all solutions of standard symmetric linear partial ... · a lie point symmetry (2.3) is a standard...

Journal of Mathematical Analysis and Applications 234, 109-122 (1999) Article ID jmaa.1999.6331, available online at http:/www.idealibrary.com on IBEki@

All Solutions of Standard Symmetric Linear Partial Differential Equations Have Classical Lie Symmetry

Philip Broadbridge

School of Mathematics and Applied Statistics, Universg of Wollongong, New South Wales 2522, Australia

and

Daniel J. Arrigo

Department of Mathematics, Uniuersig of Toledo, Toledo, Ohio 43606-3390

Submitted by M. C. Nucci

Received October 6, 1998

It is proven that every solution of any linear partial differential equation with an independent-variable-deforming classical Lie point symmetry is invariant under some classical Lie point symmetry. This is true for any number of independent variables and for equations of any order higher than one. Although this result makes use of the infinite-dimensional component of the Lie symmetry algebra due to linear superposition, it is shown that new similarity solutions, previously thought not to be classical, can be recovered prospectively by allowing symmetries to include superposition of similarity solutions already known from the finite part of the symmetry algebra. This result applies to all constant-coefficient equations and to many variable-coefficient equations such as Fokker-Planck equations. Q 1999

Key Words: Lie symmetry; nonclassical symmetries; partial differential equations; Academic Press

Fokker-Planck equation

1. INTRODUCTION

The method of reduction of variables by the Lie point symmetry algorithm is one of the best known systematic methods for simplifying and solving partial differential equations (P.D.E.’s) [6, 7, 16, 21, 22, 25, 26, 32,

109 0022-247X/99 $30.00

Copyright 0 1999 by Academic Press All rights of reproduction in any form reserved.

110 BROADBRIDGE AND ARRIGO

35, 36, 41, 431. The use of this technique has become so widespread that it is now commonly referred to as “the classical reduction method.” In 1969, Bluman and Cole [5] introduced the more general nonclassical method. This method makes use of a more general symmetry that leaves invariant only the set of a target P.D.E. solutions that also satisfy the invariant surface condition of the symmetry. For various P.D.E.’s this approach has led to a number of new solutions that could not have been found by the classical reduction method [2, 4, 14, 17, 18, 241. Recently, this technique has been generalized even further by specifying progressively smaller solution subsets that are required to be invariant under a symmetry transformation [30, 381. Each of these methods can be placed in the more general conceptual framework of variable reduction via supplementation with compatible side conditions [34] or differential constraints [33].

One cost of the extra generality of the nonclassical method is that, unlike the classical algorithm, it leads to nonlinear determining equations for the coefficients of the infinitesimal Lie symmetry generators. Neverthe- less, the nonclassical method has gained popularity since Clarkson and Kruskal [15] introduced the direct method, in which one makes a general assumption of a possible collapse of variables, without necessarily making any reference to Lie group theory. This technique is in fact closely related to the earlier nonclassical method [3, 24, 31, 33, 371.

Despite its generality, the extra complexity of the nonclassical method is not always rewarded by new similarity reductions. Some P.D.E.’s have been shown to have no nonclassical symmetries beyond the classical Lie symmetry group [l]. Furthermore, it is possible that even if nonclassical symmetries exist, they may not lead to new similarity solutions that are not invariant under some classical symmetry [l]. If, for a given P.D.E., the nonclassical analysis is fruitless, then it would be advantageous to know this in advance, as the more straightforward classical algorithm has already been successfully implemented on several convenient computer algebra systems [20]. So far, there is no known criterion for knowing ab initio whether a given P.D.E. has genuine nonclassical solutions. However, in this article we establish that an important class of P.D.E.’s has no solutions that are not invariant under some classical symmetry.

Bluman and Cole [5] originally applied the nonclassical reduction method to the linear heat equation. However, although they established that for some P.D.E.’s the nonclassical method would lead to previously inaccessi- ble solutions, they did not find any new solutions to the heat equation that could not be “included in the classical case” [51. Although the nonlinear nonclassical determining equations for the heat equation have been known for a long time, their general solution has proven to be a stumbling block. However, we expected that our recent announcement, that every solution of the linear heat equation is invariant under some classical Lie point

CLASSICAL SIMILARITY SOLUTIONS 111

symmetry [ll, would remove some of the motivation for completing this task. In fact, our observation has provided Mansfield with the key to the long-awaited general solution of the nonclassical symmetry determining relations [27].

We originally proved classical invariance of solutions of second order linear P.D.E.’s with two independent variables and with constant coefficients. In this article we provide a neater proof that immediately extends not only to linear constant-coefficient equations with derivatives of arbi- trarily high order and with any number of independent variables, but also to any linear P.D.E. with a classical symmetry that alters independent variables. For this important class of relatively simple P.D.E.’s that have a myriad of physical applications, the classical symmetry reduction algorithm may be less restrictive than was previously thought. The same will be true for the nonlinear P.D.E.’s that can be transformed to linear equations with constant coefficients simply by an invertible point transformation, since the nonlinear and linear equations will have isomorphic classical symmetry algorithms and the additional nonclassical symmetries will be in one-to-one correspondence.

In Section 2, we prove our main result, namely, that if a linear P.D.E. of order m 2 2 has a classical Lie point symmetry generator of the form

n d d r = C xi(x, U ) - + U ( X , U ) - i = 1 dX‘ dU

with X # 0, where x i (i = 1, . . . , n 2 2) are the independent variables, and u is the dependent variable, then every solution to that P.D.E. is invariant under some classical Lie point symmetry. This theorem applies to an important class of linear P.D.E.’s that has a wide variety of practical applications. This class includes not only all constant-coefficient equations but also some important variable-coefficient equations.

In Section 3, we apply the above result to a familiar Fokker-Planck equation. This example serves to illustrate that by our approach, new similarity solutions, previously believed not to be classical, can be recovered prospectively by classical point symmetry reduction.

2. LIE POINT SYMMETRIES OF LINEAR P.D.E.’S

We consider linear independent variables,

0 = Lu

= aOu(x)

mth order scalar homogeneous equations in n

m


d d d and where uil ip . . . i,(x> = 7 - -u(x).

dx 1 d X i 2 d X Z P

In the favored method of variable reduction by invariants, one seeks a compatible invariant solution expressible in the form

w ( 41,42,. . . I 4J = 0,

where {c%q,. . . , 4n} is a complete set of n independent invariants for a one-parameter Lie point transformation group on graph space

ye:(x,u) - ( % U ) , (2.2)

in infinitesimal form

x i = x i + exi(,, u ) + O( e 2 ) ;

u = u + EU(X,U) + O ( E 2 ) .

i = 1 , . . . , n (2.3a)

(2.3b)

The flow given by the one-parameter transformation (2.2) is generated by the vector field

n d d r = C xi(x, U) - + U(X, U) - i = 1 d X 1 d U

The one-parameter Lie transformation group is a classical symmetry of (2.11, provided (2.2) leaves the linear solution space of (2.1) invariant.

Invariant solutions of (2.11,

u = Icl(.), must also satisfy the invariant surface condition

n d U

d X 1 c xyx, 24) - = U(x, u), i= 1

which follows from

d

d e - [ u - q ( E ) ] = 0.

In the nonclassical method, symmetries are required to leave (2.1) invariant only after assuming Eq. (2.9, and after taking account of the differential ideal that it generates. We will make use of the well known


fact [32, 421 that for linear P.D.E.’s of order m 2 2, Xi is independent of u , and U is linear in u:

Xi = Xi(x) and U =f(x)u + g(x). (2-6) This is why, for example, in the linear theory of quantum mechanics, we do not need to consider the full generality of symmetry operators of the form (2.4) that operate nonlinearly on the wave function [lo]. We now restrict our attention to linear P.D.E.’s with point symmetries that do not leave the independent variables x i invariant.

DEFINITION. A Lie point symmetry (2.3) is a standard symmetry if it does not leave all independent variables invariant: that is, X # 0.

A differential equation is a standard symmetric equation if it has a standard symmetry.

We note that the class of standard equations is quite considerable. For example, every linear equation with constant coefficients is a standard symmetric equation since it is invariant under translations of the independent variables, generated by Xi(d/dxi) with Xi constant. In addition, for second order equations, it is known from the symmetry analysis of the canonical forms (Laplace equation, wave equation, heat equation, etc.) that there exist other symmetries with Xi depending explicitly on x. In addition there are many standard symmetric linear equations with noncon- stant coefficients, and these are ubiquitous in applications [23].

By the standard prolongation formulae [32, 71, (2.3) extends to a transformation law for higher order variables,

where

~,i,...i, = D i l D i z Di (f(x)u +g(x) - X k u k ) + X k ~ k i , . . . i , . (2.7)

This transformation law enables us to interpret

- - d d d u. l l l z , , 7 , iP as - - - U .

dZi l dZiz dZi

In (2.7) and in the following, the repeated index k is summed from 1 to n, and Di is the total xi-derivative on mth order jet space,

d d d D.= - + ui- + uik- + ..* dxi d U d u k

Having set up this notation, we can provide a concise constructive proof of our stated results.


THEOREM 1.

Prooj

Evely solution of a standard symmetric linear partial differ-

Let u(x) be a solution of a governing P.D.E. of the form (2.1).

ential equation is invariant under some classical Lie point symmetry.

Let d d

dX d U r = X ~ ( ( X ) ? + [ ~ ( x ) u + g(x)] - (2.9)

be the generating vector field of any classical Lie symmetry of the governing P.D.E. Invariance of the governing equation Lu = 0 implies

Assuming summation over repeated weakly ordered indices, this can be written as

m 0 = a”U+ C ail...iPu l l ” ’ l p .

p = 1

m

+ x k ~ , ( a 0 ) u + C XkD,(ail””p)uil...ip p = 1

(with U given as in (2.6))

= a 0 ( f ( x b + g ( x ) )

Rearranging, we have

= L ( f i + g - XkU,) + X%,( L u ) I L u = 0

= L ( f i + g - XkU,) (2.10)


(since Lu is assumed to be 0). This may be seen also from the fact that [ fu + g - Xku,l(d/du) is an evolutionary symmetry that is equivalent to

Hence, if u = +(XI is a particular solution to (2.11, then another solu- r [7, 22, 321.

tion is

(2.11)

By linear superposition, if (2.9) is a classical symmetry vector field, then so is

d

The invariant surface condition for rl is du

Xk- = f(x)u + g(x) + gl(x). dXk

(2.13)

From (2.11), it is true that u = +(x) satisfies the invariant surface condition (2.131, so that the arbitrary solution u = +(XI is invariant under the classical symmetry generated by rl.

A simple extension of the above proof to inhomogeneous P.D.E.'s allows a linear forcing term to be included in (2.1). Given any particular solution u = +Jx) to the inhomogeneous equation Lu(x) = h(x), then for a general solution u(x), u(x) - +Jx) satisfies the homogeneous equation, so that Theorem 1 still applies.

3. SIMILARITY SOLUTIONS OF A FOKKER-PLANCK EQUATION

In order to illustrate the result of Section 2, we now consider the Fokker-Planck equation

u, = u,, + xu, + u

d = - (u , +xu).

dX


Nariboli [291 found the most general Lie point symmetry generator to be

d d d T- + X - + U - ,

d t dX d U

with

T = a,, + a lez f + uZe-"

x =

U = ( c + uze-2 t - b,xet - u lx2eZ t )u + Q ,

where Q<x, t> is an arbitrary solution of (3.1). Thus, the Lie point symmetry algebra is

- u ~ ~ - ' ~ ) x + blet + bze-'

d =d5 @dm,

whered5 = span{ri; i = 1 , . . . , 5 )

d d z withr, = -, T2 = e Z t - + x - - x d t (,4 dx

d r - e - 2 t 3 - (

d r - e - f - 5 - d x '

and dm is the Abelian subalgebra generating linear superposition,

The possibility of linear superposition means that every linear equation will have an infinite dimensional symmetry subalgebra analogous to dm. Of course, without knowing the general solution of (3.11, the infinite subalgebra dm cannot be known explicitly. Hence, it is common practice to ignore it in the construction of similarity solutions. However, once these first-generation similarity solutions have been obtained by reductions with d5, then we know explicitly a finite dimensional subalgebra 4') of dm generating linear superpositions of first-generation similarity solutions and we may proceed to construct additional second-generation similarity solutions that are invariant under members of d5 @dl).

One four-parameter solution of (3.11, obtained by Pucci and Saccomandi [39], is

u = e - X Z / 2 - 2 r ( K l ( l - x2) + K2xe' + K3(3x - x3)eP t + K,e2') (3.2)


This solution cannot be recovered via a classical symmetry reduction using any member of M5. As shown by Pucci and Saccomandi, it can be recovered from a potential symmetry which is a classical symmetry of the system u, = u , u, = u, + xu.

We demonstrate here that (3.2) can indeed be found prospectively from Lie point symmetry reductions. We choose

which is obtained by symmetry reduction from T2. Now we consider the infinitesimal symmetry generator

The invariant surface condition is

e2 tu , + xe2tu, = -x2e2 tu - 2 e - ~ ' / ~ ( ~ ~ + 3 ~ , x e - ~ ) . (3 .4 )

From the method of characteristics, the general solution to (3.4) is

where F is an arbitrary twice differentiable function. Substituting (3.5) into (3 . l), we achieve reduction to the ordinary differential equation (O.D.E.)

F"( A) + 6hK3 + 2 K 1 = 0; h = xe- , .

Substituting the general solution of this O.D.E.,

F ( A) = -K3A3 - KlA2 + K 2 A + K,,

into (3.51, we recover the solution (3.2) identically. New similarity solutions, constructed in this manner, may be regarded as second-generation similarity solutions. From first- and second-generation similarity solutions q(')(x, t ) , we may construct a subalgebra

spanned by operators of the form q(2 ) (x , t ) d / d u . For example, from the previously discussed second-generation similarity solution


we may construct an infinitesimal symmetry generator

d

d U r = r2 + p( X , t ) -

Solutions which are invariant under this symmetry must satisfy the invariant surface condition

u, +xu, = -x2u + e - x z / 2 - 4 f [ K 1( 1 - x 2 ) + K3(3x - x 3 ) e P f ] (3.6)

which has the general solution

Substituting (3.7) into (3.0, we achieve reduction to the ordinary differential equation

F " ( A ) = -KlA2 - K3h3.

Hence, (3.1) has the similarity solutions (3.71, with

A4 A5 F( A) = -Kl- - K,- + K,A + K , (K;s constant).

12 20

That is,

20

+ ~ , e - ~ ' / 2 - t x + K 6 e X Z l 2 . ( 3.8)

Now we may successively define %(. + as the algebra generating linear superpositions of linear combinations of similarity solutions that are invariant under elements of 4 @%@). Even if we consider the single symmetry generator r5 E 4, as above, it becomes clear that direct integration of the invariant surface condition for d5 @a$"), followed by direct integration of the second order O.D.E. obtained by symmetry reduction from (3.0, results in similarity solutions that allow initial values u(x, 0) to


be e-"/' multiplied by a (2n + 1)th degree polynomial in x, with 2n + 2 free coefficients. Hence, an infinite dimensional solution space for (3.1) can be constructed in this manner. Furthermore, since the polynomials are dense in standard function spaces, a very versatile class of initial conditions can be incorporated in these solutions.

4. DISCUSSION

Our main result is that every solution to a linear P.D.E. with independent variable-altering Lie point symmetry is invariant under some classical Lie symmetry. This result is true if we notionally allow g(x) in the classical symmetry (2.9) to be an arbitrary solution of the original equation. That is, we include the infinite-dimensional part of the classical symmetry Lie algebra that is allowed by linear superposition. Traditionally, similarity solutions are constructed only by the so-called finite part of the symmetry algebra.

If we need to construct new similarity solutions, then we cannot logically include superposition of an arbitrary solution in the symmetry group, as we do not know all solutions. However, from a practical point of view, we can always admit to knowing some simple solutions at the outset, even if we do not know all solutions. If we include these known solutions u = g(x) in the infinitesimal symmetry operator (2.9), then solutions of the invariant surface condition may include new self-similar solutions of the governing P.D.E.

In the example of the Fokker-Planck equation discussed in the previous section, inclusion of first-generation similarity solutions, and their linear combinations, in g(x), led to new second-generation similarity solutions by the standard Lie algorithm. For this equation, repetition of this procedure leads to an infinite dimensional subspace of the function space of solutions.

In a future paper, we will demonstrate that a single symmetry of a linear P.D.E. will always lead to an infinite dimensional space of solutions, in this manner.

We are not denying that nonclassical symmetries are useful for con- structing new solutions of equations of the type (2.1). There may be solutions that are readily obtained by nonclassical symmetry reduction but which could not be invariant under any classical symmetry of the form ( 2 . 5 H 2 . 6 ) in which the seed solution g(x) could be expected to be known in advance. The same general conclusion will be true of c-integrable nonlinear P.D.E.'s [13] that can be transformed to linear equations with constant coefficients simply by a direct invertible substitution of variables. The nonlinear and equivalent linear equations will have isomorphic classi-


cal Lie symmetry algebras, and their additional nonclassical symmetries will be in one to one correspondence. Therefore, since every solution of a linear equation with constant coefficients is invariant under some classical symmetry, then the same will be true of the equivalent nonlinear equations. A directly linearizable equation will be expected to have a Lie symmetry that depends on an arbitrary solution of the equivalent linear equation [7]. However, the free solution of the linear equation may appear in the symmetry algebra as a nonlinear superposition [191. The directly linearizable equations are important because they have been very useful in modeling nonlinear physical phenomena [8, 11, 121 and in providing general insight on integrability of nonlinear equations [9, 28, 401. At this time, it is not known how to determine, from the outset, for which other classes of P.D.E.’s the nonclassical symmetry analysis will result in new solutions. However, we hope that the elementary results established here may provide a new slant on this question.

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all solutions of standard symmetric linear partial ... · a lie point symmetry (2.3) is a standard...

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