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I? =
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

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