# Journal of Nonlinear Mathematical Physics

Volume 3, Issue 1-2, May 1996

**Research Article**

## 1. Integrability of the Perturbed KdV Equation for Convecting Fluids: Symmetry Analysis and Solutions

J.M. Cerveró, O. Zurrón

Pages: 1 - 23

As an example of how to deal with nonintegrable systems, the nonlinear partial differential equation which describes the evolution of long surface waves in a convecting fluid ut + (uxxx + 6uux) + 5uux + (uxxx + 6uux)x = 0, is fully analyzed, including symmetries (nonclassical and contact transformatons),...

**Research Article**

## 2. Lie Symmetries, Infinite-Dimensional Lie Algebras and Similarity Reductions of Certain (2+1)-Dimensional Nonlinear Evolution Equations

M. Lakshmanan, M. Senthil Velan

Pages: 24 - 39

The Lie point symmetries associated with a number of (2 + 1)-dimensional generalizations of soliton equations are investigated. These include the Niznik Novikov Veselov equation and the breaking soliton equation, which are symmetric and asymmetric generalizations respectively of the KDV equation,...

**Research Article**

## 3. On Asymptotic Methods in the Theory of Differential Equations of Mathematical Physics

Mykola Shkil

Pages: 40 - 50

In 1807 Fourier suggested a original method of solving partial differential equations. The method is known to lead to ordinary differential equations containing some arbitrary parameter...

**Research Article**

## 4. Nonlinearized Perturbation Theories

Miloslav Znojil

Pages: 51 - 62

A brief review is presented of the two recent perturbation algorithms. Their common idea lies in a not quite usual treatment of linear Schrödinger equations via nonlinear mathematical means. The first approach (let us call it a quasi-exact perturbation theory, QEPT) tries to get the very zero-order approximants...

**Research Article**

## 5. Conditional and Nonlocal Symmetry of Nonlinear Heat Equation

Mykola I. Serov

Pages: 63 - 67

Conditional symmetry We investigate conditional symmetry in three directions. The first direction is a research of the Q-conditional symmetry. The second direction is studying conditional symmetry when an algebra of invariance is known and an additional condition is unknown. The third direction is the...

**Research Article**

## 6. Differential Operators, Symmetries and the Inverse Problem for Second-Order Differential Equations

P. Morando, S. Pasquero

Pages: 68 - 84

**Research Article**

## 7. Symmetry Analysis of Nonlinear PDE with A "Mathematica" Program SYMMAN

Evgenii M. Vorob'ev

Pages: 85 - 89

Computer-aided symbolic and graphic computation allows to make significantly easier both theoretical and applied symmetry analysis of PDE. This idea is illustrated by applying a special "Mathematica" package for obtaining conditional symmetries of the nonlinear wave equation ut = (u ux)x invariant or...

**Research Article**

## 8. Parasupersymmetries and Non-Lie Constants of Motion for Two-Particle Equations

Violeta Tretynyk

Pages: 90 - 95

We search for hidden symmetries of two-particle equations with oscillator-equivalent potential proposed by Moshinsky with collaborators. We proved that these equations admit hidden symmetries and parasupersymmetries which enable easily to find the Hamiltonian spectra using algebraic methods.

**Research Article**

## 9. Lie Algebras of Approximate Symmetries

Rafail K. Gazizov

Pages: 96 - 101

Properties of approximate symmetries of equations with a small parameter are discussed. It turns out that approximate symmetries form an approximate Lie algebra. A concept of approximate invariants is introduced and the algorithm of their calculating is proposed.

**Research Article**

## 10. Graph Expansions and Graphical Enumeration Applied to Semiclassical Propagator Expansions

S.A. Fulling

Pages: 102 - 110

In recent years T.A. Osborn and his coworkers at the University of Manitoba have extensively developed the well known connected graph expansion and applied it to a wide variety of problems in semiclassical approximation to quantum dynamics [2, 5, 7, 19, 21, 22, 26, 27]. The work I am reporting on attempts...

**Research Article**

## 11. Symmetry in Nonlinear Mechanics: Averaging and Normalization Procedures, New Problems and Algorithms

Alexey K. Lopatin

Pages: 111 - 129

The idea of introducing coordinate transformations to simplify the analytic expression of a general problem is a powerful one. Symmetry and differential equations have been close partners since the time of the founding masters, namely, Sophus Lie (18421899), and his disciples. To this days, symmetry...

**Research Article**

## 12. Perturbed Lie Symmetry and Systems of Non-Linear Diffusion Equations

R.J. Wiltshire

Pages: 130 - 138

The method of one parameter, point symmetric, approximate Lie group invariants is applied to the problem of determining solutions of systems of pure one-dimensional, diffusion equations. The equations are taken to be non-linear in the dependent variables but otherwise homogeneous. Moreover, the matrix...

**Research Article**

## 13. Formal Linearization and Exact Solutions of Some Nonlinear Partial Differential Equations

V.A. Baikov, K.R. Khusnutdinova

Pages: 139 - 146

**Research Article**

## 14. Symmetry Approach in Boundary Value Problems

I.T. Habibullin

Pages: 147 - 151

The problem of construction of boundary conditions for nonlinear equations compatible with their higher symmetries is considered. Boundary conditions for the sineGordon, ZhiberShabat and KdV equations are discussed. New examples are found for the JS equation.

**Research Article**

## 15. Painleve Analysis and Symmetries of the HirotaSatsuma Equation

A.A. Mohammad, M. Can

Pages: 152 - 155

The singular manifold expansion of Weiss, Tabor and Carnevale [1] has been successfully applied to integrable ordinary and partial differential equations. They yield information such as Lax pairs, Bäcklund transformations, symmetries, recursion operators, pole dynamics, and special solutions. On the...

**Research Article**

## 16. On the Poincaré-Invariant Second-Order Partial Equations for a Spinor Field

Stanislav Spichak

Pages: 156 - 159

The different second-order nonlinear partial equations are found that are invariant under the representation D(1 2, 0) D(0, 1 2) of the Poincaré group P(1, 3) and also under conformal group C(1, 3). The some exact solutions are constructed for the one of these equations.

**Research Article**

## 17. On Representations of *-Algebras in Mathematical Physics

Vasyl' L. Ostrovs'kyĭ, Yurii S. Samoilenko

Pages: 160 - 163

**Research Article**

## 18. Invariant Linear Spaces and Exact Solutions of Nonlinear Evolution Equations

Sergey R. Svirshchevskii

Pages: 164 - 169

The paper presents a survey of some new results concerning the approach to construction of explicit solutions for nonlinear evolution equations du/dt = F[u], proposed in [1, 2].

**Research Article**

## 19. Generalization of the Equivalence Transformations

S.V. Meleshko

Pages: 170 - 174

This report is devoted to generalization of the equivalence transformations. Let a system of differential equations be given. Almost all systems of differential equations have arbitrary elements: arbitrary functions or arbitrary constants.

**Research Article**

## 20. Symmetry Group of Vlasov-Maxwell Equations in Plasma Theory

V.F. Kovalev, S.V. Krivenko, V.V. Pustovalov

Pages: 175 - 180

**Research Article**

## 21. Parasupersymmetric Quantum Mechanics with Arbitrary p and N

Zoya Symenoh

Pages: 181 - 185

The generalization of parasupersymmetric quantum mechanics generated by an arbitrary number of parasupercharges and characterized by an arbitrary order of paraquantization is given. The relations for parasuperpotentials are obtained. It is shown that parasuperpotentials can be explicitly expressed via...

**Research Article**

## 22. Second-Order Differential Invariants for Some Extensions of the Poincaré Group and Invariant Equations

Irina Yehorchenko

Pages: 186 - 195

It is well-known that symmetry properties are extremely important for choosing differential equations which can be suitable for description of real physical processes. We present functional bases of second-order differential invariants for various representations of some extensions of the Poincaré group...

**Research Article**

## 23. Symmetries of Euler Equations in Lagrangian Coordinates

Victor Andreev

Pages: 196 - 201

The transition from Eulerian to Lagrangian coordinates is a nonlocal transformation. In general, isomorphism should not take place between basic Lie groups of studied equations. Besides, in the case of plane and rotational symmetric motion hydrodynamic equations in Lagrangian coordinates are partially...

**Research Article**

## 24. Representations of Algebras Associated with a Möbius Transformation

Sergei D. Silvestrov, Hans Wallin

Pages: 202 - 213

The Hilbert space representations of a class of commutation relations associated with a Möbius transformation is studied using results on convergence of continued fractions.

**Research Article**

## 25. Symmetry Properties and Reduction of the Generalized Nonlinear System of Two-Phase Liquid Equations

L.O. Tulupova

Pages: 214 - 218

Let us consider the multidimensional nonlinear system of heat equations u0 = f(v)u; v0 = u, (1) where u = u(x) R1, v = v(x) R1, x = (x0, x) R1+3, is the Laplace operator, f(v) is an arbitrary differentiable function. In this paper the classification of symmetry properties of equations (1) is investigated...

**Research Article**

## 26. Similarity Solutions for a Nonlinear Model of the Heat Equation

Effat A. Saied, Magdy M. Hussein

Pages: 219 - 225

We apply the similarity method based on a Lie group to a nonlinear model of the heat equation and find its Lie algebra.The optimal system of the model is contructed from the Lie algebra. New classes of similarity solutions are obtained.

**Research Article**

## 27. Direct Similarity Solution Method and Comparison with the Classical Lie Symmetry Solutions

Z. Jiang

Pages: 226 - 235

We study the general applicability of the ClarksonKruskal's direct method, which is known to be related to symmetry reduction methods, for the similarity solutions of nonlinear evolution equations (NEEs). We give a theorem that will, when satisfied, immediately simplify the reduction procedure or ansatz...

**Research Article**

## 28. Quasiclassical Solutions of the Schrödinger Equations as a Consequense of the Nonlinear Problem

A.A. Borghardt, D.Ya. Karpenko, D.V. Kashkakha

Pages: 236 - 240

Quasiclassic method of solving of the Schrödinger equation with quadratic Hamiltonian is used to derive solutions of Klein-Fock equation for the particle in the constant magnetic field and the jumping magnetic field.