Journal of Nonlinear Mathematical Physics

Volume 13, Issue 1, February 2006

1. A Note on Surface Profiles for Symmetric Gravity Waves with Vorticity

Mats EHRNSTROM
Pages: 1 - 8
We consider a nontrivial symmetric periodic gravity wave on a current with nondcreasing vorticity. It is shown that if the surface profile is monotone between trough and crest, it is in fact strictly monotone. The result is valid for both finite and infinite depth.

2. A Note on q-Bernoulli Numbers and Polynomials

A S HEGAZI, M MANSOUR
Pages: 9 - 18
In this paper, we define a new q-analogy of the Bernoulli polynomials and the Bernoulli numbers and we deduced some important relations of them. Also, we dduced a q-analogy of the Euler-Maclaurin formulas. Finally, we present a relation between the q-gamma function and the q-Bernoulli polynomials.

3. Deforming the Lie Superalgebra of Contact Vector Fields on S1|1 Inside the Lie Superalgebra of Superpseudodifferential Operators on S1|1

N BEN FRAJ, S OMRI
Pages: 19 - 33
We classify nontrivial deformations of the standard embedding of the Lie superalgebra K(1) of contact vector fields on the (1,1)-dimensional supercircle into the Lie supealgebra of superpseudodifferential operators on the supercircle. This approach leads to the deformations of the central charge induced...

5. Construction of Special Solutions for Nonintegrable Systems

Sergey Yu VERNOV
Pages: 50 - 63
The Painlev´e test is very useful to construct not only the Laurent series solutions of systems of nonlinear ordinary differential equations but also the elliptic and trigonmetric ones. The standard methods for constructing the elliptic solutions consist of two independent steps: transformation of...

6. Vortex Line Representation for the Hydrodynamic Type Equations

E A KUZNETSOV
Pages: 64 - 80
In this paper we give a brief review of the recent results obtained by the author and his co-authors for description of three-dimensional vortical incompressible flows in the hydrodynamic type systems. For such flows we introduce a new mixed LagrangiaEulerian description - the so called vortex line...

7. The Generalized Version of Dressing Method with Applications to AKNS Equations

Junyi ZHU, Xianguo GENG
Pages: 81 - 89
The generalized dressing method is extended to variable-coefficient AKNS equations, including a variable-coefficient coupled nonlinear Schr¨odinger equation and a variablcoefficient coupled mKdV equation. A general variable-coefficient KP equation is proposed and decomposed into the two 1+1 dimensional...

8. Symbolic Software for the Painlevé Test of Nonlinear Ordinary and Partial Differential Equations

Douglas BALDWIN, Willy HEREMAN
Pages: 90 - 110
The automation of the traditional Painlev´e test in Mathematica is discussed. The package PainleveTest.m allows for the testing of polynomial systems of nonlinear ordinary and partial differential equations which may be parameterized by arbitrary functions (or constants). Except where limited by memory,...

9. The Transition of 2-Dimensional Solitons to 1-Dimensional Ones on Hexagonal Lattices

Betti HARTMANN, Wojtek J ZAKRZEWSKI
Pages: 111 - 116
We study solitons arising in a system describing the interaction of a two-dimensional discrete hexagonal lattice with an additional electron field (or, in general, an exciton field). We assume that this interaction is electron-phonon-like. In our previous paper [4] we have studied the existence of...

10. Gauge Transformation and Reciprocal Link for (2+1)-Dimensional Integrable Field Systems

Blazej M SZABLIKOWSKI
Pages: 117 - 128
Appropriate restrictions of Lax operators which allows to construction of (2+1dimensional integrable field systems, coming from centrally extended algebra of pseuddifferential operators, are reviewed. The gauge transformation and the reciprocal link between three classes of Lax hierarchies are established.

11. Properties of the Dominant Behaviour of Quadratic Systems

A MAHARAJ, P G L LEACH
Pages: 129 - 144
We study the dominant terms of systems of Lotka-Volterra-type which arise in the the mathematical modelling of the evolution of many divers natural systems from the viewpoint of both symmetry and singularity analyses. The connections between an increase in the amount of symmetry possessed by the system...

12. Extensions of 1-Dimensional Polytropic Gas Dynamics

Boris A KUPERSHMIDT
Pages: 145 - 157
1-dimensional polytropic gas dynamics is integrable for trivial reasons, having 2 < 3 components. It is realized as a subsystem of two different integrable systems: an infinite-component hydrodynamic chain of Lax type, and a 3-component system not of Lax type.