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
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.
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...
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...
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...
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...
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,...
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
 we have studied the existence of...
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.
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...
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.