R.K. BULLOUGH, P.J. CAUDREY

Pages: 245 - 259

We draw attention to the connections recently established by others between the
classical integrable KdV and KP hierarchies in 1+1 and 2+1 dimensions respectively
and the matrix models which relate to the partition functions of 2-dimensional (1 + 1
dimensional) quantum gravity. The symmetries of the...

Jarmo HIETARINTA

Pages: 260 - 265

One of the most powerful methods for finding and solving integrable nonlinear partial
differential equations is Hirota's bilinear method. The idea behind it is to make first
a nonlinear change in the dependent variables after which multisoliton solutions of
integrable systems can be expressed as polynomials...

M. LEGARÉ

Pages: 266 - 285

The reduction by symmetry of the linear system of the self-dual Yang-Mills equations
in four-dimensions under representatives of the conjugacy classes of subgroups of the
connected part to the identity of the corresponding Euclidean group under itself is
carried out. Only subgroups leading to systems...

V.A. YATSUN, A.M. PAVLYUK

Pages: 286 - 290

Generalized self-duality equations for the supersymmetric Yang-Mills theory with a
scalar multiplet are presented in terms of component fields and superfields as well.

Victor LAHNO

Pages: 291 - 295

Classical ideas and methods developed by Sophus Lie provide us with a powerful tool for
constructing exact solutions of partial differential equations (PDE) (see, e.g., [14]). In
the present paper we apply the above methods to obtain new explicit solutions of the
nonlinear Yang-Mills equations (YME)....

Wilhelm FUSHCHYCH, Roman CHERNIHA, Volodymyr CHOPYK

Pages: 296 - 301

We prove that two nonlinear generalizations of the nonlinear Schrödinger equation are
invariant with respect to a Lie algebra that coincides with the invariance algebra of
the Hamilton-Jacobi equation.

P. NATTERMANN, H.-D. DOEBNER

Pages: 302 - 310

In this contribution we review and summarize recent articles on a family of nonlinear
Schrödinger equations proposed by G.A. Goldin and one of us (HDD) [J. Phys. A.
27, 1994, 17711780], dealing with a gauge description of the family, a classification
of its Lie symmetries in terms of gauge invariants...

Peter BASARAB-HORWATH

Pages: 311 - 318

We give ansatzes obtained from Lie symmetries of some hyperbolic equations which
reduce these equations to the heat or Schrödinger equations. This enables us to
construct new solutions of the hyperbolic equations using the Lie and conditional
symmetries of the parabolic equations. Moreover, we note...

N.A. BELOV, A.N. LEZNOV, W.J. ZAKRZEWSKI

Pages: 319 - 329

Evgenii M. VOROB'EV

Pages: 330 - 335

Nonclassical infinitesimal weak symmetries introduced by Olver and Rosenau and
partial symmetries introduced by the author are analyzed. For a family of nonlinear
heat equations of the form ut = (k(u) ux)x + q(u), pairs of functions (k(u), q(u)) are
pointed out such that the corresponding equations...

A.F. BARANNYK, Yu.D. MOSKALENKO

Pages: 336 - 340

L.M. BERKOVICH

Pages: 341 - 350

Necessary and sufficient conditions are found that the n-order nonlinear and nonautonomous ordinary differential equation could be transformed into a linear equation
with constant coefficients with the help, generally speaking, nonlocal transformation
of dependent and independent variables. These conditions...

Vladimir F. KOVALEV

Pages: 351 - 356

Results of renormgroup analysis of a quasi-Chaplygin system of equations are presented. Lie-Bäcklund symmetries and corresponding invariant solutions for different
"Chaplygin" functions are obtained. The algorithm of construction of a group on a
solution (renormgroup) using two different approaches...

Roman GAIDA, Volodymyr TRETYAK

Pages: 357 - 371

The single-time nonlocal Lagrangians corresponding to the Fokker-type action integrals are obtained in arbitrary form of relativistic dynamics. The symmetry conditions
for such Lagrangians under an arbitrary Lie group acting on the Minkowski space are
formulated in various forms of dynamics. An explicit...

Askold DUVIRYAK

Pages: 372 - 378

A relativistic two-particle model with superposition of time-asymmetric scalar and
vector interactions is proposed and its symmetries are considered. It is shown that
first integrals of motion satisfy nonlinear Poisson-bracket relations which include the
Poincaré algebra and one of the algebras so(1,3),...

V.I. ZHDANOV

Pages: 379 - 384

We study a class of explicitly Poincare-invariant equations of motion (EMs) of two
point bodies with a finite speed of propagation of interactions (combination of retarded and advanced ones) that may be considered as functional-differential equations
or differential equations with deviating argument...

S.P. ONUFRIICHUK, O.I. PRYLYPKO

Pages: 385 - 387

We study hidden symmetry of a two-particle system of equations for parastates. Invariance operators are described for various potentials.
It is a well-known fact that the systems of partial differential equations have a hidden
symmetry, which can not be observed in the classical approach of Lie [1].

S.P. ONUFRIICHUK, O.I. PRYLYPKO

Pages: 388 - 390

In this paper we find the complete set of symmetry operators for the two-particle Breit
equation in the class of first-order differential operators with matrix coefficients. A
new integral of motion is obtained.

G.A. KOTELNIKOV

Pages: 391 - 395

The infinite series of Lorentz and Poincaré-invariant nonlinear versions of the Maxwell
equations are suggested. Some properties of these equations are considered.

L.B. TUROVSKAYA

Pages: 396 - 401

Studied in this paper are real forms of the quantum algebra Uq(sl(3)). Integrable
operator representations of -algebras are defined. Irreducible representations are
classified up to a unitary equivalence.

Yu.I. SAMOILENKO, Yu.M. MALYUTA, N.N. AKSENOV

Pages: 402 - 408

V.A. TYCHYNIN

Pages: 409 - 413

A class of nonlinear wave equations is considered. Symmetry of these equations is
extended using nonlocal transformations.

Victor REPETA

Pages: 414 - 416

Group classification of the nonlinear wave equation is carried out and the conditional
invariance of the wave equation with the nonlinearity F(u) = u is found.

Igor REVENKO

Pages: 417 - 420

New exact solutions are obtained for the systems of classical electrodynamics equations.

N. VIRCHENKO

Pages: 421 - 425

Some new symmetric integral operators with kernels involving the generalized Legendre's function of the first kind Pm,n
k (z) are introduced. Some their applications are
given.

A.M. GAVRILIK, N.Z. IORGOV

Pages: 426 - 431

Generators of multiparameter deformations Uq;s1,s2,...,sn-1
(gln) of the universal enveloping algebra U(gln) are realized bilinearly by means of an appropriately generalized form of anyonic oscillators (AOs). This modification takes into account the
parameters s1, ..., sn-1 and yields usual AOs when...

Valentyn MARCHENKO

Pages: 432 - 434

Reductions and classes of new exact solutions are constructed for a class of Galileiinvariant heat equations.

Olena ROMAN

Pages: 435 - 440

We present symmetry classification of the polywave equation 2l
u = F(u). It is established that the equation in question is invariant under the conformal group C(1, n)
iff F(u) = eu
, n + 1 = 2l or F(u) = u(n+1+2l)/(n+1-2l)
, n + 1 = 2l. Symmetry reduction for the biwave equation 22
u = uk
is carried...

Halyna POPOVYCH

Pages: 441 - 446

A complete set of inequivalent two-dimensional subalgebras of the maximal Lie invariance algebra of the Euler equations is constructed. Using some of them, the Euler
equations are reduced to systems of partial differential equations in two independent
variables which are integrated in quadratures.

Lyudmila BARANNYK

Pages: 447 - 452

In this paper we obtain the maximal Lie symmetry algebra of a system of PDEs.
We reduce this system to a system of ODEs, using some rank three subalgebras of
the finite-dimensional part of the symmetry algebra. The corresponding invariant
solutions of the PDEs are obtained.

S.I. AGAFONOV

Pages: 453 - 457

Differential forms are used for construction of nonlocal symmetries of partial differential equations with conservation laws. Every conservation law allows to introduce a
nonlocal variable corresponding to a conserved quantity. A prolongation technique is
suggested for action of symmetry operators...

A.A. BORGHARDT, D.Ya. KARPENKO, N.Yu. NOSENKO

Pages: 458 - 463

Fundamental solutions (FS) with a given boundary condition on the characteristics of
relativistic problems with axial symmetry are considered. This is so-called the Goursat
problem (GP) or zero plane formalism in Dirac's terminology, or modification of the
proper time method in the Fock-Nambu-Schwinger...

M.I. SHKIL, P.F. SAMUSENKO

Pages: 463 - 467

Asymptotic formulae for resolution of L-diagonal systems of ordinary differential equations with symmetrical matrices are derived.

P.V. MARKO

Pages: 468 - 473

We describe all systems of three equations of the form 2uj = Fj(u1, u2, u3), j = 1, 3
invariant under the extended Poincaré group. As a result, we have obtained ten classes
of ~P(1, 3)-invariant nonlinear partial differential equations for real vector fields.

Ivan FEDORCHUK

Pages: 474 - 477

By means of splitting subgroups of the generalized Poincaré group P(1, 4), ansatzes
which reduce the eikonal equation to differential equations with fewer independent
variables have been constructed. The corresponding symmetry reduction has been
done. By means of the solutions of the reduced equations...

Maria SEROVA

Pages: 478 - 480

We describe a class of generalized gas dynamics equations invariant under the extended
Galilei algebra A ~G(1, n).