# Journal of Nonlinear Mathematical Physics

Volume 2, Issue 3-4, September 1995

**Research Article**

## 1. On Methods of Finding Bäcklund Transformations in Systems with More than Two Independent Variables

B. Kent Harrison

Pages: 201 - 215

Bäcklund transformations, which are relations among solutions of partial differential equationsusually nonlinearhave been found and applied mainly for systems with two independent variables. A few are known for equations like the Kadomtsev-Petviashvili equation [1], which has three independent variables,...

**Review Article**

## 2. Ansatz '95

Wilhelm Fushchych

Pages: 216 - 235

In this talk I am going to present a brief review of some key ideas and methods which were given start and were developed in Kyiv, at the Institute of Mathematics of National Academy of Sciences of Ukraine during recent years.

**Research Article**

## 3. Regular Partially Invariant Submodels of Gas Dynamics Equations

L.V. Ovsyannikov, A.P. Chupakhin

Pages: 236 - 246

The Program SUBMODELS [1] is aimed to exhaust all possibilities derived from the symmetry of differential equations for construction of submodels (i.e., systems of equations of the reduced dimension) which describe classes of exact solutions for initial equations. In the frame of this Program, our paper...

**Research Article**

## 4. Representation of Canonical Commutation Relations in a Gauge Theory, the Aharonov-Bohm Effect, and the Dirac-Weyl Operator

Asao Arai

Pages: 247 - 262

We consider a representation of canonical commutation relations (CCR) appearing in a (non-Abelian) gauge theory on a non-simply connected region in the two-dimensional Euclidean space. A necessary and sufficient condition for the representation to be equivalent to the Schrödinger representation of CCR...

**Research Article**

## 5. On the Classification of Subalgebras of the Galilei Algebras

Leonid Barannyk

Pages: 263 - 268

We investigate the structure of certain types of subalgebras of Galilei algebras and the relationship between the conjugacies of these subalgebras under different groups of automorphisms.

**Research Article**

## 6. Symmetries of Maxwell-Bloch Equations

Pantelis A. Damianou, Paschalis G. Paschali

Pages: 269 - 277

We study symmetries of the real Maxwell-Bloch equations. We give a Lax pair, biHamiltonian formulations and we find a symplectic realization of the system. We have also constructed a hierarchy of master symmetries which is used to generate nonlinear Poisson brackets. In addition we have calculated the...

**Research Article**

## 7. Symplectic Symmetries of Hamiltonian Systems

Ihor Parasyuk

Pages: 278 - 282

The goal of this paper is to describe some interesting phenomena which occur in Hamiltonian systems with symplectic (locally Hamiltonian) symmetries.

**Research Article**

## 8. Determining Equations and Differential Constraints

Oleg V. Kaptsov

Pages: 283 - 291

**Research Article**

## 9. Madelung Representation for Complex Nonlinear D'Alembert Equations in n-Dimensional Minkowski Space

N. Euler, M. Euler

Pages: 292 - 300

**Research Article**

## 10. On Lie Reduction of the Navier-Stokes Equations

Roman Popovych

Pages: 301 - 311

Lie reduction of the Navier-Stokes equations to systems of partial differential equations in three and two independent variables and to ordinary differential equations is described.

**Research Article**

## 11. Symmetry and Nonlocal Ansatzes for Nonlinear Heat Equations

Ivan Tsyfra

Pages: 312 - 318

Operators of nonlocal symmetry are used to construct exact solutions of nonlinear heat equations In [1] the idea of constructing nonlocal symmetry of differential equations was proposed. By using this symmetry, we have suggested a method for finding new classes of ansatzes reducing nonlinear wave equations...

**Research Article**

## 12. Group Analysis of Nonlinear Heat-Conduction Problem for a Semi-Infinite Body

N.A. Badran, M.B. Abdelmalek

Pages: 319 - 328

The transformation group theoretic approach is applied to present an analysis of the nonlinear unsteady heat conduction problem in a semiinfinite body. The application of oneparameter group reduces the number of independent variables by one, and consequently the governing partial differential equation...

**Research Article**

## 13. Symmetry Reduction and Exact Solutions of the EulerLagrangeBornInfeld, Multidimensional MongeAmpere and Eikonal Equations

Vasyl Fedorchuk

Pages: 329 - 333

Using the subgroup structure of the generalized Poincaré group P(1, 4), ansatzes which reduce the EulerLagrangeBornInfeld, multidimensional MongeAmpere and eikonal equations to differential equations with fewer independent variables have been constructed. Among these ansatzes there are ones which...

**Research Article**

## 14. The Singular Manifold Method: Darboux Transformations and Nonclassical Symmetries

P.G. Estévez, P.R. Gordoa

Pages: 334 - 355

We present in this paper the singular manifold method (SMM) derived from Painlevé analysis, as a helpful tool to obtain much of the characteristic features of nonlinear partial differential equations. As is well known, it provides in an algorithmic way the Lax pair and the Bäcklund transformation for...

**Research Article**

## 15. On the Spectral Theory of Operator Pencils in a Hilbert Space

Roman I. Andrushkiw

Pages: 356 - 366

Consider the operator pencil L = A - B - 2 C, where A, B, and C are linear, in general unbounded and nonsymmetric, operators densely defined in a Hilbert space H. Sufficient conditions for the existence of the eigenvalues of L are investigated in the case when A, B and C are K-positive and K-symmetric...

**Research Article**

## 16. Lie Algebras and Superalgebras Defined by a Finite Number of Relations: Computer Analysis

V.P. Gerdt, V.V. Kornyak

Pages: 367 - 373

The presentation of Lie (super)algebras by a finite set of generators and defining relations is one of the most general mathematical and algorithmic schemes of their analysis. It is very important, for instance, for investigation of the particular Lie (super)algebras arising in different (super)symmetric...

**Research Article**

## 17. Galilean-invariant Nonlinear PDEs and their Exact Solutions

Roman M. Cherniha

Pages: 374 - 383

All systems of (n+1)-dimensional quasilinear evolutional second- order equations invariant under the chain of algebras AG(1.n) AG1(1.n) AG2(1.n) are described. The obtained results are illustrated by examples of nonlinear Schrödinger equations.

**Research Article**

## 18. Two-Parameter Deformation of the Oscillator Algebra and (p, q)Analog of Two-Dimensional Conformal Field Theory

Ivan Burban

Pages: 384 - 391

The two-parameter deformation of canonical commutation relations is discussed. The self-adjointness property of the (p, q)-deformed position Q and momentum P operators is investigated. The (p, q)-analog of two-dimensional conformal field theory based on the (p, q)-deformation of the su(1, 1) subalgebra...

**Research Article**

## 19. Similarity Reductions of the Zabolotskaya-Khokhlov Equation with a Dissipative Term

Masayoshi Tajiri

Pages: 392 - 397

Similarity reductions of the Zabolotskaya-Khokhlov equation with a dissipative term to one-dimensional partial differential equations including Burgers' equation are investigated by means of Lie's method of infinitesimal transformation. Some similarity solutions of the Z-K equation are obtained.

**Research Article**

## 20. The Bäcklund and the Galilei Invariant Transformations Constructed by Similarity Variables for Soliton Equations

Shunji Kawamoto

Pages: 398 - 404

The Painlevé-test has been applied to checking the integrability of nonlinear PDEs, since similarity solutions of many soliton equations satisfy the Painlevé equation. As is well known, such similarity solutions can be obtained by the infinitesimal transformation, that is, the classical similarity analysis,...

**Research Article**

## 21. Non-Lie Symmetries and Supersymmetries

Anatolii Nikitin

Pages: 405 - 415

Appeared more than one century ago, the classical Lie approach serves as a powerful tool in investigations of symmetries of partial differential equations. In the last three decades there appear essential generalizations of this approach. They are the modern version of the Lie-Bäcklund symmetries [1],...

**Short Communication**

## 22. Galilei Invariance of the FokkerPlanck Equation with Nonlinearity

Vadym Cherkasenko

Pages: 416 - 417

Consider equation

**Research Article**

## 23. Symmetry Classification of the OneDimensional Second Order Equation of a Hydrodynamic Type

Vyacheslav Boyko

Pages: 418 - 424

The paper contains a symmetry classification of the onedimensional second order equation of a hydrodynamical type L(Lu) + Lu = F(u), where L t + ux. Some classes of exact solutions of this equation are given.