стр. 115 |

2

тогда и только тогда, когда

(m) (m)

Aab = 0, a = b, Aaa = Cm (v),

B (m) (? (1) , ? (2) , ? (1) , ? (2) ) = ? (m) Fm (v, ?) + (1 ? Cm (v))?a ?a /? (m) +

(m) (m)

1 1

[Cm (v)? (m) ]?1 ,

+ ?m Cm (v)? (m) m = 1, 2,

Cm , Fm — произвольные функции, v = (? (1) )?2 (? (2) )??1 , ? = ?xa ?xa , ?m ? C1 .

?v ?v

Доказательство теоремы аналогично доказательству теорем 1, 2, 3 в работе [1].

Следствие. Если в системе (40) функции B (m) не зависят от производных,

то система инвариантна относительно алгебры AG(1, n) только тогда, когда

имеет вид

(1)

= ?? (1) + ?1 ? (1) ln ? (1) + ? (1) F1 (v),

? 1 ?t

(41)

(2) (2) (2) (2) (2)

?2 ?t = ?? + ?2 ? ln ? +? F2 (v).

В случае системы комплексно-сопряженных уравнений шредингеровского типа

вместо системы (41) получаем уравнение

i?t = k?? = i? ln ? + ?F (|?|),

v

?? ? , k ? R1 .

где F — произвольная функция, |?| =

496 В.И. Фущич, Р.М. Чернига

1. Фущич В.И., Чернига Р.М., Симметрия и точные решения многомерных нелинейных уравнений

шредингеровского типа. I, Укр. мат. журн., 1989, 41, № 10, 1349–1357.

2. Фущич В.И., О симметрии и точных решениях многомерных волновых уравнений, Укр. мат.

журн., 1987, 39, № 1, 116–123.

3. Collins С.В., Complex potential equations. I. A tecknique for solution, Math. Proc. Cambridge Phil.

Soc. 1976, 80, 165–171.

4. Камке Э., Справочник по обыкновенным дифференциальным уравнениям, М., Наука, 1976,

576 с.

5. Градштейн И.С., Рыжик И.М., Таблицы интегралов, сумм, рядов и произведений, М., Наука,

1971, 1108 с.

6. Захаров В.Е, Шабат А.Б., Точная теория двумерной самофокусировки и одномерной автомоду-

ляции волн в нелинейных средах, Журн. эксперим. и теор. физики, 1971, 61, № 1, 118–134.

7. Дородницын В.А., Князева И.В., Свирщевский С.Р., Групповые свойства уравнения теплопрово-

дности с источником в дву- и трехмерном случаях, Дифференц. уравнения, 1983, 19, 1215–1223.

W.I. Fushchych, Scientific Works 2001, Vol. 3, 497–505.

Computer algebra application

?

for determining Lie and Lie–Backlund

symmetries of differential equations

W.I. FUSHCHYCH, V.V. KORNYAK

The application of computer algebra for determining Lie and Lie–B?cklund (LB)

a

symmetries of differential equations is considered. Algorithms for calculating the

symmetries are developed and implemented on the basis of computer algebra systems

REDUCE, AMP and FORMAC. The most effective and advanced program is written

in FORMAC. It finds LB symmetries completely automatically. In many cases the

program yields the full algebra of symmetries. If the program fails in full integration of

the determining system, it reduces the remaining determining equations to the system

in involution.

1. Introduction

The determination of point and contact Lie symmetries and Lie–B?cklund sym-

a

metries of differential equations is one of the central problems in applied mathematics

and mathematical physics. The mathematical theory is rather well developed [10, 8],

but computing of the symmetries of certain systems of differential equations requires

extremely tedious symbolic manipulations. In many cases the only possibility to cope

with the task is the application of computer algebra. There are several programs and

packages for solving the problems in this field. The REDUCE package for obtaini-

ng determining systems of point symmetries was suggested by Schwarz [11]. This

package includes several programs for different kinds of differential equations and

systems. This work was developed by adding programs for solving the determining

systems [12], and the resulting package was successfully applied to many problems

in mathematical physics. In [1] the universal REDUCE program was suggested for

computing determining systems of point and contact symmetries of arbitrary systems

of differential equations. To obtain determining equations of LB symmetries the

FORMAC and REDUCE programs have been developed [2, 3]. We have also wri-

tten an analogous AMP program. It is very difficult or impossible to handle the

LB determining systems by hand, because generally they contain several hundreds

of equations (though linear and overdetermined). We should mention, also, a recent

FORMAC package CRACKSTAR [13], which is closely related with the subject consi-

dered. This package is intended for investigation of Lie-symmetries of pde’s and

dynamical symmetries of ode’s as well as other analytic properties.

Comparing different computer algebra systems (REDUCE 2, REDUCE 3.?, AMP

6.4 and PL/I-FORMAC) we came to a conclusion that FORMAC is the most suitable

system for our purposes. Of course, it is out of date to some extent but much more

effective than REDUCE and AMP. We developed the FORMAC program LBF for

determining Lie–B?cklund symmetries of arbitrary systems of differential equations.

a

The program creates the LB determining system, integrates it as far as possible and,

J. Symbolic Computation, 1989, 7, 611–619.

498 W.I. Fushchych, V.V. Kornyak

if some part of the determining system remains, it reduces this part to the system in

involution, i.e. to system with all integrability conditions being explicit [4].

In this paper our consideration is given mostly to the general case of LB symmet-

ries, because point and contact symmetries are only special cases of LB ones. Readers

interested in the details of algorithms and programs concerning point and contact

symmetries are referred to papers mentioned above.

2. Mathematical background

We shall consider a system of s partial differential equations of kth order for m

functions u? in the n independent variables xi

F ? (xi , u? , u? ···il ) = 0,

i1

(1)

? = 1, . . . , s; ? = 1, . . . , m; i, i1 , . . . , il = 1, . . . , n; l = 1 . . . , k,

where u? ···il are jet bundle coordinates corresponding to the partial derivatives of u?

i1

with respect to xi1 , . . . , xil .

The LB group is defined as the tangent transformation group of infinite order. In

the terms of infinitesimal generators it means that coordinates of Lie-algebra depend

on the unlimited number of derivatives. The Lie-algebra vector called the LB operator

has the form

? ? ?

X = ?i + ?? ? + ?

(2)

?i1 ···il .

?u? ···il

?xi ?u i1

l?1

Summation over twice occurring indices is always understood, ? i , ? ? , ?i1 ···il depend

?

on variables xi , u? , u? ···il , and ?i1 ···il are generated recursively by

?

i1

?i = Di (? ? ) ? u? Di (? j ), ?i1 ···il = Dil (?i1 ···il?1 ) ? u? 1 ···il Dil (? j ).

? ? ?

(3)

j ji

Di is the operator of total differentiation with respect to xi

? ? ?

+ u? ? + u? 1 ···ul (4)

Di = .

i ii

?u? ···il

i

?x ?u i1

l?1

System (1) with all the differential consequences is called a differential manifold

[F ] : F ? = 0, Di (F ? ) = 0, . . . , Di1 Di2 · · · Dil (F ? ) = 0, . . . . (5)

According to the definition systems, (1) is invariant with respect to LB group, if the

differential manifold [F ] is invariant, i.e.

(6)

X[F ] = 0.

[F ]

There is a theorem staling that condition (6) is equivalent to

XF |[F ] = 0, (7)

i.e. it is sufficient to apply the X operator only to initial equations (1), but to consider

the differential consequences when transferring to the manifold.

It is easy to check that LB operators of the form

i

(8)

X? = ?? Di ,

Computer algebra application for determining symmetries 499

where ?? are arbitrary functions of the xi , u? , u? ···il variables, and to leave the

i

i1

artibrary differential manifold invariant, i.e. they do not contribute to the invariance

condition. These operators form the ideal in the Lie-algebra of all the LB operators.

Therefore, it is possible to consider, without loss of generality, the factor-algebra

of the complete Lie-algebra with respect to the above ideal. Each operator (2) is

equivalent in the factor-algebra to some operator with vanished ? i , viz.

?

X ? Y = X ? ? i Di = (? ? ? ? i u? ) + ···.

i

?u?

Thus the elements of factor-algebra may be represented in the form

? ?

X = ?? ?

(9)

+ ?i1 ···il .

?u? ···il

?u? i1

l?1

Operators (9) are called “canonical operators”. Transition to canonical operators es-

sentially simplifies the calculations, since now it is sufficient to consider m functions

? ? instead of n + m functions ? i and ? ? . Moreover, extension formulae (3) take a

simple form

?i = Di (? ? ),

? ? ?

(10)

?i1 ···il = Dil (?i1 ···il?1 ).

In terms of canonical operators the invariance conditions, i.e. the determining equa-

tions, take the form

?F ? ?F ? ?F ?

+ Di (? ? ) ? + Di2 (Di1 (? ? )) ? + · · ·

?? (11)

= 0.

?u? ?ui ?ui1 i2 [F ]

This is a system of equations with respect to ? ? . The solutions of the determining

equations depending on the derivatives of no more than kth order are called kth order

solutions. This definition allows the overdetermined system to be obtained, because

we can split the left part of (11) with respect to “free derivatives”, i.e. u? ···il for l > k.

i1

If the 1st order derivatives are not expressed by derivatives of a higher order in the

process of transition to manifold [F ], then the 1st order solutions contain point and

contact Lie symmetries. In particular, point symmetries are corresponding to the 1st

order solutions of the form

?point = ? ? (xi , u? ) ? ? j (xi , u? )u? ,

?

(12)

j

where ?, ? = 1, . . . , m; i, j = 1, . . . , n; ? j and ? ? are usual coordinates of the Lie-

algebra of point transformations. Note that the point transformation groups obtained

from the solutions of LB determining equations may be wider than the classical ones.

This may occur if some equations of the system under consideration are of an order

less than the maximum one of the system, because, when transferring to the manifold,

the LB symmetries imply the use of differential consequences, i.e. relations

Di1 · · · Dil (F ? ),

F ? = 0, Di (F ? ) = 0, ..., ...,

whereas the classical definition of point symmetries uses relations F ? = 0 only [8].

There are non-trivial LB symmetries of the 1st order, i.e. non-equivalent to point or

contact ones. It occurs, for example, for Dirac equations [14].

500 W.I. Fushchych, V.V. Kornyak

3. Description of the program and algorithms

The LBF program is a single program acting completely automatically. Its text

contains 37 internal procedures besides the main one. The total length of the program

is now 1362 lines of PL/1-FORMAC code. We shall describe the program and its

algorithms illustrating some steps by the simple example of determining 1st order LB

symmetries for one-dimensional linear heat equation. To be more concrete, let us give

the full input and output for this task with the comments on the right side.

Input:

‘HEAT EQUATION’ — comment

1 — symmetry order

(13)

‘X, T ; U ’ — independent and dependent variables

‘U T ? U XX’ heat equation UT ? UXX = 0.

—

To simplify the input and output expressions, the derivatives are represented by

concatenation of dependent and independent variables. Thus, if one uses multicharac-

ter symbols for variables one has to choose them so as to avoid ambiguities. If one

considers the system of equations, the corresponding character strings separated by

blanks must be added.

Output:

HEAT EQUATION

INDEPENDENT AND DEPENDENT VARIABLES:

X, T ; U

SYSTEM OF EQUATIONS:

1). U T ? U XX = 0

?

GENERATORS OF LIE–BACKLUND SYMMETRIES OF THE 1ST ORDER:

1).

U == C1 U X + C2 (1/2 U X + T U X) + C3 (U X X + 2 T U T ) + C4

(T U X X + T 2 U T + U (1/2 T + 1/4 X 2 )) + C5 U T + C6 U + F 1

DEPENDENCES OF FUNCTIONS:

F 1 = F 1(X, T )

REMAINING EQUATIONS:

1).

0 = F 1.(T ) ? F 1.(X, X) — equation FT ? FXX = 0

1 1

стр. 115 |