arXiv subjects
Publications and source records attributed to E. Sh. Gutshabash.
General formulas for the construction of exact solutions of the equation of the minimal surface in $R^3$, which appears in various physical problems, have been derived by the Zakharov-Shabat "dressing" method. Particular examples are considered.
A generalization of Kermack-McKendick model of epidemics to the case of inhomogeneous susceptibility of population is proposed. Some quantitative and qualitative features of epidemic process development in this situation are established.
Three integrable models - the deformed Heisenberg, Landau - Lifschits and Ishimori magnets are written in terms of the stereographic projection. The Hamiltonians of these models are obtained and certain questions related to the existence of exact solutions are studied. In particular, the stability of solitions is studied for the Heisenberg marnet, simplest stationary solutions are obtained for the Landau - Lifschits magnet, and Hamiltonians and topological charges are calculated for several known solutions of the Ishimori model
The detailed analysis of model of the hydrodynamical vortice on a plane is executed. The derivation of the corresponding equation and its simplified variant is given, a partial solutions are constructed. The question on application of Darboux transformation is considered.
The Lax representation for the nonstationary Schrödinger equation with rather arbitrary potential is proposed. Some examples of the construction of exact solutions are given by means of Darboux Transformation method.
The integration procedure based on the the generalized Darboux transform is suggested for the Ishimori magnet model. Exact solutions are constructed for the model of background of spiral structures. The possibility of phase transition in the system is hypothesized.
Two approaches to solution of the two-dimensional Helmholtz equation with a "wave number" are proposed. The results can be applied both in numerical areas of physics and in the theory of nonlinear equations. The first approach is based on the requirement of the covariance of equation under the generalized Darboux transformation (Moutard transformation). It allows to construct a new solution of equation, using a given initial solution of the equation. Simultaneously we obtain the "dressing" relation for the "wave number". The simplest examples of the approach are considered in detail. In the second approach the Green-Cauchy formula (the $\bar \partial$-method) is applied to reduce the solution of the equation to the solution of a system of singular integral equations.
Gauge transformation properties for an associated linear system of model Ishimori's magnet model have been discussed. Explicit formulas for the gauge transformation matrix have been obtained. Darboux Transformation has been suggested and appropriate dressing relations have been found.
The two-dimensional generalization of the Polyelectolite model of DNA is proposed. It is reduced to the boundary problem for nonlinear completely integrable equation $\sh$-Gordon. In the linearisible version the exact solution is constructed and its asymptotic is found. The soliton solutions of nonlinear equation are calculated that allowed tells about the possibility of the structural phase transition in the considered system (DNA+polyelectrolite) on the temperature.
Spiral-logarithmic structure is suggested as a stationary solution of a modified equation for the Heisenberg model, and the single- and N-soliton solutions are constructed on this base.
The application of the Darboux Transformation method to the integrable model of Cylindrically Symmetrical Chiral field has been considered. The associated linear system of matrix equations has been proposed and the properties of symmetrie for its solutions has been obtained. The necessary form of Darboux Transformation has been found and formal one- and N-soliton solutions have been constructed. With the use of Pohlmayer's Transformation the equation sin-Gordon type have been given and the hypothesis about its integrability has been deduced.
We propose a nonlinear $σ$-model in a curved space as a general integrable elliptic model. We construct its exact solutions and obtain energy estimates near the critical point. We consider the Pohlmeyer transformation in Euclidean space and investigate the gauge equivalence conditions for a broad class of elliptic equations. We develop the inverse scattering transform method for the $\sinh$-Gordon equation and evaluate its exact and asymptotic solutions.