max{x,0,0,t} |J2 |2 = 0.038; (b) |J2 |2 (0, y, 0, t) with max{0,y,0,t} |J2 |2 = 0.036; (c) |J2 |2 (0, 0, z, t) with max{0,0,z,t} |J2 |2 = 0.038. The parameters given by Eq. (4.3.20) with a3 (t) = 0.

and potential applications in nonlinear optics and BECs. Similarly we can also obtain three-dimensional higher-order self-similar rogue wave solutions in terms of transformation (4.3.10).

4.4 Generalized Inhomogeneous Higher-Order Nonlinear Schrödinger Equation with Modulating Coefficients We know that the NLSE possesses very important significant. When the optical pulses become shorter (e.g., 100 fs [194]), higher-order dispersive and nonlinear effects, such as third-order dispersion (TOD), self-steepening (SS), and the self-frequency shift (SFS) (alias the perturbation terms) arising from the stimulated Raman scattering become significant in the study of ultra-short optical pulse propagation. With the above-mentioned aim, Kodama and Hasegawa [204, 205] presented the higher-order NLS equation

4.4 Generalized Inhomogeneous Higher-Order Nonlinear Schrödinger

i

𝜕8 " 𝜕2 8 + g|8|2 8 + 𝜕z 2 𝜕t2 𝜕3 8 𝜕(|8|2 8) 𝜕|8|2 + a3 8 ] + iA8, = i: [a1 3 + a2 𝜕t 𝜕t 𝜕t

153

(4.4.1)

reduced from the Maxwell equation ∇×∇× E=–

1 𝜕2 D , D = : ∗ E, c2 𝜕t2

(4.4.2)

where c being the speed of light [205], the real-valued parameters ", g, a1 , a2 , a3 and A represent GVD, SPM, SS, SFS and the gain or loss terms, respectively. Equation (4.4.1) contains many types of integrable model, such as the Hirota equation [206], Sasa–Satsuma equation [207], KN-type derivative NLS equation [208], and CLL-type derivative NLS equation [209]. Equation (4.4.1) has been found to admit many types of solution (see, e.g., [214–218]). Recently, controllable rogue waves of Eq. (4.4.1) with all coefficients depending only on space z have been considered [219], but one more condition was missed to support the obtained results, that is to say, the sum of the parameters related to SS and the SFS should be zero (a2 + a3 = 0). In the following, we study the generalized model of equation Eq. (4.4.1) with space/time-modulated GVD, SPM, and gain or loss terms, space-modulated TOD, SS and SFS, and more terms such as space/time-modulated external potential, linear group velocity, and differential gain or loss term [194, 220] i

𝜕8 𝜕2 8 = "(z, t) 2 + [V(z, t) + i𝛾(z, t)] 8 + g(z, t)|8|2 8 𝜕z 𝜕t 3 𝜕 8 𝜕(|8|2 8) 𝜕|8|2 +i [!1 (z) 3 + !2 (z) + !3 (z)8 ] 𝜕t 𝜕t 𝜕t 𝜕8 , + [,(z) + i3(z, t)] 𝜕t

(4.4.3)

where z is the normalized propagation distance along the optical fiber, t is the retarded time, 8 ≡ 8(z, t) denotes the slowly varying envelope amplitude of the electric field measured in units of the square root of the power at position z in the optical fiber and at time t. "(z, t), g(z), !1 (z), !2 (z) and !3 (z) are all real-valued functions of the listed variables and stand for GVD SPM TOD SS and SFS arising from stimulated Raman scattering, respectively [194]. V(z, t) and 𝛾(z, t) are the external potential and gain or loss distribution, respectively, ,(z) denotes the differential gain or loss parameter [220], and 3(z, t) is related inversely to the group velocity of the modes (a walk-off effect) [194, 221]. Equation (4.4.3) is associated with a variational principle $L /$8∗ = 0 with the Lagrangian density L = i(88∗z – 8z 8∗ ) – 2"|8t |2 + g|8|4 + 2(V + i𝛾)|8|2 +!1 (8t 8∗tt – 8∗t 8tt ) + (, + i3)(8t 8∗ – 88∗t ) +[!2 (|8|)t + !3 88∗t ]|8|2 ,

(4.4.4)

154

4 The Rogue Wave Solution and Parameters Managing

where 8∗ stands for the complex conjugate of the electric field 8, and the subscript denotes the partial derivative with respect to the variables (z, t). Equation (4.4.3) describes many types of nonlinear model, such as the NLS equation with varying coefficients, the derivative NLS equation with varying coefficients, the Hirota equation with varying coefficients, the Sasa–Satsuma equation with varying coefficients, and the higher-order NLS equation without three-photon nonlinear absorption.

4.4.1 Symmetry Reductions – Transformation and Hirota Equation In general, Eq. (4.4.3) is not integrable since these varying coefficients strongly affect the wave propagation of optical pulses in a self-similar manner. In order to study the exact analytical solutions of Eq. (4.4.3), we need to look for some integrability conditions using methods such as the symmetric reduction approach. Based on the Lie symmetry infinitesimal generator method [222, 223], these similarity reductions can be obtained using the third-order propagation Pr(3) (7) of the vector field 7=T

𝜕 𝜕 𝜕 𝜕 + Z + (R + (I 𝜕t 𝜕z 𝜕8R 𝜕8I

(4.4.5)

acting on Eq. (4.4.3), i.e., 󵄨 Pr(3) (7)F 󵄨󵄨󵄨󵄨{F=0} = 0,

(4.4.6)

where 8(z, t) = 8R (z, t) + i8I (z, t) with 8R , 8I ∈ ℝ, the variables T, Z, (R , and (I are all undetermined functions of z, t, 8I , 8R , F = {[–i𝜕z + "𝜕t2 + (V + i𝛾) + g|8|2 + i(!1 𝜕t3 + !3 𝜕t (|8|2 )) 󵄨 +(, + i3)𝜕t ]8 + !2 𝜕t (8|8|2 )}󵄨󵄨󵄨󵄨8=8 +i8 , R I 𝜕 𝜕 (3) z z 𝜕 t t 𝜕 + (I + (R + (I Pr (7) = 7 + (R 𝜕8R,z 𝜕8I,z 𝜕8R,t 𝜕8I,t 𝜕 𝜕 𝜕 𝜕 +(Rtt + (Itt + (Rttt + (Ittt . 𝜕8R,tt 𝜕8I,tt 𝜕8R,ttt 𝜕8I,ttt Based on the above Eq. (4.4.6), the similarity variables and transformations of Eq. (4.4.3) can be found by solving the characteristic equation [222, 223] dt dz d8R d8I = . = = T Z (R (I

(4.4.7)

To study multi-soliton solutions of Eq. (4.4.3), including rogue wave solutions, we need to reduce it to some integrable (or easily solved) differential equation (e.g., the Hirota equation [206] or the Sasa–Satsuma equation [207]), which possesses multi-soliton

4.4 Generalized Inhomogeneous Higher-Order Nonlinear Schrödinger

155

solutions or even rogue waves). In the following we consider the symmetry (reduction) transformation [189, 224] 8(z, t) = 1(z)ei>(z,t) J[4(z, t), '(z)]

(4.4.8)

connecting the solutions of Eq. (4.4.3) with those of the following Hirota equation [206] with constant coefficients i

𝜕J 𝜕J 𝜕3 J 𝜕2 J = – 2 + G|J|2 J + 2√2i- ( 3 + 3|J|2 ), 𝜕' 𝜕4 𝜕4 𝜕4

(4.4.9)

where the physical field J(', 4) is a function of the variables 4(z, t) and '(z), which are the new temporal and spatial coordinates, respectively, and G, - are both real-valued constants. Since the main goal of this paper focuses on rogue waves of Eq. (4.4.3), we choose G < 0 (e.g. G = –1), which corresponds to the attractive case. We know that the NLSE possesses two basic transformations, i.e., the scaling and gauge transformations [225]. But high-order NLSE (4.4.9) with - ≠ 0 differs from the NLSE (2.5.1). It possesses the following proposition: Equation (4.4.9) possesses the following two modified basic transformations leaving itself invariant: i) Scaling-parameter transformation ' → !2 ',

4 → !4,

J → J/!,

- → !-,

(4.4.10)

leaving itself invariant for any SPM parameter G ≠ 0, where ! is a real-valued constant. ii) “Gauge” transformation ' → k3 ',

4 → k4 + +',

J → k2 Jei(p4+q') ,

(4.4.11)

leaving Eq. (4.4.9) invariant only for the self-focusing SPM parameter G = –1, which just makes Eq. (4.4.9) possess rogue wave solutions, where k = 1 + 6√2-p, + = –kp(1 + k), q = –p2 (2√2-p + 1) with p being a real-valued constant. Based on the above-mentioned method, the substitution of Eq. (4.4.8) into Eq. (4.4.3) with J(', 4) satisfying Eq. (4.4.9) yields the following system of partial differential equations: !1 4t 4tt = 0,

!2 + !3 = 0,

(4.4.12a)

"4tt – 3!1 (4tt >t + 4t >tt ) + ,4t = 0, 'z +

42t ("

– 3!1 >t ) = 0,

34t + 2"4t >t + !1 (4ttt –

(4.4.12b) (4.4.12c)

34t >2t )

– 4z = 0,

1z – 1[">tt – 3!1 >t >tt + ,>t + 𝛾] = 0,

(4.4.12d) (4.4.12e)

156

4 The Rogue Wave Solution and Parameters Managing

V + >z – ">2t + !1 (>3t – >ttt ) – 3>t = 0,

(4.4.12f)

– 2√2-'z = 0,

(4.4.12g)

1 (g – !2 >t ) – G'z = 0,

(4.4.12h)

!2 1 4t – 6√2-'z = 0.

(4.4.12i)

!1 43t 2

2

Generally speaking, the equations in system (4.4.12) may not be compatible with each other; however, one can find suitable constraints for these coefficients "(z, t), g(z, t), V(z, t), 𝛾(z, t), !j (t), ,(z) and 3(z, t), such that system (4.4.12) is compatible. This requirement leads us to the following procedure: – First of all, we solve Eqs (4.4.12a)–(4.4.12c) to obtain the similarity variables 4(z, t), '(z) and the phase >(z, t) that subject to the GVD parameter "(z, t) and the differential gain or loss term ,(z). – Second, it follows from Eqs (4.4.12d)–(4.4.12i) that we can determine the amplitudes 1(z), the external potential V(z, t), SPM g(z, t), GVD "(z, t), TOD !1 (z) 3(z, t) SS !2 (z) and SFS !3 (z) in terms of the obtained variables '(z), 4(z, t) and >(z, t). – Finally, we may establish a “bridge” between Eqs (4.4.3) and (4.4.9) (also called a Lie–Bäcklund transformation).

4.4.2 Determining Similarity Variables and Controlled Coefficients For the considered Eq. (4.4.3) in the presence of TOD, i.e., !1 (z) ≢ 0, it follows from Eq. (4.4.12a) that the new temporal variable 4(z, t) should be of the form 4(z, t) = 41 (z)t + 40 (z),

(4.4.13)

where 41 (z) and 40 (z) are functions of z. The substitution of Eq. (4.4.13) into Eq. (4.4.12b) yields (3!1 >tt – ,)4t = 0,

(4.4.14)

4t = 0;

(4.4.15)

3!1 >tt – , = 0.

(4.4.16)

which leads to two cases, i.e.,

otherwise

From Eqs (4.4.15) and (4.4.13), it follows that 4 = 40 (z). Substituting them into equation (4.4.12d), it deduces that 4 should be a constant; this is inconsistent with our

4.4 Generalized Inhomogeneous Higher-Order Nonlinear Schrödinger

157

initial hypothesis. Therefore, the rest conditions (4.4.16) should be satisfied. From this condition we can obtain that – (I) , ≡ 0, > = >1 (z)t + >0 (z); – (II) , ≢ 0, > = 6!,(z) t2 + >1 (z)t + >0 (z). 1

Notation. Actually, case (I) can be derived from case (II). To illustrate the obtained results conveniently, we divide them into two cases. When Eq. (4.4.3) involves or does not involve the gain–loss term ,(z), system (4.4.12) possesses the following solutions: Case I. In the absence of the differential gain or loss term ,(z), i.e., ,(z) ≡ 0. 4 = 41 (z)t + 40 (z), '=

> = >1 (z)t + >0 (z),

√2 ∫ ! (s)431 (s)ds, 1 = 10 exp [∫ 𝛾(s)ds] , 4- 0 1 0 z

(4.4.17a)

z

" = !1 (z) [3>1 (z) –

41 (z) ], 2√2-

(4.4.17b) (4.4.17c)

g=

!1 (z)421 (z) G4 (z) [3>1 (z) + 1 ] , 2 1 (z) 2√2-

(4.4.17d)

3=

41̇ (z)t + 40̇ (z) 4 (z) + !1 (z)>1 (z) [ 1 – 3>1 (z)] , √241 (z)

(4.4.17e)

V=[

> (z)4̇ 0 (z) >1 (z)4̇ 1 (z) – >̇ 1 (z)] t + 1 41 (z) 41 (z) +!1 (z)>21 (z) [

!3 (z) = –!2 (z) = –

41 (z) – >1 (z)] – >̇ 0 (z), 2√2-

3!1 (z)421 (z) , 12 (z)

(4.4.17f) (4.4.17g)

where >j (z) (j = 0, 1), 4j (z) (j = 0, 1), 𝛾(z) and !1 (z) are free differentiable functions of space, and 10 is a constant. It is easy to see that in the absence of the differential gain or loss term ,(z), the GVD parameter "(z, t), SPM parameter g(z, t) and gain or loss term 𝛾(z, t) are only functions of space z. It follows from Eqs (4.4.17b)–(4.4.17d) that the gain or loss terms 𝛾(z) can be used to manipulate the amplitude 1(z), SS parameter !2 (z) and SPM parameter g(z). The TOD parameter !1 (z) is used to control the variable '(z), the GVD parameter "(z), SFS parameter !2 (z), SPM parameter g(z), 3(z) and potential V(z, t). Notice that the coefficients of the first-degree term in 4(z, t) and the phase >(z, t) differ from the ones in which they must be constants [219], since we consider two more terms in Eq. (4.4.3), i.e., the group velocity term 3(z, t) and the external potential V(z, t). The varying parameters 41 (z) and >1 (z) will excite complicated structures which may be useful to control the propagation of optical ultra-short pulses.

158

4 The Rogue Wave Solution and Parameters Managing

Case II.

In the presence of the differential gain or loss term ,(z), i.e., ,(z) ≢ 0. 4 = 41 (z)t + 40 (z), ' = >=

√2 z ∫ ! (s)431 (s)ds, 4- 0 1

,(z) 2 t + >1 (z)t + >0 (z), 6!1 (z)

𝛾=–

,2 (z) t + 𝛾0 (z), 3!1 (z)

41 (s) ) + 𝛾0 (s)] ds} , 6√24 (z) " = ,(z)t + !1 (z) [3>1 (z) – 1 ] , 2√2421 (z) G!1 (z)41 (z) g = 2 [,(z)t + 3!1 (z)>1 (z) + ], 1 (z) 2√2,2 (z) 2 t + 31 (z)t + 30 (z), 3=– 3!1 (z)

(4.4.18a) (4.4.18b) (4.4.18c)

z

1 = 10 exp {∫ [,(s) (>1 (s) –

(4.4.18d)

0

V=–

,3 (z) 3 t + v2 (z)t2 + v1 (z)t + v0 (z), 27!21 (z)

!3 (z) = –!2 (z) = –

3!1 (z)421 (z) , 12 (z)

(4.4.18e) (4.4.18f) (4.4.18g) (4.4.18h) (4.4.18i)

where >j (z) (j = 0, 1), 4j (z) (j = 0, 1), 𝛾0 (z) and !1 (z) are arbitrary differentiable functions of space z, 10 is a constant. Parameters 31 (z), 30 (z), v2 (z), v1 (z) and v0 (z) in Eqs (4.4.18g) and (4.4.18h) are given by 4 ̇ (z) ,(z) √ [ 241 (z) – 12->1 (z)] + 1 , 641 (z) 4 ̇ (z) !1 (z)>1 (z) 30 (z) = [41 (z) – 3√2->1 (z)] + 0 , √241 (z) 31 (z) =

v2 (z) =

,(z)41̇ (z) ,2 (z) 41 (z) ,(z) [ ] + , – >1 (z)] – [ 3!1 (z) 6√26!1 (z) z 3!1 (z)41 (z)

v1 (z) = ,(z)>1 (z) [

(4.4.19a) (4.4.19b)

(4.4.20)

3! (z)41̇ (z)>1 (z) + ,(z)40̇ (z) 41 (z) – >1 (z)] – >1̇ (z) + 1 , (4.4.21) √ 3!1 (z)41 (z) 3 2-

v0 (z) = !1 (z)>21 (z) [

4 ̇ (z)>1 (z) 41 (z) – >1 (z)] – >0̇ (z) + 0 . 41 (z) 2√2-

(4.4.22)

It is easy to see that in the presence of the differential gain or loss term ,(z) ≢ 0, the GVD parameter "(z, t), SPM parameter g(z, t) and gain or loss term 𝛾(z, t) are all functions of both z and t, which differ from the usually considered higher-order NLS

4.4 Generalized Inhomogeneous Higher-Order Nonlinear Schrödinger

159

equation with varying coefficients. The differential gain or loss term ,(z) can be used to modulate the phase >(z, t), gain or loss term 𝛾(z, t), the amplitude 1(z), the GVD parameter "(z, t), the SS parameter !2 (z), SPM parameter g(z, t), 3(z, t) and potential V(z, t). The phase >(z, t) is a second-degree polynomial in t with coefficients being functions of z, which is similar to that in the solutions of the NLS equation with varying coefficients, but the external potential V(z, t) is a third-degree polynomial in t, with coefficients being functions of z, which differs from that in the solutions of the NLS equation with varying coefficients. These have never been given in the previous research for the variable coefficient NLS-type equations [189, 199]. Notice that the solutions in Case I can be found directly from the solutions in Case II with ,(z) = 0, but we here list Case I in order to clearly point out the variation of GVD "(z), SPM g(z), 𝛾(z), 3(z, t) and V(z, t). Thus, these chosen differentiable functions in the amplitude 1(z), GVD "(z, t), SPM g(z, t), V(z, t), 𝛾(z, t), TOD !1 (z), SS !2 (z), SFS !3 (z), >(z, t) and new variables ('(z), 4(z, t)) can excite a wide range of nonlinear wave structures of Eq. (4.4.3), such as periodic wave equations, multi-soliton solutions and even rogue wave solutions.

4.4.3 Darboux Transformation for the Hirota Equation To study the self-similar rogue wave solution of Eq. (4.4.3), we need to know the rogue wave solution of Hirota equation (4.4.9) with G = –1, which can be obtained through Darboux transformation [226] Jn+1 = Jn –

4iqn+1 p∗n+1 (n = 0, 1, 2, ...) |pn+1 |2 + |qn+1 |2

(4.4.23)

with the proper initial (“seed”) solution J0 = ei' , where p∗n+1 denotes the complex conjugation of pn+1 . The characteristic functions (pn , qn ) and the solution Jn of Eq. (4.4.9) satisfy the simplified Lax Pair in which the characteristic is chosen as (+ = i) [226] [

– 1 pn+1 ] = [ i √2 qn+1 ' Jn √ 2

[

i J∗ p √2 n ] [ n+1 ] , 1 qn+1 √2

M M12 p pn+1 ] = [ 11 ] [ n+1 ] , qn+1 4 M21 –M11 qn+1

where Mij are given by M11 = √2-(J∗ J4 – JJ∗4 ) – (2 + i/2)|J|2 – 4- – i, M12 = 2i-J∗44 –

J∗4 (4i- – 1) + J∗ (2i-|J|2 + 4i- – 1), √2

M21 = 2i-J44 +

J4 (4i- – 1) + J(2i-|J|2 + 4i- – 1), √2

where J∗ denotes the complex conjugate of the electric field J.

(4.4.24a) (4.4.24b)

160

4 The Rogue Wave Solution and Parameters Managing

4.4.4 Optical Rogue Wave Solutions 4.4.4.1 The First-Order Optical Rogue Wave Solutions Here we consider the lowest-order rational solutions of Eq. (4.4.9) [224], which serve as prototypes of rogue waves. As a result, we obtain the first-order self-similar (nonstationary) rogue wave solutions of Eq. (4.4.3) in the form 81 (z, t) = 1(z) {1 –

4 + 8i'(z) } × exp[i'(z) + i>(z, t)] 1 + [√24(z, t) + 12-'(z)]2 + 4'2 (z) (4.4.25)

on the basis of the obtained similarity reduction transformation (4.4.8), where the new variables 4(z, t) and '(z), the amplitude 1(z), and the phase >(z, t) are given by systems (4.4.17) and (4.4.18). For Case I, 4j (z), >j (z) (j = 0, 1), !1 (z), 𝛾(z) are used to modulate the nonstationary rogue wave solution (4.4.25) and coefficients of Eq. (4.4.3); whereas for Case II, 4j (z), >j (z) (j = 0, 1, 2), !1 (z), ,(z), 𝛾0 (z) are used to control the nonstationary rogue wave solutions (4.4.25) and coefficients of Eq. (4.4.3). To understand the obtained self-similar optical rogue wave solution (4.4.25) and the coefficient of Eq. (4.4.3) better, we must make sure that TOD !1 (z), SS !2 (z), SFS !3 (z), GVD "(z) for Case I (or the coefficient of GVD "(z, t) in time for Case II), SPM g(z) for Case I (or the coefficient of SPM g(z, t) in time for Case II), 𝛾(z) for Case I (or the coefficient of 𝛾(z, t) in time for Case II), the coefficients of 3(z, t) in time, and the coefficients of the external potential V(z, t) in time in system (3) are bounded for realistic cases. For illustrative purposes, we choose these free parameters in Case I in the form 10 = 1.0, - = 0.6, 41 (z) = dn(z, k1 ), 40 (z) = cn(z, k2 ), !1 (z) = c1 dn(z, k3 ), 𝛾(z) = c2 sn(z, k4 )dn(z, k4 ), >1 (z) = sn(z, k5 ), >0 (z) = cn(z, k6 ),

(4.4.26)

where sn(z, k), cn(z, k) and dn(z, k) stand for the respective Jacobi elliptic functions, kj ∈ (0, 1) (j = 1, 2, ..., 6) are their moduli, and cj (j = 1, 2) are real-valued constants. For the fixed parameters (4.4.26), Fig. 4.15 depicts the profiles of GVD "(z), SS !2 (z), SPM g(z), and 3(z, t), V(z, t), >(z, t). Figure 4.16 exhibits the evolution of intensity distributions |81 (z, t)|2 of the rogue wave fields given by Eq. (4.4.25). We can see that the solutions are localized in time and keep localization infinitely in space, and are generated from the varying coefficients and differ from the usual rogue wave solutions [175, 191, 200, 225, 226]). The solutions may be useful for experimentalists, who can modulate these coefficients to generate different rogue wave phenomena in nonlinear optics.

161

4.4 Generalized Inhomogeneous Higher-Order Nonlinear Schrödinger

(a)

(b)

0.4

β(z)

(c)

1.0

0.2

0.8

α2(z)

0.8

0

g(z)

0.4

0.6

–0.2

0

0.4

–0.4

0.2

–0.6 –20

–10 z 0

10

20

(d) σ 8

–0.4

–20

–10

z 0

10

20

–20

(e)

(f)

V

φ

10

10

0

0

0

–8 15

–10 15 t

0

0 –15 –15

–15

15 t

z

0

0

–10

z0

10

–10 15

z

–15 15

t

20

15

0

0 –15 –15

z

Figure 4.15: (a) GVD "(z) (4.4.17c); (b) SS !2 (z) (4.4.17g); (c) SPM g(z) (4.4.17d); (d) 3(z, t) (4.4.17e); (e) The potential V(z, t) (4.4.17f); (f) The phase >(z, t) (4.4.17a). The parameters are given by Eq. (4.4.26) with G = –1, k1 = k3 = 0.9, k2 = 0.6, k4 = 0.8, k5 = 0.5, k6 = 0.7, c1 = 0.2, c2 = 0.3.

20

20 z0 –20 –20

0t

Figure 4.16: Profile of the wave intensity distribution defined by the solution (4.4.25) with max{z,t} |81 |2 ≃ 16.2, the parameters are given by Eq. (4.4.26) with k1 = k3 = 0.9, k2 = 0.6, k4 = 0.8, k5 = 0.5, k6 = 0.7, c1 = 0.2, c2 = 0.3.

On the other hand, if we choose the free parameters in another form 41 (z) = 1 + 0.1 sin(z), 40 (z) = cos(z), !1 (z) = 0.2 + 0.1 sin(z), - = 0.1,

(4.4.27)

and 10 , 𝛾(z) and >j (z) (j = 0, 1) are the same as those given by Eq. (4.4.26), then the evolution of the intensity distribution of the rogue wave solutions (4.4.25) will be changed. Figure 4.17 displays the profile of the rogue wave solution (4.4.25). The

162

4 The Rogue Wave Solution and Parameters Managing

15

15 z

0

Figure 4.17: Profile of the wave intensity distribution defined by the solution (4.4.25) with max{z,t} |81 |2 ≃ 6.4, the parameters are given by Eq. (4.4.27) with k4 = 0.5, c2 = 0.1.

0t –15 –15

solutions are localized both in time and almost in space, thus almost revealing the usual rogue wave features [226]. For Case II, we choose different gain or loss terms ,(z) = ,0 cn(z, k7 )dn(z, k7 ), 𝛾0 (z) = c3 sn(z, k8 )dn(z, k8 ),

(4.4.28)

where cj (j = 3, 4) are real-valued constants. Figure 4.18 depicts the profiles of ,(z), GVD "(z, t), SS !2 (z), SPM g(z, t), 3(z, t), V(z, t), >(z, t) given by system (4.4.18). Figure 4.19 exhibits The evolution of intensity distributions of the rogue wave fields given by Eq. (4.4.25), where the parameters are determined by Eqs (4.4.26) and (4.4.28).

(b)

(c)

0.8

0.2

2

0.6

0

–6 –15

0.2 –0.4 –5

–15

z

5

(e)

t

0 –15

15

–5

z

Ψ

0

80 40 0 –40 –80 15 –15

2

–30 –60 15 t 0

0 15 –15

z

–90 –15 t

0

0 15 –15

z

15 t 0

z

(g)

–2 –6 –15

0 15 –15

σ

–10 –15

15

0

15

5

(f) γ 6

0

–2

α2

0.4

–0.2

g 10

6

0 15 –15

z

(h)

15 t

0

0 15 –15

V 400 200 0 –200 –400 –15

15 0

z

μ

(d)

β

1.0

0.4

t

0 15 –15

z

(a)

Figure 4.18: (a) Profiles of the differential gain or loss term ,(z) given by equation Eq. (4.4.28); (b) SS !2 (z) (4.4.18i); (c) GVD "(z, t) (4.4.18e); (d) SFS g(z, t) (4.4.18f); (e) The gain or loss term 𝛾(z, t) given by (4.4.18c); (f) 3(z, t) (4.4.18g); (g) The phase >(z, t) given by (4.4.18b); (h) The potential V(z, t) given by (4.4.18h). The parameters are given by Eq. (4.4.28) with G = –1, k1 = k3 = 0.9, k2 = 0.6, k4 = 0.8, k5 = 0.9, k6 = 0.7, c1 = 0.2, c2 = 0.3, ,0 = 0.5.

163

4.4 Generalized Inhomogeneous Higher-Order Nonlinear Schrödinger

25

15 z

0

0 –25 –15

t

Figure 4.19: Profile of the wave intensity distribution of the first-order optical rogue wave solution (4.4.25) with max{z,t} |81 |2 ≃ 30.7. The parameters are determined by Eqs (4.4.26) and (4.4.28) with k1 = k3 = 0.9, k2 = 0.6, k4 = 0.8, k5 = 0.9, k6 = 0.7, c1 = 0.2, c2 = 0.3, ,0 = 0.5.

4.4.4.2 Second-Order Optical Rogue Wave Solutions Based on the transformation (4.4.8) and the second-order rogue wave solution of higher-order Hirota equation (4.4.9) [226], we can obtain the second-order self-similar rogue wave solutions of Eq. (4.4.3) in the form [224] 82 (z, t) = 1(z) [1 –

P(', 4) + i'Q(', 4) i['+>(z,t)] , ]e H(', 4)

(4.4.29)

where the functions P(', 4), Q(', 4) and H(', 4) are P(', 4) = 4844 + 1152√2-'43 + 14442 [4'2 (36-2 + 1) + 1] +576√2-'4[12'2 (12-2 + 1) + 7] – 36 +192'4 [216-2 (6-2 + 1) + 5] + 864'2 (44-2 + 1), Q(', 4) = 9644 + 2304√2-'43 + 9642 [4'2 (108-2 + 1) – 3] +1152√2-'4[4'2 (36-2 + 1) + 1] – 360 +384'4 (36-2 + 1)2 + 192'2 (108-2 + 1), H(', 4) = 846 + 288√2-'45 + 1244 [4'2 (180-2 + 1) + 1] +96√2-'43 [12'2 (60-2 + 1) – 1] +642 {16'4 [216-2 (30-2 + 1) + 1] – 24'2 (60-2 + 1) + 9} +72√2-'4[16'4 (36-2 + 1)2 – 8'2 (108-2 – 1) + 17] +64'6 (36-2 + 1)3 – 432'4 (52-2 + 1)(12-2 – 1) +36'2 (556-2 + 11) + 9,

(4.4.30)

(4.4.31)

(4.4.32)

in which the new variables 4(z, t), '(z), the amplitude 1(z) and the phase >(z, t) are given by system (4.4.17) or (4.4.18), and - is a real-valued constant. These parameters !1 , !2 , ,, 41 , 40 , 𝛾, 61 , 60 and - can be used to control the wave propagation of second-order self-similar rogue wave solution (4.4.29) and the coefficients of Eq. (4.4.3). Similar to the above cases, we choose three kinds of parameters given by Eqs (4.4.26)–(4.4.28) to study the intensity distribution of wave propagation of the

164

4

4 The Rogue Wave Solution and Parameters Managing

|Ψ2|

2 0 80 40

6 0 z –40 –80 –6

4

0

t

Figure 4.20: Profile of the wave intensity distribution of the second-order optical rogue wave solution (4.4.29) with max{z,t} |82 | ≃ 4.9. The parameters are determined by equation (4.4.26) with k1 = k3 = 0.9, k2 = 0.6, k4 = 0.8, k5 = 0.5, k6 = 0.7, c1 = 0.2, c2 = 0.03, - = 6.0.

|Ψ2|

2 0 8 z

0

0

6

Figure 4.21: Profile of the wave intensity distribution of the second-order optical rogue wave solution (4.4.29) with max{z,t} |82 | ≃ 4.7. The parameters are determined by Eq. (4.4.27) with k4 = 0.5 and c2 = 0.02, - = 5.0.

10

Figure 4.22: The parameters are determined by Eq. (4.4.29) with max{z,t} |82 | ≃ 5.3. The parameters are determined by Eq. (4.4.27) with k1 = k3 = 0.9, k2 = 0.6, k4 = 0.8, k5 = 0.9, k6 = 0.7, c1 = 0.2, c2 = 0.05, ,0 = 0.01, - = 2.0.

t

–8 –6

5 |Ψ2| 3 1 0 40 z

0

0 –40

–10

t

second-order self-similar rogue wave solution (4.4.29) (see Figs 4.20-4.22). Figure 4.20 shows that the second-order rogue wave solution (4.4.29) is localized in time and not localized in space, since the modulated parameters are periodic functions of space. In the other case, Fig. 4.21 illustrates that the rogue wave solution (4.4.29) is localized both in space z and in time; however the complicated structure differs from the usual rogue wave features, in particular one near the origin. For the presence of the differential gain or loss term ,(z) ≠ 0, Fig. 4.22 illustrates that the self-similar rogue wave solution is localized both in space and in time.

4.5 Two-Dimensional Binary Mixtures of Bose–Einstein—Condensates

165

4.5 Two-Dimensional Binary Mixtures of Bose–Einstein—Condensates 4.5.1 Two-Component Gross–Pitaevskii Equations For the binary mixtures of BECs composed of two hyperfine states, say of the |F = 1, mf = –1⟩ and |F = 2, mf = 1⟩ of 87 Rb confined at different vertical positions by timedependent parabolic traps and modulated by the time-dependent electromagnetic coupling and time-dependent gain or loss distributions, in the mean-field approximation, we use the two-component Lagrangian density for the described macroscopic wave functions J1 and J2 [196–198, 227, 228]

L = iℏ (J1

2 𝜕J∗1 𝜕J∗ G (t) ℏ2 + J2 2 ) + ∑ |∇Jj |2 + 11 |J1 |4 𝜕t 𝜕t 2mj 2 j=1

+G12 |J1 |2 |J2 |2 + +

2 G22 (t) |J2 |4 + ∑[Vext,j (r, t) + iℏAj (t)]|Jj |2 2 j=1

(4.5.1)

ℏK(t) 2 ∗ ∑ J J – ℏB(t)|J2 |2 , 2 j=1 j 3–j

where Jj ≡ Jj (r, t) (j = 1, 2) represent two species in the mixture, J∗j stands for the complex conjugation of Jj , r ∈ ℝ2 , ∇2 = 𝜕x2 + 𝜕y2 is the 2D divergence operator with 𝜕x = 𝜕/𝜕x, mj denotes the mass of the atom of the j-th condensate, the external potentials Vext,j (r, t) are both the real-valued functions of time and spatial coordinates, and the nonlinear interaction parameters Gij (t) = 20ℏ2 aij (t)/mij denote effective atom–atom interactions with mij = mi mj /(mi + mj ) being the reduced mass, aij (t) = aji (t) being the s-wave scattering lengths for binary collisions between atoms in internal states |i⟩ and |j⟩ (aij (t) > 0 (< 0) corresponds to repulsive (attractive) atom–atom interaction, which can be controlled by varying magnetic fields [196]). Gjj (t) and G12 (t) stand for the intraspecies and interspecies interactions, respectively. The gain or loss distributions Aj (t) are real-valued functions of time [229]. The time-dependent term K(t) denotes the linear electromagnetic coupling, and B(t) stands for the time-dependent chemicalpotential difference [196, 197, 230]. The total number of atoms in the binary mixture of BECs is ∫(|J1 |2 + |J2 |2 )d2 r = N. On the basis of the Lagrangian density (4.5.1) and Euler–Lagrange equation (alias the variational principle) $L /$J∗j = 0 (j = 1, 2), it follows that 2D inhomogeneous two-component GP equations with time-dependent parameters are of the form [196, 227, 228]

166

4 The Rogue Wave Solution and Parameters Managing

iℏ

𝜕J1 ℏ2 ∇2 + Vext,1 (r, t) + iℏA1 (t) + G11 (t)|J1 |2 =[– 𝜕t 2m1 +G12 (t)|J2 |2 ]J1 +

iℏ

ℏK(t) J2 , 2

(4.5.2a)

𝜕J2 ℏ2 ∇2 + Vext,2 (r, t) + iℏA2 (t) + G12 (t)|J1 |2 =[– 𝜕t 2m2 +G22 (t)|J2 |2 ]J2 +

ℏK(t) J1 – ℏB(t)J2 . 2

(4.5.2b)

Under the scaling transformation r → √ℏ/(m9⊥ )r, t → 9–1 ⊥ t and m1 = m2 = m, system (4.5.2) reduces as the coupled GP equation [231] i

𝜕81 1 = [ – ∇2 + V1 (r, t) + i𝛾1 (t) + g11 (t)|81 |2 𝜕t 2 +g12 (t)|82 |2 ]81 + 9(t)82 ,

i

(4.5.3a)

𝜕82 1 = [ – ∇2 + V2 (r, t) + i𝛾2 (t) + g12 (t)|81 |2 𝜕t 2 +g22 (t)|82 |2 ]82 + 9(t)81 ,

(4.5.3b)

where the wave functions 8j (r, t), the external potentials Vj (r, t), the nonlinear interactions gij (t), the gain or loss distributions 𝛾j (t) and the couplings 9(t) are related to the vector fieldsJj (r, t), the external potential Vext,j (r, t), the nonlinear interactions Gij (t), the gain or loss distributions Aj (t) and the coupling fields K(t) in system (4.5.2). Notice that during the above-mentioned procedure, the two terms Vext,2 (r, t)J2 and –ℏB(t)J2 (i.e., [Vext,2 (r, t) – ℏB(t)]J2 ) are related to V2 (r, t)82 in system (4.5.3). System (4.5.3) contains three types of interactions, i.e., attractive case (gij (t) < 0), repulsive case (gij (t) > 0) and other case (i.e., the mixture case). 4.5.2 Symmetry Reduction Analysis In general, it is difficult to solve system (4.5.3) because of its nonlinearities and varying coefficients. To analytically find some physically interesting solutions of system (4.5.3), we need to seek for some integrability conditions for these coefficients in system (4.5.3) from the symmetry point of view. System (4.5.3) possesses many types of similarity reductions based on the Lie group transformation method. Here, to seek for its multi-soliton solutions including vector rogue wave solutions, we need to reduce system (4.5.3) to one system of nonlinear partial differential equations with constant coefficients in (1 + 1) dimensional, which is easy to solve. Thus, we use the general similarity transformations [231]

4.5 Two-Dimensional Binary Mixtures of Bose–Einstein—Condensates

̂

̂ t), 4(t)] ̂ (j = 1, 2) 8j (r, t) = 1̂ j (t)ei>j (r,t) 6j ['(r,

167

(4.5.4)

to reduce the system (4.5.3) into the following coupled NLS equation: i𝜕4̂ 61 = –𝜕'2̂ 61 + (G11 |61 |2 + G12 |62 |2 )61 + -1 62 ,

(4.5.5a)

i𝜕4̂ 62 = –𝜕'2̂ 62 + (G12 |61 |2 + G22 |62 |2 )62 + -2 61 ,

(4.5.5b)

̂ t), and 4(t), ̂ Gji are all realwhere 6j (',̂ 4)̂ are the functions of two new variables '(r, valued constants, and -j are the constant couplings. System (2.3.1) corresponds to the focusing (Gji < 0), defocusing (Gji > 1) and mixed (other cases) coupled NLS equations [233]. Based on the above-mentioned method, we have the following system of partial differential equations: ∇2 '̂ = 0, 24̂ t – |∇'|̂ 2 = 0,

(4.5.6a)

'̂ t + ∇>̂ j ⋅ ∇'̂ = 0 (j = 1, 2),

(4.5.6b)

21̂ j,t + 1̂ j [∇2 >̂ j – 2𝛾j (t)] = 0 (j = 1, 2),

(4.5.6c)

2

29(t)1̂ 3–j = -j 1̂ j |∇'|̂ , (j = 1, 2),

(4.5.6d)

2gji (t)1̂ 2j

(4.5.6e)

2

– Gji |∇'|̂ = 0 (i, j = 1, 2),

2Vj (r, t) + |∇>̂ j |2 + 2>̂ j,t = 0 (j = 1, 2).

(4.5.6f)

It follows from system (4.5.6) that we find that these linear relations for the coefficients: 𝛾1 (t) = 𝛾2 (t), 1̂ 1 (t) = c0 1̂ 2 (t) (c0 = const), and g11 (t) : g22 (t) : g12 (t) = G11 : G22 : G12 . We do not consider this type of the reduction here. In the following, we consider the similarity transformations for the two-species condensates 8j (r, t) in the form [189, 199, 231] 8j (r, t) = 1j (t)ei>j (r,t) I['(r, t), 4(t)] (j = 1, 2)

(4.5.7)

with the amplitudes 1j (t) and phases >j (r, t) being real-valued functions of the indicated variables. We require I['(r, t), 4(t)] to satisfy(1+1)-D classical NLS equation i𝜕4 I(', 4) = –𝜕'2 I(', 4) + G |I(', 4)|2 I(', 4),

(4.5.8)

which possesses lots of kinds of solutions. In order to control boundary conditions as infinity, we impose the natural constraints for the new variable '(r, t) (or after the transformation '(r, t) → '(r, t) – '0 (t)) ' → 0 (r → 0)

'→∞

(r → ∞).

(4.5.9)

168

4 The Rogue Wave Solution and Parameters Managing

Substitute transformation (4.5.7) into system (4.5.3) and demand I['(r, t), 4(t)] to satisfy the (1+1)-D classical NLS Eq. (4.5.8), and after applying relatively simple algebra we obtain the system of partial differential equations ∇2 ' = 0, 24t – |∇'|2 = 0, 't + ∇>j ⋅ ∇' = 0, 2

(4.5.10a)

21j,t (t) + 1j (t)[∇ >j – 2𝛾j (t)] = 0,

(4.5.10b)

2[gjj (t)12j (t)

(4.5.10c)

2Vj (r, t) +

+ g12 (t)123–j (t)] – G |∇'|2 = 0, |∇>j |2 + 2>j,t + 29(t)13–j (t)1–1 j (t)

= 0,

(4.5.10d)

where j = 1, 2. Notice that system (4.5.10) consists of 10 differential equations with 14 unknown variables '(r, t), 4(t), >j (r, t), 1j (t), and gij (t), Vj (r, t), 𝛾j (t) and 9(t) which may generate some types of solutions including some free functions. In general, system (4.5.10) may not be compatible with each other; however, one can find some proper constraints for the external potentials Vj (r, t), the intraspecies and interspecies interactions gij (t), and the gain or loss 𝛾j (t) such that system (4.5.10) can be solvable. First of all, we solve Eq. (4.5.10a) subject to the boundary condition (4.5.9) such that we obtain similarity variables '(r, t), 4(t) and the different phases >j (r, t) corresponding to the different physical fields 8j (r, t). Second, it follows from Eqs (4.5.10b)–(4.5.10d) that we can determine the amplitudes 1j (t), the external potentials Vj (r, t) , and nonlinear interactions gij (t) in terms of the above-obtained variables '(r, t), 4(t) and >j (r, t). Finally, we can establish a powerful “bridge” between the (2+1)-D two-component inhomogeneous GP Eq. (4.5.3) with variable coefficients and classical NlSE (4.5.8).

4.5.3 Determining Similarity Variables and Controlled Coefficients By expression (4.5.10a), we have the similarity variables '(r, t), 4(t) and phases >j (r, t) in the form t

'(r, t) = c(t) ⋅ r – ∫ c(s) ⋅ aj (s)ds,

(4.5.11a)

>j (r, t) = r D(t) r + aj (t) ⋅ r + >j0 (t),

(4.5.11b)

0

4(t) =

t

1 ∫ |c(s)|2 ds, 2 0

(4.5.11c)

where D(t) = diag(Dx (t), Dy (t)) and D3 (t) = –ċ3 (t)/[2c3 (t)] (3 = x, y, an over-dot denotes the derivative with respect to time). c(t) = (cx (t), cy (t)), aj (t) = (ajx (t), ajy (t)) and >j0 (t) are all time-dependent real-valued functions. Moreover, the frequencies c(t) and the frequency shifts aj (t) should satisfy the compatibility condition c(t) ⋅ [a1 (t) – a2 (t)] = 0.

(4.5.12)

4.5 Two-Dimensional Binary Mixtures of Bose–Einstein—Condensates

169

Actually Eq. (4.5.12) possesses lots of kinds of solutions (e.g., we can choose a1 (t) = a2 (t) and any c(t)). The two variables cx (t) and cy (t) denote the inverse of the widths of the localized matter-waves in x- and y-directions, respectively. t –1 [,(t)c–1 x (t), ,(t)cy (t)] with ,(t) = ∫0 c(s) ⋅ aj (s)ds denotes the center of mass. It follows from Eq. (4.5.11b) that the phases are consistent with that in the Gaussian variational analysis in time-dependent BECs [232]. On the basis of Eqs (4.5.11) and (4.5.10b)–(4.5.10d), we can determine the amplitudes 1j (t), external potentials Vj (r, t) and nonlinear interaction gjj (t) 1j (t) = 1j0 $j (t)√|cx (t) cy (t)|, g11 (t) = g22 (t) =

G |c(t)|

2

21210 $21 (t) |cx (t) cy (t)| 2

G |c(t)| 21220 $22 (t) |cx (t) cy (t)|

Vj (r, t) = r A(t) r + bj (t) ⋅ r –

(4.5.13a) 12 $22 (t) – g12 (t) 20 , 1210 $21 (t) 12 $2 (t) – g12 (t) 210 12 , 120 $2 (t) 1(3–j)0 $3–j (t) 1 |a (t)|2 – >̇ 0j (t) – 9(t) , 2 j 1j0 $j (t)

(4.5.13b) (4.5.13c) (4.5.13d)

where j = 1, 2, 1j0 are nonzero integration constants, and t

$j (t) = exp [∫ 𝛾j (s)ds] .

(4.5.14)

0

We have introduced the time-dependent 2 × 2 diagonal matrix A(t) = diag(Ax (t), Ay (t)) with the entries A3 (t) =

ċ2 (t) c̈3 (t) – 32 , 2c3 (t) c3 (t)

(4.5.15)

and the vector functions bj (t) = (bjx (t), bjy (t)) with bj3 (t) =

aj3 (t)ċ3 (t) c3 (t)

– ȧ j3 (t) (j = 1, 2).

(4.5.16)

It follows from Eqs (4.5.13a)–(4.5.13b) that the gain or loss terms 𝛾j (t) and frequencies c3 (t) can be used to manipulate the amplitudes 1j (t) of the vector matter-wave solutions, the intracomponent nonlinearities (g11 (t), g22 (t)) are related to the intercomponent one g12 (t). Equation (4.5.13d) shows that the external potentials Vj (r, t) just satisfy the condition that the potentials in two-component BECs are chosen as the time-dependent harmonic trapping potentials [196, 227]. Moreover, we find that the linear coupling 9(t) is only related to the potentials and that the frequencies c3 (t) are used to control the coefficients of second-degree terms about spaces in Vj (r, t). It is ready to see that the superfluid velocities (alias the velocity fields) vj (r, t) related to the above-obtained self-similar solutions (4.5.7) have the form

170

4 The Rogue Wave Solution and Parameters Managing

vj (r, t) = ∇>j (r, t) = –[x (ln |cx (t)|)t , y(ln |cy (t)|)t ] + aj (t),

(4.5.17)

where j = 1, 2. The velocity fields vj (r, t) given by Eq. (4.5.17) may have one of the four types of velocity fields: [x, y], [x, –y], [–x, y], [–x, –y] (under the scaling translation transformations (x → -1 x – -2 , y → -3 y – -4 ) with -j ∈ ℝ). Figure 4.23(a)–(c) displays the profiles of the velocity field v1 (r, t) given by Eq. (4.5.17) for the chosen parameters (4.5.20) at the different time t = 1, 25, 30. Therefore, it follows from Eq. (4.5.17) that the time-dependent divergences (alias the flux density, which is the amount of flux entering or leaving a point) of two velocity fields vj (r, t) are of the same form div vj (r, t) = ∇ ⋅ vj (r, t) = ∇2 >j (r, t) = –[ ln |cx (t) cy (t)|]t ,

(4.5.18)

which measures flux per unit area and is the parameter of only time. The timedependent divergences are modulated by frequencies c3 (t). It follows from Eq. (4.5.18) that we have the following proposition: (i) The flux density is increasing (called a source (fountain) meaning that the fluid flows outward) as [ ln |cx (t) cy (t)|]t < 0, positive divergence or flux density); (ii) It is decreasing (called a sink, meaning that the fluid flows inward) as [ ln |cx (t) cy (t)|]t > 0, negative divergence or flux density; (iii) It has no source or sink (called divergence free) as ln |cx (t) cy (t)| = const. (i.e., vj (r, t)), in which the velocity fields vj (r, t) given by Eq. (4.5.18) are called incompressible. Figure 4.23(d) displays the profile of the divergence div v1 (r, t) given by Eq. (4.5.18) for parameters given by Eq. (4.5.20).

4.5.4 Types of Nonlinear Interaction On the basis of the solution of Eq. (4.5.8), these chosen functions in the amplitude 1j (t), nonlinearities gij (t), external potentials Vj (r, t) and the phases >j (r, t) can generate abundant vector matter-wave solution structures of system (4.5.3), such as period wave equations, multi-soliton solutions and multi-rogue wave solutions. Here, we only consider vector rogue wave solutions of system (4.5.3) from rogue waves of Eq. (4.5.8), which implies that the nonlinear interaction in Eq. (4.5.8) should be attractive, i.e., G < 0. This condition (G < 0) and (4.5.10c) (or Eq. (4.5.13b)) generate four different cases for the intracomponent interactions gij (t) (see Table 4.1) to excite the vector rogue wave phenomena in system (4.5.3). This differs from one in the single time-dependent NLS equation [175, 189, 199–201].

4.5 Two-Dimensional Binary Mixtures of Bose–Einstein—Condensates

5

(a)

y 0

–5 –5

5

171

(b)

y 0

–5

x 0

(t=1)

5

5

x 0

(t=25)

5 (d)

0.5 Divergence

(c)

–5

y 0

0

–0.5

–5 –5

x 0

(t=30)

5

–10

t 0

10

Figure 4.23: (a)-(c) The profiles of the velocity field v1 (r, t) given by (4.5.17) with parameters given by Eq. (4.5.20) with kx = 0.1, ky = 0.9, (a): t = 1, (b): t = 25, (c) t = 30. (d) The profile of the divergence divv1 (r, t) given by Eq. (4.5.18) versus time for the parameters given by Eq. (4.5.20) with kx = 0.1, ky = 0.9.

Table 4.1 The existent conditions of vector rogue wave solutions for the intracomponent nonlinearities gij (t) in system (4.5.3). The sign +(.) stands for the repulsive (attractive) atom–atom interaction in binary BECs Case

g11 (t)

g22 (t)

g12 (t)

I II III

+ + –

+ – +

– – –

IV

–

–

g12 (t) < Min {

G |∇'|2 – 2g11 (t)121 (t)

2122 (t) 2 G |∇'| – 2g22 (t)122 (t) 2121 (t)

,

}

4.5.5 Self-Similar Vector Rogue Wave Solution 4.5.5.1 The First Order Self-Similar Vector Rogue Wave Solution Based on the above-obtained similarity transformation (4.5.7) and the first-order rational solutions of the self-focusing NLS equation [175], we obtain the first-order self-similar vector matter rogue wave solutions of system (4.5.3) in the form [231]

172

4 The Rogue Wave Solution and Parameters Managing

t

∫0 𝛾1 (s)ds √|cx (t)cy (t)| 8(1) 1 (r, t) = 110 e

× [1 –

2 4 + 8i!2 4(t) ] ei[>1 (r,t)+! 4(t)] , 2 2 4 2 1 + 2! ' (r, t) + 4! 4 (t)

(4.5.19a)

t

∫0 𝛾2 (s)ds √|cx (t)cy (t)| 8(1) 2 (r, t) = 120 e

× [1 –

1+

2 4 + 8i!2 4(t) ] ei[>2 (r,t)+! 4(t)] , + 4!4 42 (t)

2!2 '2 (r, t)

(4.5.19b)

where the nonzero parameter ! denotes the scaling, and the self-similar variables '(r, t), 4(t) and the phases >j (r, t) are given by Eqs (4.7.12a,b). The parameters (r, t)| of vector rogue 𝛾j (t), c3 (t) and aj (t) are used to modulate the amplitudes |8(1) j waves (4.5.19). Some functions of time appearing in Eq. (4.5.19) lead to the abundant structures of vector rogue wave phenomenon of system (4.5.3). In what follows, we choose these free parameters to study the evolution of rogue wave solution, to make sure that these nonlinearities gij (t) and coefficients of the external potentials Vj (r, t) in time in system (4.5.3) are bounded for realistic cases. Therefore, we choose these free parameters in the form 110 = 0.5, 120 = 0.8, g12 (t) = g0 sin(t), 9(t) = 90 sin(t), c3 (t) = aj3 (t) = dn(t, k3 ), 𝛾1 (t) = 𝛾10 sn(t, k)dn(t, k), 𝛾2 (t) = 𝛾20 (t) sin(t) cos(t),

(4.5.20)

>j0 (t) = 0, where sn, cn and dn stand for the respective Jacobi elliptic functions, k3 , k ∈ (0, 1) are their moduli, and g0 , 90 , 𝛾j0 are constants. Figure 4.24 depicts the profiles of intracomponent nonlinearities g11 (t), g22 (t), and intercomponent nonlinearity g12 (t) given by Eqs. (4.5.13b) and (4.5.20), the gain or loss terms 𝛾j (t) given by Eq. (4.5.20), and the external potential V1 (r, t) given by Eq. (4.5.13d). Notice that another potential V2 (r, t) is similar to V1 (r, t), and we only plot the profile of V1 (r, t) here. Figure 4.25 exhibits the amplitude |8(1) (r, t)| of the j first-order rogue wave solution (4.5.19). We can see that the classical rogue wave solution [175, 191, 225] can be transformed into a significantly more complicated evolution along the t-axis. On the other hand, we may choose free parameters as the following form: cx (t) = 0.8 + c1 sin2 (t), cy (t) = 1.2 + c2 sin(t) cos(t), aj3 (t) = 0,

(4.5.21)

1j0 , g12 (t), 9(t), >j0 (t) and 𝛾j (t) are determined by Eq. (4.5.20), then the evolution of amplitude distributions of the 2D vector rogue wave solutions (4.5.19) will be changed.

173

4.5 Two-Dimensional Binary Mixtures of Bose–Einstein—Condensates

(a) gij 10

(b) γj 0.4

0

0.2

–10 0

–20 –30

–0.2

–40

–0.4

–50 –15

t

0

15

–10

(c)

t

0

10

(d)

V1

V1 80

4 2 0 –2 –4 20

40 0 –40 –80 15

20 0

x0

15 0

y0

t

–20 –20

t

–15 –15

Figure 4.24: (a) Profiles of the intracomponent nonlinearities g11 (t) (dashed line) and g22 (t) (dashed-dotted line), and intercomponent nonlinearity g12 (t) (solid line) given by Eqs (4.5.13b) and (4.5.20); (b) The gain or loss terms 𝛾1 (t) (solid line) 𝛾2 (t) (dashed line); (c)-(d) The external potential V1 (r, t) given by Eq. (4.5.13d) with (c): y = 0; (d) x = 0 versus time for the parameters are given by Eq. (4.5.20) with kx = 0.1, ky = k = 0.9, g0 = 4, 90 = 1, 𝛾10 = 𝛾20 = 1.

Figure 4.26 displays the profiles of intracomponent nonlinearities g11 (t), g22 (t), and intercomponent nonlinearity g12 (t) given by Eqs (4.5.13b) and (4.5.20), and the external potential V1 (r, t) given by Eq. (4.5.13d) versus time for the parameters defined by Eq. (4.5.21). Figure 4.27 displays the evolution of amplitude of the first-order rogue wave solution (4.5.19). The solutions are localized in both time and space, thus almost revealing the usual “rogue wave” features [175]. 4.5.5.2 Second-Order Vector Matter Rogue Wave Solutions In general, fixing the transformation (4.5.7) and the coefficient (4.5.3) of equations (see (4.5.11) and (4.5.13)), together with the rogue wave solution of classical NLSE, we can obtain the second-order self-similar vector rogue wave solutions of system (4.5.3) in the form [175, 191] t

∫0 𝛾1 (s)ds √|cx (t)cy (t)| 8(2) 1 (r, t) = 110 e

× [1 +

P(', 4) – i!2 4(t)Q(', 4) i[>1 (r,t)+!2 4(t)] , ]e H(', 4)

(4.5.22a)

174

4 The Rogue Wave Solution and Parameters Managing

(a) |Ψ(1) 1 |

(b) 3 |Ψ(1) 1 |

4

t=5 2

30

2

1 0

0 10 x

0 –10

t=0

t

0

–30

(c)

–20

0 x

20

(d)

|Ψ2(1)|

2 |Ψ (1)| 2

t=5

2 30

1 0

0 10 x

0 –10

1 t=0

t 0

–30

–20

0 x

20

Figure 4.25: Profiles of the wave amplitude distributions of the first-order vector rogue wave (1) (1) (1) solutions (4.5.19). (a) |8(1) 1 (x, 0, t)|; (b) |81 (x, 0, t)| t = 0, 5; (c) |82 (x, 0, t)|; (d) |82 (x, 0, t)| t = 0, 5. The parameters are determined by Eq. (4.5.20) with kx = 0.1, ky = k = 0.9, 𝛾10 = 𝛾20 = 1.0, ! = 0.5.

(a) 20 gij 0 –20 –40

(b)

(c)

V1 4 0 –4

V1 4

15

–60 –20

0 15 0

x 0 0

–15 –15

20

t

25

–4 15 0

y0

t

–15 –25

Figure 4.26: (a) Profiles of a intracomponent nonlinearities given by Eq. (4.5.13b), g11 (t) (dashed line), g22 (t) (dashed-dotted line) and g12 (t) (solid line), (b)–(c):the external potential V1 (r, t) given by (4.5.13d) with (b) y = 0 and (c) x = 0. The parameters are determined by Eq. (4.5.21) with kx = 0.1, ky = 0.9, k = 0.9, g0 = 4, 90 = 1, 𝛾10 = 𝛾20 = 1, c1 = 0.02, c2 = 0.01. t

∫0 𝛾2 (s)ds √|cx (t)cy (t)| 8(2) 2 (r, t) = 120 e

× [1 +

P(', 4) – i!2 4(t)Q(', 4) i[>2 (r,t)+!2 4(t)] ]e , H(', 4)

(4.5.22b)

where these functions P(', 4), Q(', 4), and H(', 4) are given by the following quasipolynomials:

4.5 Two-Dimensional Binary Mixtures of Bose–Einstein—Condensates

(a) (1) |Ψ1 |

(b) 1.6 |Ψ(1)| 1

1.6

175

t=4

1.2

1.2 30

0.8 0.4 0 10

0 t x

0.8 0.4 0

0 –10

–30

(c) (1) |Ψ2 |

(d)

1.6

1.2

1.2

t=0 –20

0 x

(1) |Ψ2 |

0.8

t=4

30

0.8

0.4

0.4 0 10

0 x

0 –10

–30

t

0

20

t=0 –20

0 x

20

Figure 4.27: Profiles of the wave amplitudes of the first-order vector rogue wave solutions (4.5.19). (1) (1) (1) (a) |8(1) 1 (x, 0, t)|; (b) |81 (x, 0, t)| with t = 0, 5; (c) |82 (x, 0, t)|; |82 (x, 0, t)| with t = 0, 5. The parameters are determined by Eqs (4.5.20) and (4.5.21) with kx = 0.1, ky = k = 0.9, 𝛾10 = 𝛾20 = 1, c1 = 0.02, c2 = 0.01, ! = 0.6.

1 3 3 P(', 4) = !2 (– !2 '4 – 6!4 '2 42 – 10!6 44 – '2 – 9!2 42 ) + , 2 2 8 15 Q(', 4) = !2 (!2 '4 + 4!4 '2 42 + 4!6 44 – 3'2 + 2!2 42 ) – , 4 1 1 2 1 H(', 4) = !2 ( !4 '6 + !6 '4 42 + !8 '2 44 + !10 46 + !2 '4 12 2 3 8 9 3 9 33 3 + !6 44 – !4 '2 42 + '2 + !2 42 ) + 32 , 2 2 16 8 where the scaling ! is a nonzero parameter, and the variables '(r, t), 4(t) and the phase >j (r, t) are given by Eqs (4.7.12a,b). The parameters 1j0 , 𝛾j (t) and c3 (t) are used (r, t)| of the second-order vector rogue wave solution of to modulate the amplitude |8(2) j Eq. (4.5.22). The free parameters in the vector rogue wave solution (4.5.22) could yield abundant rogue wave solutions (4.5.3). We choose the parameters given by Eq. (4.5.20) with !3 (t) = 0 and Eq. (4.5.21), and Figs 4.28 and 4.29 illustrate the wave amplitude distributions |8(2) (r, t)| of the interacj tions of the second-order vector rogue wave solutions given by Eq. (4.5.22). It is easy to find that the vector field propagations for the interaction of two rogue waves in this

176

4 The Rogue Wave Solution and Parameters Managing

(a)

(b)

|Ψ(2) 1 | 60

(2) |Ψ1 |

t=7

0.8

5 3 1 0

1.2

0

0.4

t

10 x

0

0 –10

–60

t=0 –20

0 x

20

(d)

(c) (2) |Ψ2 |

2

|Ψ2(2)|

t=0

40 1.5 3 2

1 0

1 0 10

t

x

0.5 0

0 –10

–40

t=12 –20

0 x

20

Figure 4.28: Profiles of the wave amplitudes of the second-order vector rogue wave solutions (2) (2) (2) (4.5.22). (a) |8(2) 1 (x, 0, t)|; (b) |81 (x, 0, t)| with t = 0, 7; (c) |82 (x, 0, t)|; (d) |82 (x, 0, t)| with t = 0, 12 defined by the solutions for !3 (t) = 0 and other parameters given by Eq. (4.5.20) with kx = 0.1, ky = k = 0.9, 𝛾10 = 𝛾20 = 1, ! = 1.

case are more complicated. Figure 4.28 shows that the vector rogue wave solutions given by Eq. (4.5.22) are localized in both x, y and not localized in time because of the modulated parameters. In the other case, Fig. 4.29 displays that the vector rogue wave solutions given by Eq. (4.5.22) are localized both in spaces and time; however, these complicated structures differ from the usual rogue wave feature [175, 191, 234].

4.6 Two-Dimensional Nonlocal Nonlinear Schrödinger Equation 4.6.1 Two-Dimensional Nonlocal Nonlinear Model The model under the consideration is to focus on the two-dimensional nonlocal NLS equation [235, 237] i

∞ 𝜕2 8 𝜕8 = – 2 – 38 ∫ |8|2 dy, 𝜕t 𝜕x –∞

(4.6.1)

where 8 ≡ 8(x, y, t) is a two-dimensional field envelope and 3 > 0 is the nonlinearity coefficient. Based on the bilinear transformation [236], Eq. (4.6.1) with 3 = 2 was given

4.6 Two-Dimensional Nonlocal Nonlinear Schrödinger Equation

(a)

177

(b)

(2) 1.6 |Ψ1 |

(2) 2 |Ψ1 |

t=0

1.2 1.5 0.8 20

0.4 0 20

t=2

0.5

0t x0

1

0 –20

–20 –20

(c)

0 x

20

(d)

2.5 |Ψ (2)| 2 2

3

|Ψ2(2)|

t=0

1.5 2

1

20

0.5 0 20

t=3

1 0t

x0 –20–20

0 –20

0 x

20

Figure 4.29: Profiles of the wave amplitudes of the second-order vector rogue wave (2) (2) (2) solutions (4.5.22). (a) |8(2) 1 (x, 0, t)|; (b) |81 (x, 0, t)| with t = 0, 2; (c) |82 (x, 0, t)|; (d) |82 (x, 0, t)| with t = 0, 3 defined by the solutions for !3 (t) = 0 and other parameters given by Eqs (4.5.20) and (4.5.21) with kx = 0.1, ky = 0.9, k = 0.9, 𝛾10 = 0.1, 𝛾20 = 0.15, c1 = 0.02, c2 = 0.01, ! = 0.8.

in terms of Gram-type determinant [237]. Equation (4.6.1) is a nonlinear differentialintegral equation. In what follows, we consider the rogue wave solution for this model.

4.6.2 Two-Dimensional Variable Separation Reduction To construct the analytical rogon-like solutions of Eq. (4.6.1), we employ the twodimensional similarity transformation [238] 8(x, y, t) = g(y) ei>(x,y,t) 6('(x, t), 4(t)),

(4.6.2)

to Eq. (4.6.1) and obtain i4t

∞ 𝜕6 𝜕2 6 𝜕6 + '2x 2 + 3|6|2 6 ∫ |g(y)|2 dy + 'xx 𝜕4 𝜕' 𝜕' –∞ 𝜕6 2 – (>t + >x )6 + i>xx 6 = 0, +i('t + 2>x 'x ) 𝜕'

(4.6.3)

where g(y), '(x, t), 4(t), >(x, y, t) and 6(', 4) are functions of the indicated variables to be determined.

178

4 The Rogue Wave Solution and Parameters Managing

If we require '(x, t), 4(t), >(x, y, t) and g(y) satisfy the following constraints >xx = 0,

'xx = 0, >t + >2x ∞ 3∫

4t =

= 0, 2

|g(y)| dy =

–∞

't + 2>x 'x = 0, '2x ,

(4.6.4a) (4.6.4b)

G'2x ,

(4.6.4c)

where G is a constant and sign(G) = sign(3), then Eq. (4.6.3) reduces to the onedimensional classical NLS equation with constant coefficients i

𝜕6(', 4) 𝜕2 6(', 4) + + G |6(', 4)|2 6(', 4) = 0, 𝜕4 𝜕'2

(4.6.5)

which possesses many kinds of solutions, involves rogue wave solution and so on. It follows from system (4.6.4) that after some algebra, these two new independent variables ('(x, t), 4(t)) and the phase >(x, y, t) are given by '(x, t) = !(x – 2kt), 4(t) = !2 t, >(x, y, t) = kx – k2 t + c(y),

(4.6.6)

and the wave function g(y) satisfies the following integral equation: ∫

∞

|g(y)|2 dy =

–∞

G!2 , 3

(4.6.7)

where !, k are free parameters, and c(y) is a free function of space y. Equation (4.6.7) shows that |g(y)|2 is a bounded real constant on the whole coordinate axes (–∞, ∞). For the special case 3 = G!2 , Eq. (4.6.7) is just the normalization condition for the wave function g(y), i.e., ∫

∞

|g(y)|2 dy = 1.

–∞

There, in fact, exist many types of functions for g(y) to solve Eq. (4.6.7) (e.g. the Hermite–Gaussian function, the hyperbolic secant function, etc.). Therefore for the chosen function g(y) given by Eq. (4.6.7), we can obtain the rogue wave-like solutions of Eq. (4.6.1) in terms of the similarity transformation (4.6.2) with Eq. (4.6.6). 4.6.3 Two-Dimensional Rogue Wave-Like Solution We choose the wave function g(y) as the Hermite (also called Hermite–Gaussian) function [239] g(y) =

1 √n! 2n √0

2 2

Hn (9y) e–9

y /2

,

(4.6.8)

4.6 Two-Dimensional Nonlocal Nonlinear Schrödinger Equation

179

where Hn (9y) is the Hermite polynomial [239], 9 ≠ 0 is a constant. Substituting Eq. (4.6.8) into Eq. (4.6.7), we can obtain the following condition: ∫

∞

!2 (G = 1), i. e., 3 = !2 |9| > 0. 3

|g(y)|2 dy = |9|–1 =

–∞

(4.6.9)

Based on the obtained similarity transformation (4.6.2) with Eq. (4.6.6) and the firstorder rogue wave solution of the NLSE (4.6.5), we arrive at the first-order rogue wavelike solutions of Eq. (4.6.1) in the form [238] 81 (x, y, t) =

Hn (9y) √n! 2n √0

2 2

e–9

y /2

2

×ei[kx+(!

[1 –

4 + 8i!2 t ] 1 + 2!2 (x – 2kt)2 + 4!4 t2

–k2 )t+c(y)]

,

(4.6.10)

whose intensity can be written as 2 2

H 2 (9y)e–9 |81 (x, y, t)| = n n n! 2 √0 2

y

2

[2!2 (x – 2kt)2 + 4!4 t2 – 3] + 64!4 t2 2

[1 + 2!2 (x – 2kt)2 + 4!4 t2 ]

,

(4.6.11)

which involves four free parameters n, 9, !, k to control the different types of roguelike wave propagations. For the given parameters ! = 9 = 1, n = 0 and k = 0, 1, Figs 4.30 and 4.31 display the cross-sections of the density distribution (4.6.11) (x, t, y = 1)–, (x, y, t = 0)– and (t, y, x = 0)–space, respectively. Figures 4.32 and 4.33 exhibit the cross-sections of the density distribution (4.6.11) for the other parameters ! = 9 = 1, k = 0 and n = 1, 2, in (x, t, y = 1)–, (x, y, t = 0)–, and (t, y, x = 0)– space, respectively. In a similar way, we can obtain the second-order rogue wave-like solution of Eq. (4.6.1) [238] 82 (x, y, t) =

1 √n! 2n √0

2 2

Hn (9y) e–9

2

×ei[kx+(!

y /2

[1 +

–k2 )t+c(y)]

P2 (x, t) + iQ2 (x, t) ] R2 (x, t) ,

(4.6.12)

with these functions P2 (x, t), Q2 (x, t) and R2 (x, t) being the polynomial forms of variables x, t 1 P2 (x, t) = – !4 (x – 2kt)4 – 6!6 t2 (x – 2kt)2 – 10!8 t4 2 3 3 – !2 (x – 2kt)2 – 9!4 t2 + , 2 8 Q2 (x, t) = –!2 t [!4 (x – 2kt)4 + 4!6 t2 (x – 2kt)2 + 4!8 t4 15 –3!2 (x – 2kt)2 + 2!4 t2 – ] , 4

(4.6.13)

180

4 The Rogue Wave Solution and Parameters Managing

(a) 4

(b) 4

(c) 4

t

x

t

–4 –4

x

4

–4 –4

y

4

–4

(d) 4

(e) 4

–4 (f) 4

t

x

t

–4 –4

x

4

–4 –4

y

4

y

4

–4 –4

4

y

Figure 4.30: Cross-sections of the intensity distribution |81 |2 (4.6.11): three-dimensional distribution (top row), density plot (bottom row), ! = 9 = 1.0, n = 0, k = 0. (a), (d) (x, t)-plane with y = 1; (b), (e) (x, y)-plane with t = 0; (c), (f) (y, t)-plane with x = 0.

(a) 4

(b)

(c)

4

4

t

x

t

–4 –4

x

4

–4 –4

y

4

–4

(d) 4

(e) 4

–4 (f) 4

t

x

t

–4 –4

x

4

–4 –4

y

y

4

–4 4

–4

y

4

Figure 4.31: Cross-sections of the intensity distribution |81 |2 (4.6.11): three-dimensional distribution (top row), density plot (bottom row), ! = 9 = 1.0, n = 0 with k = 1. (a), (d) (x, t)-plane with y = 1; (b), (e) (x, y)-plane with t = 0, (c), (f): (y, t)-plane with x = 0.

181

4.6 Two-Dimensional Nonlocal Nonlinear Schrödinger Equation

(a) 4

(b) 4

(c) 4

t

x

t

–4

–4

x

4

–4 –4

y

–4

4

(e)

(f)

4

4

4

t

x

t

(d)

–4 –4

x

y

4

–4

y

4

–4

–4 4

–4

–4

4

y

Figure 4.32: Cross-sections of the intensity distribution |81 |2 (4.6.11): three-dimensional distribution (top row), density plot (bottom row), ! = 9 = 1.0, n = 1 k = 0. (a), (d) (x, t)-plane with y = 1; (b), (e) (x, y)-plane with t = 0; (c), (f) (y, t)-plane with x = 0.

(a) 4

4

4

t

x

t

–4 –4

(b)

x

4

(d) 4

t

–4 –4

x

4

–4 –4

(c)

y

4

–4 –4

(e) 4

(f) 4

x

t

–4 –4

y

4

–4 –4

y

4

y

4

Figure 4.33: Cross sections of the intensity distribution |81 |2 (4.6.11): three-dimensional distribution (top row), density plot (bottom row), ! = 9 = 1.0, n = 2 with k = 0 (a), (d) (x, t)-plane with y = 1; (b), (e) (x, y)-plane with t = 0; (c), (f): (y, t)-plane with x = 0.

182

4 The Rogue Wave Solution and Parameters Managing

R2 (x, t) =

1 6 1 ! (x – 2kt)6 + !8 t2 (x – 2kt)4 + !10 t4 (x – 2kt)2 12 2 2 1 9 3 + !12 t6 + !4 (x – 2kt)4 + !8 t4 – !6 t2 (x – 2kt)2 3 8 2 2 9 2 33 3 + ! (x – 2kt)2 + !4 t2 + , 16 8 32

whose intensity can be written as |82 (x, y, t)|2 =

Hn2 (9y) –92 y2 [R2 (x, t) + P2 (x, t)]2 + Q22 (x, t) e , n! 2n √0 R22 (x, t)

(4.6.14)

which involves four free parameters n, 9, !, k to manage the different types of rogue wave-like propagations. For the fixed parameters ! = 9 = 1, n = 0 and k = 0, 1, Figs 4.34 and 4.35 depict the cross-sections of the density distribution (4.6.14) (x, t, y = 1)–, (x, y, t = 0)– and (t, y, x = 0)– space, respectively. For the fixed parameters ! = 9 = 1, k = 0 and n = 1, 2, Figs 4.36 and 4.37 exhibit the crosssections of the density distribution (4.6.11) (x, t, y = 1)–, (x, y, t = 0)– and (t, y, x = 0)– space, respectively.

(a) 4

4

(c) 4

t

x

t

–4 –4

(b)

x

4

(d) 4

t

–4 –4

x

4

–4 –4

y

4

–4 –4

(e) 4

(f) 4

x

t

–4 –4

y

4

–4 –4

y

4

y

4

Figure 4.34: Cross sections of the intensity distribution |81 |2 (4.6.11): three-dimensional distribution (top row), density plot (bottom row), ! = 9 = 1.0, n = 0, k = 0. (a), (d) (x, t)-plane with y = 1; (b), (e) (x, y)-plane with t = 0; (c), (f): (y, t)-plane with x = 0.

4.6 Two-Dimensional Nonlocal Nonlinear Schrödinger Equation

(a) 4

(b) 4

(c) 4

t

x

t

–4 –4

x

4

–4 –4

y

4

–4 –4

(d) 4

(e) 4

(f) 4

t

x

t

–4 –4

x

4

–4 –4

y

4

–4 –4

y

183

4

4

y

Figure 4.35: Cross sections of the intensity distribution |81 |2 (4.6.11): three-dimensional distribution (top row), density plot (bottom row), ! = 9 = 1.0, n = 0 with k = 1. (a), (d) (x, t)-plane with y = 1; (b), (e) (x, y)-plane with t = 0; (c), (f): (y, t)-plane with x = 0.

(a) 4

(b)

(c)

4

4

t

x

–4 –4

x

4

–4 –4

t

y

4

–4 –4

(d) 4

(e) 4

(f) 4

t

x

t

–4 –4

y

4

y

4

–4 x

4

–4

y

4

–4 –4

Figure 4.36: Cross-sections of the intensity distribution |81 |2 (4.6.11): three-dimensional distribution (top row), density plot (bottom row), ! = 9 = 1.0, n = 1 with k = 0. (a), (d) (x, t)- plane with y = 1; (b), (e) (x, y)-plane with t = 0; (c), (f) (y, t)-plane with x = 0.

184

4 The Rogue Wave Solution and Parameters Managing

(a) 4

(b) 4

t

x

–4 –4

x

4

(d) 4

4

t

y

4

(e) 4

t

–4 –4

–4 –4

(c)

4

–4 –4

y

4

y

4

(f) 4

t

x

x

–4 –4

y

4

–4 –4

Figure 4.37: Cross sections of the intensity distribution |81 |2 (4.6.11): three-dimensional distribution (top row), density plot (bottom row), ! = 9 = 1.0, n = 2 with k = 0 (a), (d) (x, t)-plane with y = 1; (b), (e) (x, y)-plane with t = 0; (c), (f) (y, t)-plane with x = 0.

4.7 The Generalized Ablowitz–Ladik–Hirota Lattice with Variable Coefficients 4.7.1 Discrete Nonlinear Physical Model As the prototypical discretizations of the continuum NLS equation, the discrete NLS equation [240] i8n,t + (8n+1 – 28n + 8n–1 ) + :|8n |2 8n = 0

(4.7.1)

and the Ablowitz–CLadik (AL) lattice [241, 242] i8n,t + (8n+1 + 8n–1 )(1 + ,|8n |2 ) + v8n = 0

(4.7.2)

have been studied extensively in the field of nonlinear science. The former is nonintegrable, but has some interesting applications on physics [243–247]. The latter is integrable and possesses an infinite number of conservation laws [241, 242], as well as having been depicted as an effective lattice to study properties of the intrinsic localized modes [248]. Moreover, the nonintegrable discrete NLS equation can also be regarded as a perturbation of the integrable AL lattice [249]. In addition, the Salerno

4.7 The Generalized Ablowitz–Ladik–Hirota Lattice with Variable Coefficients

185

model (SM) has also been presented, interpolating between the nonintegrable discrete NLS equation (, = 0) and the integrable AL lattice (: = 0) in the form [250–256] i8n,t + (8n+1 + 8n–1 )(1 + ,|8n |2 ) + :|8n |2 8n + v8n = 0,

(4.7.3)

which can be derived on the basis of a variational principle $LSM /$8∗n = 0 from the Lagrangian density LSM = ∑ i(8∗n 8n,t – 8n 8∗n,t ) + 4Re(8∗n 8n+1 ) n

+,(8n+1 + 8n–1 )8∗n |8n |2 + :|8n |4 + 2v|8n |2 , where Re(⋅) denotes the real part of a complex variable, 8n ≡ 8n (t) stands for the complex field amplitude at the nth site of the lattice, the parameter , is the intersite nonlinearity and corresponds to the nonlinear coupling between nearest neighbors, : measures the intrinsic onsite nonlinearity, and v describes the inhomogeneous frequency shift. The SM has been applied in biology [250] and Bose–Einstein condensates [251]. Recently, the discrete NLS equation (4.7.2) (, = 0, : = 1, and v = –2,) has been verified to support discrete RWs by numerical simulation and statistical analysis [257]. Specially, it has also been shown that exact discrete RWs can exist in the AL lattice (4.7.2) (i.e., for the case , = 1, : = 0 and v = –2, in Eq. (4.7.2)) on the basis of the limit cases of their multi-soliton solutions [259–261]. We here address the generalized Ablowitz–Ladik–Hirota ALH lattice with variable coefficients modeled by the following lattice: iIn,t + [D(t)In+1 + D∗ (t)In–1 ](1 + g(t)|In |2 ) –2vn (t)In + i𝛾(t)In = 0,

(4.7.4)

which can be derived in terms of a variational principle $LALH /$8∗n = 0 from the following Lagrangian density: LALH = ∑n i(I∗n In,t – In I∗n,t ) + 4Re(D(t)8∗n In+1 ) +g(t)(D(t)In+1 + D∗ (t)In–1 )I∗n |In |2 – 2[2vn (t) – i𝛾(t)]|8n |2 ,

(4.7.5)

where In ≡ In (t) stands for the complex field amplitude at the nth site of the lattice, the complex-valued function D(t) is the coefficient of tunnel coupling between sites and can be rewritten as D(t) = !(t) + i"(t) with !(t) and "(t) being differentiable, real-valued functions, g(t) stands for the time-modulated intersite nonlinearity, vn (t) is the space- and time-modulated inhomogeneous frequency shift, and 𝛾(t) denotes the time-modulated effective gain or loss term. In fact, this nonlinear lattice model (4.7.4) contains many special lattice models, such as – AL lattice (!(t) = const., "(t) = vn (t) = 𝛾(t) = 0, g(t) = const) [240, 241], – the AL equation with an additional term accounting for dissipation (!(t) = const, "(t) = vn (t) = 0, 𝛾(t) = const, g(t) = const) [251],

186

– – –

4 The Rogue Wave Solution and Parameters Managing

the discrete Hirota equation (!(t) = const, "(t) = const, vn (t) = 𝛾(t) = 0, g(t) = const) [261], the generalized AL lattice (!(t) = const, "(t) = 𝛾(t) = 0, g(t) = const) [262], the discrete modified KdV equation (!(t) = vn (t) = 𝛾(t) = 0, "(t) = const, g(t) = const) [263].

4.7.2 Differential-Difference Similarity Reductions and Constraints We search for a proper similarity transformation connecting solutions of Eq. (4.7.4) with those of the following discrete Hirota equation with constant coefficients [261] iIn,t + [+In+1 + +∗ In–1 ](1 + |In |2 ) – 2Re(+)In = 0,

(4.7.6)

where In ≡ In (4) is a physical field of space n and time 4 ≡ 4(t), and the complexvalued parameter + can be rewritten as + = a + ib with a, b ∈ ℝ. The discrete Hirota model (4.7.6) contains some special physical models, such as the AL lattice for the case (a = 1 and b = 0) [241, 242] and the discrete mKdV equation (a = 0, b = 1) [263]. It has been shown in Ref. [264] that the discrete Hirota equation (4.7.6) is in fact an integrable discretization of the three-order NLS equation [265] iqt + a(qxx + |q|2 q) – ib(qxxx + 6|q|2 qx ) = 0,

(4.7.7)

which plays an important role in nonlinear optics [194]. In order to reduce Eq. (4.7.4) to (4.7.6), we here apply the similarity transformation in the form [266] Jn (t) = 1(t)ei>n (t) In [4(t)],

(4.7.8)

where the amplitude 1(t) and the phase >n (t) are both real-valued functions of indicated variables to be determined. To conveniently substitute ansatz (2.4.2) into Eq. (4.7.4) and to further balance the phases in every term, i.e., Jn (t), Jn+1 (t) and Jn–1 (t), we should firstly know the explicit expression of the phase in space >n (t). Similar to the phases in the continuous GP (NLS) equations with variable coefficients [189, 199], we here consider the case that the phase is expressed as a quadratic polynomial in space with coefficients being functions of time in the form >n (t) = p2 (t)n2 + p1 (t)n + p0 (t).

(4.7.9)

Based on symmetry analysis, we simply balance the coefficients of these terms Jn (t), Jn+1 (t) and Jn–1 (t), such that we find that the phase in transformation (2.4.2) should be a first-degree polynomial in space n with coefficients being functions of time, namely

4.7 The Generalized Ablowitz–Ladik–Hirota Lattice with Variable Coefficients

>n (t) = p1 (t)n + p0 (t),

187

(4.7.10)

where p0,1 (t) are functions of time to be determined. We substitute transformation (2.4.2) into Eq. (4.7.4) and require for the new physical field In [4(t)] to satisfy Eq. (4.7.6), and then we can obtain that ̇ + 𝛾(t)1(t) = 0, 1(t)

(4.7.11a)

̇ – !(t) cos p1 (t) + "(t) sin p1 (t) = 0, a4(t)

(4.7.11b)

[b"(t) + a!(t)] sin p1 (t) – [b!(t) – a"(t)] cos p1 (t) = 0,

(4.7.11c)

̇ = 0, 2vn (t) + ṗ 1 (t)n + ṗ 0 (t) – 2a4(t)

(4.7.11d)

2

g(t)1 (t) = 1.

(4.7.11e)

where the dot denotes the derivative with respect to time. For the system (4.7.11), we can solve them by the following steps: First, we solve Eqs (4.7.11a)-(4.7.11c) to obtain the functions 1(t), 4(t) and p1 (t). And then, based on the above-obtained functions 1(t), 4(t) and p1 (t), we can determine the external potential vn (t) and nonlinearity g(t) by solving Eqs (4.7.11d) and (4.7.11e). 4.7.3 Determining the Similarity Transformation and Coefficients It follows from Eqs (4.7.11a) to (4.7.11c) that we can write the variables 1(t), p1 (t) and 4(t) t

1(t) = 10 exp [– ∫ 𝛾(s)ds] ,

(4.7.12a)

c1

p1 (t) = tan–1 [

b!(t) – a"(t) ], a!(t) + b"(t)

(4.7.12b)

t

4(t) = (a2 + b2 )–1/2 ∫ [!2 (s) + "2 (s)]1/2 ds,

(4.7.12c)

c2

where 10 is an integration constant. Now it follows from Eqs (4.7.11d) and (4.7.11e), the external potential vn (t) and nonlinearity g(t) vn (t) = V1 (t)n + V0 (t),

(4.7.13a)

t 1 g(t) = 2 exp [2 ∫ 𝛾(s) ds] , 10 0

(4.7.13b)

where the coefficients of external potential vn (t) are 1 V1 (t) = – ṗ 1 (t), 2

(4.7.14a)

188

4 The Rogue Wave Solution and Parameters Managing

V0 (t) = –

ṗ 0 (t) !2 (t) + "2 (t) +[ ] 2 a2 + b2

1/2

.

(4.7.14b)

It follows from Eqs. (4.7.13a)-(4.7.14b) that the gain–loss term 𝛾(t) can modulate the intersite nonlinearity g(t), and the tunnel coupling D(t) = !(t) + i"(t) can modulate inhomogeneous frequency shift vn (t)). This means that only two varying coefficients (e.g., 𝛾(t) and D(t) = !(t) + i"(t)) are left free. Moreover, it follows from Eq. (4.7.13a) that the intersite nonlinearity g(t) is always positive, i.e., the attractive intersite nonlinearity. For the inhomogeneous frequency shift vn (t) given by Eq. (4.7.14b) (1), when !(t) ≠ c"(t) with c being a constant, the frequency shift vn (t) is a linear function of space n with coefficients being functions of time. (2) When the inhomogeneous frequency shift depends on only time, i.e., vn (t) ≡ v(t), which means that ṗ 1 (t) = 0 (see Eq. (4.7.11d)), there exist two cases to be discussed: (i) If p1 (t) ≠ 0, in which we have p1 (t) = p1 = const ≠ 0, then this means that >n (t) is still a linear function of the discrete space n, i.e., >n (t) = p1 n + p0 (t). In this case, the variable function, 4(t) and the inhomogeneous frequency shift vn (t) are given by t

4(t) =

∫0 !(s)ds

, a cos(p1 ) + b sin(p1 ) ṗ (t) a!(t) – 1 , vn (t) = a cos(p1 ) + b sin(p1 ) 2

(4.7.15a) (4.7.15b)

and 1(t), g(t) are same as Eqs (4.7.12a) and (4.7.13a). (ii) If p1 (t) = 0, which means that the relation for the coefficients in Eq. !(t) = (a/b)"(t) is required and >n (t) is only a function of time, i.e., >n (t) ≡ p0 (t), then the variable function 4(t) and the inhomogeneous frequency shift vn (t) are given by 4(t) =

1 t ∫ !(s)ds, a 0

v(t) = !(t) –

ṗ 0 (t) , 2

(4.7.16)

and 1(t), g(t) are the same as Eqs (4.7.12a) and (4.7.13a), where 𝛾(t), !(t) and p0 (t) are free functions of time, and a, b, 10 are all free parameters.

4.7.4 Nonautonomous Discrete Rogue Wave Solutions and Interaction In general, we have a large degree of freedom in choosing the coefficients of similarity transformation (2.4.2) and Eq. (4.7.4). As a consequence, we can obtain an infinitely large family of exact solutions of the generalized ALH lattice with variable coefficients given by Eq. (4.7.4) in terms of exact solutions of the discrete Hirota equation (4.7.6) and transformation.

189

4.7 The Generalized Ablowitz–Ladik–Hirota Lattice with Variable Coefficients

4.7.4.1 Nonautonomous Discrete One-Rogue Wave Solutions Based on the similarity transformation (2.4.2) and the first-order rogue wave solution Hirota equation (4.7.6) [259], we present the first-order nonautonomous discrete rogue wave solution (4.7.4) [266] t

J(1) n (t) = 10 √, exp {– ∫ 𝛾(s)ds + i[>n (t) + >̂ n (4)]} 0

4(1 + ,) [1 + 4i,√a2 + b2 4(t)] ], × [1 – 1 + 4,n2 + 16,2 (1 + ,)(a2 + b2 )42 (t) [ ]

(4.7.17)

where the phase >̂ n (4) is defined by b >̂ n (4) = 24(t) [(1 + ,)√a2 + b2 – a] – n tan–1 ( ) , a

(4.7.18)

, is a positive parameter, the variable 4(t) is given by Eq. (4.7.12c), and the phase >n (t) = p1 (t)n + p0 (t) with p1 (t) is given by (4.7.12b), p0 (t) being an arbitrary differentiable function of time. To illustrate the wave propagation of the obtained nonautonomous discrete onerogue wave solution (4.7.17), we can choose these free parameters in the form !(t) = c1 sin(2t), "(t) = c2 cos(t), 𝛾(t) = 𝛾0 sin(t) cos2 (t), a = b = , = 10 = 1.0,

(4.7.19)

where 𝛾0 , c1,2 are constants. Figure 4.38 depicts the profiles of the coefficient V1 (t) = –(1/2)ṗ 1 (t), the nonlinearity g(t), and the gain or loss term 𝛾(t) (4.7.19) for the fixed parameters (4.7.19) and (a)

(b)

(c)

1 3.5 V1(t)

0.5

0.3

g(t)

γ(t)

3 2.5

0

0

2 –0.5 1.5 –1.0 0

6

t

12

18

1 0

–0.3 6

t

12

18

0

4

8 t

12

16

Figure 4.38: (a) Profiles of the coefficient V1 (t) = –(1/2)ṗ 1 (t)of the first-degree term of the external potential vn (t) given by Eq. (4.7.13a); (b) Nonlinearity g(t) (4.7.13b); (c) The gain or loss term 𝛾(t). The parameters were given by Eq. (4.7.19) with 𝛾0 = c1 = c2 = 1.

190

4 The Rogue Wave Solution and Parameters Managing

(a)

(b) 6

15

(c) 50 t=0

t

30

0

0 10

–15 –3

n

0

–6 –4

3

n 0

0 –4

4

t=0.2

–2

0 n

2

4

2 (1) 2 Figure 4.39: (a) Profiles of the intensity distribution |J(1) n (t)| with max(n,t) |Jn (t)| = 4.7; (b) The (1) 2 (1) 2 density distribution |Jn (t)| ; (c) The intensity distributions |Jn (t)| at t = 0, 0.2. The parameters are determined by Eq. (4.7.19) with 𝛾0 = c1 = c2 = 1.

(a)

(b) 20

30

(c) 50 40

t

t

t=0

30

0

0

20 10

–30 –2

n

0

2

–20 –2

n

0

2

0 –2

n

t=2 0

3

2 (1) 2 Figure 4.40: (a) Profiles of the intensity distribution |J(1) n (t)| with max(n,t) |Jn (t)| = 38; (b) The (1) 2 (1) 2 density distribution |Jn (t)| ; (c) The intensity distributions |Jn (t)| at t = 0, 2. The parameters are determined by Eq. (4.7.19) with 𝛾0 = 2, c1 = 0.01, c2 = 0.02.

𝛾0 = c1 = c2 = 1. The evolution of the intensity distribution |J(1) n (t)| for the first-order rogue wave solution given by Eq. (4.7.17) is illustrated in Fig. 4.39. The discrete rogue wave solution is localized both in space and time, thus revealing the usual discrete rogue wave features. However, if we fix the coefficient 𝛾0 = 1 of the gain or loss term and adjust the coefficients c1 = 0.01 and c2 = 0.02, then the evolution of the intensity distribution for the first-order rogue wave solution is changed (see Fig. 4.40), and it follows from Fig. 4.40 that the first-order discrete rogue wave solution in this case is localized in space and keeps the localization longer in time than usual rogue waves. Moreover, it follows from Figs. 4.39c and 4.40c that the amplitude of the first-order discrete rogue-wave solution decreases as time increases, and the amplitude in Fig. 4.39 is decreased faster than that in Fig. 4.40 as time increases. 4.7.4.2 The Second-Order Discrete Rogue Wave Solution Based on the transformation (2.4.2) and the second-order rogue wave solution of Hirota equation (4.7.6) [259], we can obtain the second-order nonautonomous discrete rogue wave solution of Eq. (4.7.4) in the form [266]

191

4.7 The Generalized Ablowitz–Ladik–Hirota Lattice with Variable Coefficients

t

J(2) n (t) = 10 √, exp {– ∫ 𝛾(s)ds + i[>n (t) + >̂ n (4)]} 0

Pn(2) (N, T) + i√ T(4) Q(2) (N, T) ] 1+, n [ × [1 – 12(1 + ,) ], Hn(2) (N, T) [ ]

(4.7.20)

(2) where , is a positive constant, the functions Pn(2) (N, T), Q(2) n (N, T) and Hn (N, T) are given by

Pn(2) (N, T) = 5T 2 + 6(N + 2, + 3)T + N 2 + (6 – 4,)N – 3(4, + 1),

(4.7.21a)

2 2 Q(2) n (N, T) = T + 2(N + 1)T + N – (16, + 6)N Hn(2) (N, T) = T 3 + 3(N + 8, + 9)T 2 + N 3 + (3 – 2 2

(4.7.21b)

– 3(8, + 5), 8,)N

2

+3(N – 6N – 16,N + 48, + 72, + 33)T +(27 + 24, + 16,2 )N + 9,

(4.7.21c)

where N(n) = 4,n2 , T(4) = 16,2 (1 + ,)(a2 + b2 )42 , and 4 ≡ 4(t) is given by Eq. (4.7.12c), the part phase >n (t) = p1 (t)n + p0 (t) with P1 (t) is given by Eq. (4.7.12b), p0 (t) being an arbitrary differentiable function of time, and the phase >̂ n (4) is given by Eq. (4.7.18). Similarly, we can choose these free parameters given in Eq. (4.7.19) except for , = 1/16. Figures 4.41 and 4.42 depict the evolution of intensity distribution. Figure 4.41 shows that the second-order nonautonomous discrete rogue wave solution is localized both in space and time with the parameters 𝛾0 = 1, c1 = 2, c2 = 1, but if we fix the coefficient 𝛾0 = 1 of the gain or loss term and adjust the coefficients c1 = 0.2, c2 = 0.1, then the evolution of the intensity distribution for the second-order discrete rogue wave solution is changed (see Fig. 4.42). It follows from the figure that the second-order nonautonomous discrete rogue wave solution is localized in space and keeps the localization longer in time than usual rogue waves. Moreover, it follows from Figs. 4.41c and 4.42c that the amplitude of the second-order discrete rogue-wave solution decreases as (a)

(b)

(c)

20

30

2.5 t=0

2.0

t

t

0

0

1.5 1.0 0.5

–30 –10

n 0

10

–20 –10

n 0

10

0 –8

–4

t=1.5 0 n

4

8

2 (2) 2 Figure 4.41: (a) Profiles of the intensity distribution |J(2) n (t)| with max(n,t) |Jn (t)| = 2.11; (b) The 2 (2) 2 density distribution |J(2) (t)| ; (c) The intensity distributions|J (t)| at t = 0, 1.5. Here we choose the n n parameters as 𝛾0 = 2, c1 = 0.2, c2 = 0.1 in Eq. (4.7.19).

192

4 The Rogue Wave Solution and Parameters Managing

(a)

(b)

(c)

20

30

2.5 t=0

2.0

t

t

0

0

1.5 1.0 0.5

–30 –10

n 0

10

–20 –10

n 0

10

0 –8

–4

t=1.5 0 n

4

8

2 (2) 2 Figure 4.42: (a) Profiles of the intensity distribution |J(2) n (t)| with max(n,t) |Jn (t)| = 1.65; (b) The (2) 2 (2) 2 density distribution |Jn (t)| ; (c) The density distribution |Jn (t)| at t = 0, 0.6. The parameters are determined by Eq. (4.7.19) with 𝛾0 = 1, c1 = 2, c2 = 1.

time increases, and the amplitude in Fig. 4.41 decreases faster than that in Fig. 4.42 as time increases. In conclusion, we have studied the self-similar rogue wave solution of lots of time modulation nonlinear physical models. The approach may also be extended to other discrete nonlinear lattices with variable coefficients for studying their discrete rogue wave solutions and wave propagation. For instance, the other coupled discrete model, multi-component GP/NLS equations and high-order ones.

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