# Journal of Nonlinear Mathematical Physics

Volume 3, Issue 3-4, September 1996

**Research Article**

## 1. Symmetries of the Classical Integrable Systems and 2-Dimensional Quantum Gravity: a `Map'

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...

**Research Article**

## 2. Gauge symmetry and the generalization of Hirota's bilinear method

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...

**Research Article**

## 3. Symmetry Reductions of the Lax Pair of the Four-Dimensional Euclidean Self-Dual Yang-Mills Equations

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...

**Research Article**

## 4. Generalized Self-Duality for the Supersymmetric Yang-Mills Theory with a Scalar Multiplet

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.

**Research Article**

## 5. On Poincaré-Invariant Reduction and Exact Solutions of the Yang-Mills Equations

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).

**Research Article**

## 6. On Unique Symmetry of Two Nonlinear Generalizations of the Schrödinger Equation

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.

**Research Article**

## 7. Gauge Classification, Lie Symmetries and Integrability of a Family of Nonlinear Schrödinger Equations

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...

**Research Article**

## 8. A Symmetry Connection Between Hyperbolic and Parabolic Equations

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 that...

**Research Article**

## 9. Discrete Symmetry and Its Use to Find Multi-Soliton Solutions of the Equations of Anisotropic Heisenberg Ferromagnets

N.A. Belov, A.N. Leznov, W.J. Zakrzewski

Pages: 319 - 329

**Research Article**

## 10. Weak and Partial Symmetries of Nonlinear PDE in Two Independent Variables

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 admit...

**Research Article**

## 11. Conditional Symmetry and Exact Solutions of the Multidimensional Nonlinear d'Alembert Equation

A.F. Barannyk, Yu.D. Moskalenko

Pages: 336 - 340

**Research Article**

## 12. The Method of an Exact Linearization of n-order Ordinary Differential Equations

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...

**Research Article**

## 13. Group and Renormgroup Symmetry of Quasi-Chaplygin Media

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 is...

**Research Article**

## 14. Symmetries of the Fokker-Type Relativistic Mechanics in Various Forms of Dynamics

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...

**Research Article**

## 15. Symmetries of the Relativistic Two-Particle Model with Scalar-Vector Interaction

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),...

**Research Article**

## 16. Relativistic Two-Body Problem: Existence and Uniqueness of Two-Sided Solutions to Functional-Differential Equations of Motion

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...

**Short Communication**

## 17. Symmetry of a Two-Particle Equation for Parastates

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].

**Short Communication**

## 18. On Symmetry of the Generalized Breit Equation

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.

**Research Article**

## 19. Nonlinear Maxwell Equations

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.

**Research Article**

## 20. *-Representations of the Quantum Algebra Uq(sl(3))

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.

**Research Article**

## 21. Three-Generation Distler-Kachru Models

Yu.I. Samoilenko, Yu.M. Malyuta, N.N. Aksenov

Pages: 402 - 408

**Research Article**

## 22. Nonlocal Symmetry of Nonlinear Wave Equations

V.A. Tychynin

Pages: 409 - 413

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

**Short Communication**

## 23. Conditional and Lie Symmetry of Nonlinear Wave Equation

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.

**Research Article**

## 24. On Exact Solutions of the Lorentz-Maxwell Equations

Igor Revenko

Pages: 417 - 420

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

**Research Article**

## 25. On Some Generalized Symmetric Integral Operators of Buschman-Erdelyi's Type

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.

**Research Article**

## 26. Multiparameter Deformations of the Algebra gln in Terms of Anyonic Oscillators

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...

**Short Communication**

## 27. On Symmetry Reduction of Nonlinear Generalization of the Heat Equation

Valentyn Marchenko

Pages: 432 - 434

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

**Research Article**

## 28. Symmetry Analysis of the Multidimensional Polywave Equation

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...

**Research Article**

## 29. On Reduction of the Euler Equations by Means of Two-Dimensional Algebras

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.

**Research Article**

## 30. Symmetry Reduction for a System of Nonlinear Evolution Equations

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.

**Research Article**

## 31. Application of Differential Forms to Construction of Nonlocal Symmetries

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 on...

**Research Article**

## 32. Fundamental Solutions of the Axial Symmetric Goursat Problem

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...

**Research Article**

## 33. Derivation of asymptotical formulas for resolution of systems of differential equations with symmetrical matrices

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.

**Research Article**

## 34. On the ten classes of scale-invariant nonlinear wave equations for vector fields

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.

**Short Communication**

## 35. Symmetry Reduction and Exact Solutions of the Eikonal Equation

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...

**Short Communication**

## 36. Symmetry Properties of Generalized Gas Dynamics 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).