Transformation Methods for Nonlinear Partial Differential Equations [1 ed.] 9810209339, 9789810209339

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TRANSFORMATION METHODS FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

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TRANSFORMATION METHODS FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

Dominic G. B. Edelen & Jian-hua Wang Lehigh University

World Scientific Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Tottcridge, London N20 8DH

TRANSFORMATION METHODS FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Copyright © 1992 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form orbyany means, electronic or mechanical, including photocopying, recording orany information storage and retrieval system now known or to be invented, without written permission from the Publisher.

ISBN 981-02-0933-9

Printed in Singapore by Utopia Press.

FORWARD

The use of transformation methods to solve systems of partial differential equations in the classical manner (i.e., by symmetry and similarity methods) is severely limited by the Backlund theorem. pseudogroup of transformation

This theorem states that a group or

that carries solutions of a system of partial

differential equations into solutions of that system is the prolongation of a group of point transformation that acts on the Cartesian product of the domain space of the independent variables and the range space of the dependent variables. One way to circumvent this limitation is to discard the notion that the transformations form a group or pseudogroup. This has been successfully pursued in published studies of what are now termed Backlund transformations.

The purpose of this work is to

present several essentially different alternatives. The approach we will take is primarily geometric in nature.

Full use is

therefore made of E. Cartan's exterior calculus. We will use the standard notation of the exterior calculus: A

for exterior multiplication,

d

for exterior differentiation,

J

for inner multiplication,

£

for Lie differentiation.

These operations lead to remarkable simplifications, both in the theory and in actual calculations.

The reader is referred to [1 - 7] for proofs of the following

results involving these operations. Let M be a given manifold of finite dimension, let

a, (3 and

7

be exterior differential forms over

M (i.e., elements of the

graded algebra A(M) of exterior differential forms over M), let U and V be vector fields over M (i.e., derivations on the ring

AU(M) of C 0 0 functions over

M), let f belong to A°(M), let a = degree of a, and let b = degree of 3. have

We

VI

aAj9 = ( - l ) a 7 A a ,

aA(/? + 7) =

/or any choice of the generating function f^-VJC1

(7.7) that belongs to A (Ki) .

Proof. We follow the line of argument given in [7, 8]. Since Lie differentiation and exterior differentiation commute, a vector field V G T(Ki) can be an isovector of the contact ideal C, if and only if there exist N functions

{A9j(xy, q ' , r-'j | 1

< a, P < N} such that (7.8)

£ v C a = A^C^.

Computation of the Lie derivative of Ca = d q a — rj* dx 1 , for any V of the form (7.1), and substitution into (7.8) lead to the conditions (7.9)

d v a - r f dv1 - v? dx1 = Ag (dq^ - r?dx") .

A resolution of these relations on the basis {dx1, d q a , dr^} for A (Ki) shows that (7.9) will be satisfied if and only if (7.10) (7.11) (7.12)

d{va - r p i V J - v f =t - A g r f , dp" flj£va-rj*ajvj.=

- r j * c y =* A^ , 0.

25 Noting that V J C a = v a - r j V , (7.11) yields (7.13)

A« = < y v j c « ) ,

and we obtain (7.3). Now that we know the required evaluation of A*o, (7.10) can be solved for

Vj* so as to obtain (7.2)

with

Z- defined by (7.4). The only

conditions that remain to be satisfied are (7.12), which we write as (7.14)

9jv° = rj^vJ .

The left-hand sides of (7.14) are the components of the gradient of va with respect to the variables r r , and hence the system (7.14) entails the integrability conditions (d^ck

- d^d^)va

= 0. Thus, (7.14) demand satisfaction of the conditions 8 1, we must have 5 m v

= 0, and hence v = f ( y , q^), where the fs

are

arbitrary smooth functions of their indicated arguments. A substitution of 3 m v K = 0 into (7.14) gives dfva where the fs

= 0.

We must therefore have va = f"(xJ, q^),

are arbitrary smooth functions of their indicated arguments. This

establishes (7.5). When N = 1, (7.15) reduce tp

26 5kvm =

gm^lt

j

from which we infer the existence of a function £(xJ, q , rj) such that

v m =5f £ , A substitution of this result back into (7.14) gives v

= r- d\£ — f, and hence

(7.6) is established. D An understanding of these results can be gleaned as follows. Let

DIF(G)

be the group of diffeomorphisms

'x1 = X V , q^) ,

(7.17)

'qQkQV/)

of the underlying graph space G. The Lie algebra of DIF(G) is T(G); namely, all vector fields of the form

v G = vi(xJ,q^)ai + v a ( x J,q^a a .

(7.i8)

However, for N > 1, (7.1) and (7.5) show that any V € ISOfC-^] has the form V = V Q + Zj< V Q \ Ca> &a

(7.19) because V J C a = V Q J C " .

In fact, (7.19) shows that for N > 1, any element V

of ISO[Ci] is uniquely determined by the corresponding element V/-i of T(G). On the other hand, similar results hold for N = 1 only when the function £ is an affine function of the r's. Theorem 7.2 7/N > 1, then the Lie algebra ISO[C-.] of isovectors of the first order contact ideal is generated by the Lie algebra T(G) of DIF(G) through the relation (7.20)

V = V G + Z i < V G J C a > 4 , V G e T(G) .

27 This result is true /or N = 1 only when the generating function £ g A (K-.) is affine in the variables {rj*}. This result can be thought of in several different ways. First, we can view (7.20) as defining a lifting of any element element

V

of

ISO[Cx] C T(Kj).

diffeomorphisms of

of

T(G)

ISO[C1]

to a corresponding

is the Lie algebra of

Ki that preserves the contact 1-forms,

viewed as a lifting of contact 1-forms of

V/-,

Since

this lifting can be

DIF(G) to the subgroup of DIF(Ki) that preserves the

Ki.

In fact, Theorem 7.2 tells us that the this is the only

lifting of DIF(G) to a subgroup of DIF(Kj) that preserves the contact 1-forms. This is the view taken in the earlier literature [13-15] where it is referred to as the first group extension. More recent investigations from the jet bundle point of view [17, 18, 31] necessarily confine their considerations to groups that preserve the fibration of the jet bundle as an inductively fibered bundle with base admissible groups become prolongations of [17], we use the notation

pr^ '(T(G))

G.

Accordingly, all the

DIF(G) (see [31]). Following Olver for the Lie algebra of the k

order

prolongation of the Lie algebra of DIF(G). Theorem 7.2 can thus be stated in the equivalent form that ISO[Ci] = p r ^ ^ T ( G ) ) for N > 1. On the other hand, for • (1) N = 1, Theorem 7.2 shows that ISO[Ci] contains pr v '(T(G)) as a subalgebra that obtains only when the generating function £ is an affine function of the r's. Thus, for N = 1, the restriction of the Lie algebra ISOfCJ to the Lie subalgebra prv(1)'(T(G)) will exclude all elements of ISO[Ci] that are generated by functions £ € A (K-i) that are not affine in the r's. Theorem 6.1 shows that isovectors of the fundamental ideal form a Lie subalgebra of ISO[Ci]. Accordingly, we see that

ISO[3] is a Lie subalgebra of

pr^ '(T(G)) for N > 1. Thus, restriction to first prolongations does not exclude isovectors of the fundamental ideal when N > 1. On the other hand, when N = 1,

restriction to prolongations can eliminate isovectors of the fundamental ideal

because p r ^ ^ G ) )

is a proper Lie subalgebra of ISO[C^]. It seems to be an act

of faith in much of the current literature that ISO[3] C pr^ '(T(G)), so that nothing is missed by the restriction to Lie algebras of prolongations of

T(G).

That this surmise is false in a number of important problems with N = 1 is shown in the next section.

28 The fact that ISOfC-J = p r ^ ' ( T ( G ) ) for N > 1 is another way of stating the Backlund Theorem [19, 20, 21]: the most general diffeomorphism of K-, that preserves the contact structure is a prolongatiqn of an element of DIF(G) (i.e., is the first group extension of a smooth, invertible point transformation of graph space). Although the Backlund Theorem does not hold for N = 1, it is still true that

ISO[C|] is a proper subalgebra of the Lie algebra T(Ki)

of DIF(Ki) for

N = 1. The Backlund Theorem is the principal obstruction to obtaining solutions of PDE by transformation-theoretic methods | because it drastically restricts the available choices of elements of DIF(Ki). It is clear that this obstruction can be circumvented only by a drastic reformulation of underlying structure; that is, the contact ideal CTJ of A(KT-J) will have to be replaced by other ideals of A(K-rj). This will be accomplished in chapters 3 and 4 by using the notion of Cartan annihilators of an ideal. The resulting transformation-theoretic methods for solving systems of PDE are presented in chapter 5. The significant differences that occur when

N = 1 can be understood as

follows. When N = 1, the contact manifold K, is of dimension 2n + 1 and the contact 1-form is given by (7.21)

C^dqi-r^dx1.

Hence (7.22)

dC 1 = - drj- A dx*

is a closed 2-form of maximal rank on Ki . Accordingly, Ki can be viewed as the manifold S

x IR with local coordinates {x1, r- , q }, where S

manifold with fundamental 2-form

u> = — dC*

■K = rj A dx1 — dq 1 = — C 1 (i.e., u = dn),

is a symplectic

and symplectic potential 1-form The only difference between the

results obtained here and those of a classical symplectic manifold is that we have included the range space of the symplectic gauge function

q

as an additional

coordinate function for Ki . The form of an isovector of Ci with N = 1, namely

29

(7.23)

V = ( 4 0 ^ + (^dh ~ fl8i - ^d\ j

q

= (9i10Zi-Zi5i1-(5ql> shows the close relation with Hamiltonian vector fields on a symplectic manifold. In fact, if £ does not depend on the argument q and V is allowed to act on a function £ that does not depend on q , then (7.23) gives

v = (d\)d{ - (d i 1 can be obtained by an analysis along the lines given in the proof of Theorem 7.1. The calculations can become confused, however, if the appropriate order is not maintained throughout the analysis. We therefore refer the reader to Suhubi's paper [28] where a careful and concise proof of the following results is given. Theorem 7.3 A vector field (7.24)

V = viflj + vada

+ vf 4

v

+ ••■ +

Pr-iDa«

lD

in T(KT->) is an isovector field of the contact ideal Cp if and only if v? = z [ ° ) < v a > - r ? z [ 0 ) < v J > ,

(7.25)

v i? - - i• i■ = z i( k W *i - ... > - r ?i . .- i. -i r

k

1 < k < D — 1 , where

r

k

r

k

j

: Zi[ k ) < v J > ,

30

(7.26)

Zf°) = 5 i

+

rfa^,

^ , -^- i) +4vr k « and for N > 1, v1 = 4(j,

(7.27)

q13) , v a *= f*(j,

for any smooth choice of the n + N functions

q13)

{f1, £*} of their indicated arguments,

while for N = 1,

v» = d[ , v1 = i}ai - (

(7.28)

for any smooth function ^(x^, q , r - ) . For N > 1, these results show that any isovector field of C Q is the Dth order prolongation of a vector field on the graph manifold

G (see [17, 18, 31]);

that is, ISO[CD] = pr D (T(G)). On the other hand, for N = 1, we see that an isovector field of Cj> is the (D — 1)

prolongation of ISO[Ci] from Ki to Krj ,

in the sense of a prolongation from a vector field on the first jet bundle to a vector field on the D

jet bundle. The marked difference between isovector fields for N

= 1 and for N > 1 thus continues to hold no matter what the order of the contact manifold happens to be.

8. SYMMETRIES THAT ARE NOT PROLONGATIONS For the particularly simple case where ri = 2, N = 1, the fundamental ideal of A(Ki) with balance 2-form B x = dP(rj, i\) 1^^+

dQ(rJ, x\) A n2

(i.e., an equation of conservation in 2-dimensions) has been shown [21] to admit an oo-dimensional Abelian Lie algebra of isovectors that pr^

;I

(T(G)). Any isovector in this Lie algebra is of the form

does not belong to

31

v = (d\od1 + {d\i)d2 + {v\d\( + v\d\i - o s q - z^&al

- z2dl,

where £(rj, ri) is any solution of the linear, second order PDE

(dlQ)d\d\t + {d\Y)d\d\i + (sfp +a}Q)a}sff - o. This example disproves the conjecture that fundamental ideals of A(Ki) only have isovectors that belong to prv(1)'(T(G)). We obtain similar results for the ndimensional Monge-Ampere equation in this section. Consideration is necessarily restricted to the case

N = 1.

We therefore

suppress the Greek indices and hence a local coordinate system on Ki is given by {x\ q, r- | 1 < i < n } . The space Ki

is therefore of dimension 2 n + l . For

simplicity, we also introduce the notation

(8-1)

flq

=£, *=£,

and the contact ideal is generated by the single contact 1-form (8.2)

C 1 = dq - rjdx 1 .

The generalized Monge-Ampere equation in

n dimensions is encoded by the n-

form (8.3)

H-^ = dr-^ A dr 2 A • ■ • A dr n ,

and hence the fundamental ideal is given by (8.4) because dHi = 0.

3 = I { C \ dC 1 , H : }

32 Since IS0[3] C ISO[Ci], we start with a generic element (8.5)

V = (dk) d-x + (rj