# Journal of Nonlinear Mathematical Physics

Volume 4, Issue 1-2, May 1997

**Research Article**

## 1. Gauge Transformations for a Family of Nonlinear Schrödinger Equations

Gerald A. Goldin

Pages: 6 - 11

An enlarged gauge group acts nonlinearly on the class of nonlinear Schrödinger equations introduced by the author in joint work with Doebner. Here the equations and the group action are displayed in the presence of an external electromagnetic field. All the gauge-invariants are listed for the coupled...

**Research Article**

## 2. Proper-Time Relativistic Dynamics and the Fushchych-Shtelen Transformation

Tepper L. Gill, James Lindesay, M.F. Mahmood, W.W. Zachary

Pages: 12 - 27

We report on a new formulation of classical relativistic Hamiltonian mechanics which is based on a proper-time implementation of special relativity using a transformation from observer proper-time, which is not invariant, to system proper-time which is a canonical contact transformation on extended phase-space....

**Research Article**

## 3. Symmetries of Separating Nonlinear Schrödinger Equations

George Svetlichny

Pages: 28 - 35

We review here the main properties of symmetries of separating hierarchies of nonlinear Schrödinger equations and discuss the obstruction to symmetry liftings from (n)particles to a higher number. We argue that for particles with internal degrees of freedom, new multiparticle effects must appear at each...

**Research Article**

## 4. On the Self-Similar Solutions of Generalized Hydrodynamics Equations and Nonlinear Wave Patterns

V.A. Danylenko, V.A. Vladimirov

Pages: 36 - 43

Solutions of the system of dynamical equations of state and equations of the balance of mass and momentum are studied. The system possesses families of periodic, quasiperiodic and soliton-like invariant solutions. Self-similar solutions of this generalized hydrodynamic system are studied. Various complicated...

**Research Article**

## 5. On New Galilei- and Poincare-Invariant Nonlinear Equations for Electromagnetic Field

Wilhelm Fushchych, Ivan Tsyfra

Pages: 44 - 48

Nonlinear systems of differential equations for E and H which are compatible with the Galilei relativity principle are proposed. It is proved that the Schrödinger equation together with the nonlinear equation of hydrodynamic type for E and H are invariant with respect to the Galilei algebra. New Poincare-invariant...

**Research Article**

## 6. On Integration of the Nonlinear d'Alembert-Eikonal System and Conditional Symmetry of Nonlinear Wave Equations

Renat Z. Zhdanov

Pages: 49 - 61

We study integrability of a system of nonlinear partial differential equations consisting of the nonlinear d'Alembert equation 2u = F(u) and nonlinear eikonal equation uxµ uxµ = G(u) in the complex Minkowski space R(1, 3). A method suggested makes it possible to establish necessary and sufficient compatibility...

**Research Article**

## 7. Functional Algebras and Dimensional Reduction in the LPDEs Integration Problem

A.V. Shapovalov, I.V. Shirokov

Pages: 62 - 68

**Research Article**

## 8. Variational Symmetry in Non-integrable Hamiltonian Systems

Umeno Ken

Pages: 69 - 77

We consider the variational symmetry from the viewpoint of the non-integrability criterion towards dynamical systems. That variational symmetry can reduce complexity in determining non-integrability of general dynamical systems is illustrated here by a new non-integrability result about Hamiltonian systems...

**Research Article**

## 9. The Symmetry Reduction of Nonlinear Equations of the Type □u + F(u, u1)u0 = 0 to Ordinary Differential Equations

Leonid Barannyk, Halyna Lahno

Pages: 78 - 88

The reduction of two nonlinear equations of the type □u+F(u, u1)u0 = 0 with respect to all rank three subalgebras of a subdirect sum of the extended Euclidean algebras A ~E(1) and A ~E(3) is carried out. Some new invariant exact solutions of these equations are obtained.

**Research Article**

## 10. Symmetries of Vector Exterior Differential Systems and the Inverse Problem in Second-Order Ostrograds'kii Mechanics

R.Ya. Matsyuk

Pages: 89 - 97

Symmetries for variational problems are considered as symmetries of vector-valued exterior differential systems. This approach is applied to equations for the classical spinning particle.

**Research Article**

## 11. Conditional Symmetry and Exact Solutions of a Nonlinear Galilei-Invariant Spinor Equation

Andrey Andreytsev

Pages: 98 - 101

Reduction of a nonlinear system of differential equations for spinor field is studied. The ansatzes obtained are shown to correspond to operators of conditional symmetry of these equations.

**Research Article**

## 12. Symmetry Reduction of Poincaré-Invariant Nonlinear Wave Equations

A.F. Barannyk, Yu.D. Moskalenko

Pages: 102 - 106

Reduction of multidimensional Poincaré-invariant equations to ordinary differential equations and 2-dimensional equations is considered.

**Research Article**

## 13. New Spherically Symmetric Solutions of Nonlinear Schrödinger Equations

Roman M. Cherniha

Pages: 107 - 113

New soliton-like spherically symmetric solutions for nonlinear generalizations of the Schrödiner equation are constructed. A new nonlinear projective invariant Schrödiner equation is suggested and formulae of multiplication of its solutions are found.

**Short Communication**

## 14. Conditional Invariance and Exact Solutions of a Nonlinear System

Pavlo Myronyk, Natalia Bubenchikova

Pages: 114 - 116

The Lie and Q-conditional invariance of one nonlinear system of PDEs of the thirdorder is searched. The ansatze have been built which reduce the PDEs system to ODEs. The classes of exact solutions of the given system are obtained. The relation between the Korteweg-de Vries equation and Harry-Dym equation...

**Research Article**

## 15. Contact Transformations in Classical Mechanics

Yurij Yaremko

Pages: 117 - 123

Transformations of coordinates of points in an infinite-dimensional graded vector space, the so-called contact transformations, are examined. An infinite jet prolongation of the extended configuration space of N spinless particles is the subspace of this vector space. The dynamical equivalence among...

**Research Article**

## 16. Continuity Equation in Nonlinear Quantum Mechanics and the Galilei Relativity Principle

Wilhelm Fushchych, Vyacheslav Boyko

Pages: 124 - 128

Classes of the nonlinear Schrödinger-type equations compatible with the Galilei relativity principle are described. Solutions of these equations satisfy the continuity equation.

**Short Communication**

## 17. Reduction and Exact Solutions of the Monge-Ampere Equation

Oleg Leibov

Pages: 129 - 131

On the basis of a subgroup structure of the Poincaré group P(1, 3) the ansatzes which reduce the MongeAmpere 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...

**Research Article**

## 18. Symmetry Properties and Exact Solutions of the Fokker-Planck Equation

Valery Stohny

Pages: 132 - 136

Symmetry properties of some Fokker-Planck equations are studied. In the one-dimensional case, when symmetry groups turn out to be six-parameter ones, this allows to find changes of variables to reduce such Fokker-Planck equations to the one-dimensional heat equation. The symmetry and the family of exact...

**Short Communication**

## 19. Conditional Symmetry of Equations of Nonstationary Filtration and of the Nonlinear Heat Equation

Alla Vorobyova

Pages: 137 - 140

Conditional symmetry of the nonlinear gas filtration equation is studied. The operators obtained enabled to constract ansatzes reducing this equation to ordinary differential equations and to obtain its exact solutions.

**Research Article**

## 20. The Algebra A ~P (1, 3) Invariants and Their Application to the Theory of Born-Infeld Field

Iryna A. Mishchenko

Pages: 141 - 145

The algebra A ~P (1, 3) invariants were found. These invariants allowed to reduce the Born-Infeld equation. After the reduction some solutions of the equation were found.

**Short Communication**

## 21. Reduction and Some Exact Solutions of the Multidimensional Liouville Equation

I.I. Yuryk

Pages: 146 - 148

Exact solutions of the multidimensional Liouville equation are constructed.

**Short Communication**

## 22. On Classes of Lie Solutions of MHD Equations, Expressed via the General Solution of the Heat Equation

Victor Popovych

Pages: 149 - 151

Large classes of Lie solutions of the MHD equations describing the flows of a viscous homogeneous incompressible fluid of finite electrical conductivity are constructed. These classes contain a number of arbitrary functions of time and the general solutions of the heat equation.

**Short Communication**

## 23. On the Symmetry of Some Nonlinear Generalization of a Vector Subsystem of the Maxwell Equations

Volodymyr Smalij

Pages: 152 - 154

The problem of studying the maximal Lie symmetry of some nonlinear generalization of the vector subsystem of the Maxwell equations is completely solved.

**Research Article**

## 24. A Model of Control of an Equation for Two-Electron Interaction

V.I. Sokolov

Pages: 155 - 160

Quantum Schrödinger equation, describing dynamical spin-interaction of two electrons with external magnetic field, is considered as an object for cybernetic research. Indeed, because of having a possibility to change external magnetic field, we can influence the interaction of particles. The algorithm...

**Research Article**

## 25. On Relativistic Mass Spectra of a Two-Particle System

Volodymyr Tretyak, Volodymyr Shpytko

Pages: 161 - 167

A relativistic two-particle system with time-asymmetric scalar and vector interactions in the two-dimensional space-time is considered within the frame of the front form of dynamics using the dynamical symmetry approach. The mass-shell equation may be represented in terms of the nonlinear canonical realization...

**Research Article**

## 26. Point Symmetries of Controlled Systems and Their Applications

Victor I. Lehenkyj

Pages: 168 - 172

A problem of finding point symmetries of controlled systems is discussed, basic theorems and algorithms are formulated. The application to some problems of flight dynamics is suggested.

**Short Communication**

## 27. Mathematical Simulation of Heat Transfer in Relaxing Media

V.M. Bulavatsky, I.I. Yuryk

Pages: 173 - 174

We find a numerically-analytical solution of a boundary problem for the third-order partial differential equation, which describes the mass and heat transfer in active media.

**Research Article**

## 28. On Finite-Gap Elliptic Solutions of the KdV Equation

A.M. Korostil'

Pages: 175 - 179

We present a simple and general method for calculation of finitegap elliptic solutions of the KdV equation as an isospectral deformation of Schrödinger potential based on their representation by rational functions of the elliptic Weierstrass functions.

**Research Article**

## 29. Nonlinear Quantum Dynamical Equation for Self-Acting Electron

V.P. Oleinik

Pages: 180 - 189

From the action principle, the quantum dynamical equation is obtained both relativistically and gauge invariant, which is analogous to the Dirac equation and describes behaviour of an arbitrary number of self-acting charged particles. It is noted that solutions of this equation are indicative of the...

**Research Article**

## 30. States of a Charged Particle with a Tensor-Like Mass in External Constant Magnetic Field

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

Pages: 190 - 194

Solutions of the Schrödinger equation for a particle with a tensor-like mass are considered. It is shown that the problem of determination of the coherent states in this case is reduced to integration of the nonlinear system of ordinary differential equations.

**Research Article**

## 31. On the Braided FRT-Construction

Yurij Bespalov

Pages: 195 - 205

A fully braided analog of the Faddeev-Reshetikhin-Takhtajan construction of a quasitriangular bialgebra A(X, R) is proposed. For a given pairing C, the factor-algebra A(X, R; C) is a dual quantum braided group. Corresponding inhomogeneous quantum group is obtained as a result of generalized bosonization....

**Short Communication**

## 32. On Quantum Systems of Particles with Singular Magnetic Interaction

W.I. Skrypnik

Pages: 206 - 208

For systems of particles with singular magnetic interation for special choice of a selfadjoint extension of the Hamiltoniam equilibrium reduced density matrices are calculated in the thermodynamic limit for simplest pair magnetic potentials.

**Research Article**

## 33. Two-Point Boundary Optimization Problem for Bilinear Control Systems

Alla V. Vinogradskaya

Pages: 209 - 213

This paper presents a new approach to the optimization problem for the bilinear system x = {x, } (1) based on the well-known method of continuous parametric group reconstruction using of its structure constants defined by the Brockett equation z = {z, }. (2) Here x is the system state vector, {·, ·}...

**Short Communication**

## 34. One-Dimensional Discontinuous Flows in Relativistic Magnetohydrodynamics

V.I. Zhdanov, P.V. Titarenko

Pages: 214 - 217

We consider discontinuous flows of relativistic magnetic fluid with a general equation of state that is not supposed to be normal in the sense of Bethe and Weyl. The criteria of admissibility of the shock waves without a supposition of the relativistic version of the convexity condition are obtained....

**Short Communication**

## 35. Radiative Friction in the Lorentz-Dirac Equation and its Decomposition in the Interaction Constant

A.A. Borghardt, D.Ya. Karpenko

Pages: 218 - 220

The nonlinear Lorentz-Dirac equation of motion for charged particle, if one takes into account radiative friction can be written in dimensionless variables. Then, there is a possibility of introducing the constant of fine structure and following approximate solving it. This result may be used for more...

**Research Article**

## 36. Completely Integrable Generalized C. Neumann Systems on Several Symplectic Submanifolds

Zhijun Qiao

Pages: 221 - 230

New completely integrable generalized C. Neumann systems on several symplectic submanifolds are presented, and the relations between the generalized C. Neumann systems and the spectral problems are further discussed in this paper. In particular, a new eigenvalue problem is proposed in Part 3.3.

**Research Article**

## 37. Sound Attenuation in a Circular Duct of a Viscous Medium in the Absence of Mean Flow

Mina B. Abd-El-Malek, Sarwat N. Hanna

Pages: 231 - 240

An analytical method to study the effect of viscosity of a medium and the wave number on sound propagation and sound attenuation numbers in circular ducts has been presented. The method is based on the variation of parameters of the solution corresponding to the case of inviscid acoustic waves in circular...

**Short Communication**

## 38. The Scenario of Appearance of Periodic and Chaotic Modes in a Model with Control over Biosynthetic Processes

T.P. Kovalenko

Pages: 241 - 244