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Bµ (? )xµ + f2 (? ) = 0 (20)

and Aµ , Bµ , f1 , f2 are arbitrary functions of one variable satisfying
?
Aµ Aµ = ?1, Aµ B µ = 0, Aµ B µ = 0, Bµ B µ = 0. (21)

The abtive theorems give us some general information about the solutions of (18)
in particular cases. Of course, these results express the solution in terms of other
functions, but now we know how to generate these functions: we have to choose
Aµ , Bµ , f1 , f2 appropriately so as to define both ? and then w (as we do below
in a particular case). In this way, we have a systematic way of obtaining solutions
of (18). Our approach to the solution of the nonlinear d’Alembert equation is based
on a change of variable as in (2), and a decomposition of the equation (4) for the
new variable w into ‘component’ equations (5), (6) and (7). Equation (5) involves
the change of variable and the nonlinearity F (u), whereas (7) provides us with a
system which can be dealt with using theorems 1–3. The d’Alembert equation with
nonlinearity sin u was discussed in [4], where a change of variable together with a
decomposition of the ensuing equation was also used. The result obtained in [4] can
be obtained with our results, as follows. In (20), (21) put

Aµ = ?µ , Bµ = ?µ ,

where ?µ = ?µ ? ?µ , ?µ , ?µ , ?µ are constant vectors satisfying

?µ ?µ = ??µ ? µ = ??µ ? µ = 1, ?µ ? µ = ?µ ? µ = ?µ ? µ = 0.

Equation (20) then defines ? through

f2 (? ) = ??µ xµ
412 P. Basarab-Horwath, W.I. Fushchych, M. Serov

and on choosing f2 invertible we obtain the solution of (19)

w = ?µ xµ + f (?µ xµ ).

When F (u) = ? sin u we obtain

u = 4 arctan exp[?µ xµ + f (?µ xµ )], (22)

where f is an arbitrary differentiable function. Equation (22) is the solution obtained
in [4]. As can be seen, our method gives a useful way of obtaining exact solutions of
nonlinear equations.
WF acknowledges support by the Swedish Natural Sciences Research Council,
grant number R-RA 9423-307.

1. Fushchych W.I., Zhdanov R.Z., Revenko I.V., General solutions of the nonlinear wave equation and
the eikonal equation, Ukr. Math. J., 1991, 43, 1420–1460 (in Russian).
2. Fushchych W.I., Zhdanov R.Z., Revenko I.V., Compatibility and solutions of the nonlinear d’Alem-
bert and Hamilton equations, Preprint, Institute of Mathematics, Ukrainian Academy of Sciences,
Kiev, 1990 (in Russian).
3. Fushchych W.I., Zhdanov R.Z., Yehorchenko I.A., On the reduction of nonlinear multidimensional
wave equations and compatibility of the d’Alembert–Hamilton system, J. Math. Anal. Appl., 1991,
161, 352–361.
4. Ouroushev D., Martinov N., Grigorov A., An approach for solving the two-dimensional sine-Gordon
equation, J. Phys. A: Math. Gen., 1991, 24, L527–L528.
W.I. Fushchych, Scientific Works 2002, Vol. 4, 413–414.

Symmetry analysis. Preface
W.I. FUSHCHYCH

Till now there are no general methods of investigation of arbitrary nonlinear partial
differential equations (PDE). However if a nonlinear equation is beautiful, that is to
say it possesses a non-trivial symmetry, than it is possible to obtain rather wide
and rich information about its solutions; to carry out reduction of multidimensional
equations to ordinary differential equations [1], to construct classes of exact and
approximate solutions, investigate asymptotic of special classes of solutions, etc. [1–
6].
It is important to point out that beautiful equations are these ones which are
widely used in mathematical and theoretical physics, and in applied mathematics. It is
connected with the fact that mathematical models of real processes must be of such a
form that e.g. conservation laws of energy, momentum, angular momentum of motion
or relativity principles [1, 3] and other important principles of physics are satisfied in
these models. The beauty (or symmetry, or approximate symmetry) of an equation has
a lot of forms: local (or Lie, as we call it in [2, 6]), nonlocal (non-Lie [2]), discrete,
non-group, non-algebraic. Therefore it is not simple to give a mathematically correct,
effective and general enough definition of beauty of an equation.
The principal ideas and methods of investigation of group (local) properties of
partial differential equations PDE are developed by Sophus Lie. These methods enable
to study group properties of an arbitrary partial differential equation. The above
methods form a part of modern theory of differential equations called on L.V. Ov-
syannikov and N.Kh. Ibragimov suggestion “Group analysis of PDE” [4, 5].
Since PDE prove rather often to possess symmetry that cannot be presented in
terms of Lie groups or Lie algebras, we use a more general term “Symmetry analysis”
suggested in [2, 6]. Symmetry analysis is the aggregate of mathematical methods for
investigating local, geometry, non-geometry, discrete, inner and dynamic symmetries
of PDE.
The present collection contains papers by participants of the Seminar “Symmetry
analysis of mathematical physics equations” (Institute of Mathematics of the Academy
of Sciences of Ukraine) in which two scientific directions are considered:
1. Conditional symmetry of equations of nonlinear mathematical physics.
2. Local, non-local symmetry and construction of classes of exact solutions of
nonlinear PDE.
In conclusion, I adduce several beautiful second-order PDE, that have not been
investigated yet
a(u, ?, 2u) + b(u, ?, u? u? u?? ) + c(u, ?, det uµ? ) = F (u, ?),
2 2 2
?u ?u ?u
? ? u? u = ? ? ··· ?
?
,
?x0 ?x1 ?xn
in Symmetry Analysis of Equations of Mathematical Physics, Kyiv, Institute Mathematics, 1992, P. 5–
6.
414 W.I. Fushchych

?u ?u ? 2 u
u? u? u?? =
u = u(x0 , x1 , . . . , xn ), ,
?x? ?x? ?x? ?x?

?2u
det uµ? ? det ;
?xµ ?x?

?2E ?2H ?E ?H
? v 2 ?E = 0, ? v 2 ?H = 0, v = v E, H, , ;
2 2
?t ?t ?x ?x

?vµ
v 2 ? v? v ? = v 0 ? v 1 ? v 2 ? v 3 ;
2 2 2 3
v? = 0,
?x?

? ?
+ vk vl = Fl (v1 , v2 , v3 ), l = 1, 2, 3;
?t ?xk

?2u ?2u
= F (u, ?, 2u);
?xµ ?x? ?xµ ?x?

? ?u ? ? ?u ?
+ + u = F (u).
?t ?xb ?xb ?t ?xa ?xa

In the above formulae a, b, c, F , F1 arbitrary smooth

1. Fushchych W., Shtelen V., Serov M., Symmetry and exact solutions of equations of nonlinear
mathematical physics, Kiev, Naukova Dumka, 1989 (in Russian).
2. Fushchych W.I., A new method of study of the group properties, Dokl. AN USSR, 1979, 246, ¹ 4,
846–850.
3. Fushchych W.I., Nikitin A.G., Symmetry of equations of quantum mechanics, Moscow, Nauka, 1990
(in Russian).
4. Ovsyannikov L.V., Group analysis of differential equations, Moscow, Nauka, 1978.
5. Olver P., Applications of Lie groups to differential equations, New York, Springer, 1986.
6. Fushchych W.I., On symmetry and exact solutions of multidimensional nonlinear wave equations,
Ukr. Math. J., 1987, 39, ¹ 1, 116–123.
W.I. Fushchych, Scientific Works 2002, Vol. 4, 415–431.

Conditional symmetry of equations
of nonlinear mathematical physics
W.I. FUSHCHYCH

1. Introduction. In this paper we present some results on conditional symmetry
of nonlinear equations of mathematical and theoretical physics which were obtained
in the Institute of Mathematics of Ukrainian Academy of Science. The term and the
concept of “conditional symmetry of equation” or “conditional invariance” had been
introduced in [1–10].
Speaking about the conditional symmetry of an equation, we mean of the symmet-
ry of some subset of its solutions. To be a constructive one, such general definition
needs, some more details. To study conditional symmetry means to give analytical
description of conditions (constraints) for the set of solutions of an equation under
study picking out subsets having wider (or another) symmetry properties than the
whole set of solutions. Having carried out such description one can obtain solutions
which cannot be obtained within the framework of the classical Vie approach (as it
is known, in the Lie approach reduction, of the multi-dimensional partial differential
equation (PDE) to equations with less number of independent variables is carried out
by means of symmetry of the set of its solutions in a whole).
Euler, Bateman, Lie, Smirnov and Sobolev (1932) and many other classics used
implicitly symmetry of subsets of solutions for linear d’Alembert and Laplace equa-
tions to construct their exact solutions. Not long ago Bluman and Cole [11] suggested
the “non-classical method of solutions invariant under group” for the linear heat
equation. Olver and Rosenau (1986) [12] constructed solutions of the one-dimensional
nonlinear acoustics equation

u00 = ? 2 u/?t2 , u11 = ? 2 u/?x2 (1)
u00 = uu11 ,

which cannot be obtained by means of Lie method. Clarkson and Kruskal suggested
“new method of invariant reduction of the Boussinesq equation”
1
(2)
u00 + u11 + u1111 = 0.
2
Conclusion 1. Using the concept of “conditional symmetry of PDE” we can obtain
the above results within the framework of the unified symmetry approach.
Conclusion 2. The majority of linear and nonlinear equations of mathematical
physics: d’Alembert, Maxwell, Schr?dinger, Dirac, heat, acoustics, KdV equations
o
possess some conditional symmetry.
Note 1. All solutions of the Boussinesq equation (2) constructed by Clarkson and
Kruskal had been obtained independently by Levi and Winternitz [14], and by Fu-
shchych and Serov [10], using the concept of conditional symmetry.
in Symmetry Analysis of Equations of Mathematical Physics, Kyiv, Institute Mathematics, 1992, P. 7–
27.
416 W.I. Fushchych

Let us consider some PDE
L(x, u, u, u, . . . , u) = 0, (3)
s
12

where u = u(x), x ? R(n + 1), u(x) ? R, u is the set of s-th order partial derivatives
s
of u(x).
According to Lie, the equation (3) is invariant under the first-order differential
operator
? ?
X = ? µ (x, u) (4)
+ ?(x, u)
?xµ ?u
if the following condition is satisfied:

? (5)
X L = ?L XL = 0,
s s L=0

where X is the s-th prolongation of the operator X, ? = ?(x, u, u, u, . . . , u) is some
s
12
s
differential expression.
Let us designate by the symbol Q = {Q, . . . , Q} a collection of operators not
1 k
belonging to the invariance algebra (IA) of the equation (3), i.e. Q ? IA.
Definition 1 [2, 5]. We say that the equation (3) is conditionally-invariant under
the operators Q if there exists some additional condition
L1 (x, u, u, u, . . . , u) = 0 (6)
s
12

to be compatible.
The additional condition (6) picks out some subset from the whole set of solutions
of the equation (3). It appears that for many important equations of mathematical
physics such subsets admit the wider symmetry than the whole set of solutions. Such
subsets are to be constructed.
Let the operator Q act on the equation (3) as follows:
(7)
Q L = ?0 L + ?1 L1
s
or
QL = 0,
Lu = 0
s L1 u = 0

where ?0 , ?1 are some differential expressions depending on x, u, u, u, . . . , u, Q is the
s
12 s
s-th prolongation of the operator Q. Then the invariance condition reads
(8)
Q L1 = ?2 L + ?3 L1 ,
s

where ?2 , ?3 are some differential expressions.
The principal problem of our approach is to describe in explicit form equations of
the form (6) which extend symmetry of the equation (6).
The principal and difficult problem can be essentially simplified if one chooses the
following nonlinear first-order PDE as an additional condition (6):
(9)
Qu = 0,
Conditional symmetry of equations of nonlinear mathematical physics 417

where
?µ ? ?/?xµ , ?u ? ?/?u.
Q = j µ (xµ , u)?µ + z(xµ , u)?u , (10)
In this case, the invariance condition for the system of equations (3), (9) takes the
form
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