Sidorenko Yu. M.
Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1097–1115
By using the binary Darboux transformations, we construct scattering operators for a Dirac system with special potential depending on 2n arbitrary functions of a single variable. It is shown that one of the operators coincides with the scattering operator obtained by Nyzhnyk in the case of degenerate scattering data. It is also demonstrated that the scattering operator for the Dirac system is either obtained as a composition of three Darboux self-transformations or factorized by two operators of binary transformations of special form. We also consider several cases of reduction of these operators.
Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1531-1550
A class of nonlinear nonlocal mappings that generalize the classical Darboux transformation is constructed in explicit form. Using as an example the well-known Davey–Stewartson (DS) nonlinear models and the Kadomtsev–Petviashvili matrix equation (MKP), we demonstrate the efficiency of the application of these mappings in the (2 + 1)-dimensional theory of solitons. We obtain explicit solutions of nonlinear evolution equations in the form of a nonlinear superposition of linear waves.
Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced system
Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 78–97
We present a spatially two-dimensional generalization of the hierarchy of Kadomtsev-Petviashvili equations under nonlocal constraints (the so-called 2-dimensionalk-constrained KP-hierarchy, briefly called the 2d k-c-hierarchy). As examples of (2+1)-dimensional nonlinear models belonging to the 2d k-c KP-hierarchy, both generalizations of already known systems and new nonlinear systems are presented. A method for the construction of exact solutions of equations belonging to the 2d k-c KP-hierarchy is proposed.
Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 252–263
We investigate integrable reductions in the Davey-Stewartson model and introduce the hierarchy of the matrix Burgers equations. By using the method of nonlocal reductions in linear problems associated with the hierarchy of the Davey-Stewartson-II equations, we establish a nontrivial relation between these equations and a system of matrix Burgers equations. In an explicit form, we present reductions of the Davey-Stewartson-II model that admit linearization.
Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 220-235
We construct a hierarchy of Poisson Hamiltonian structures related to an “elliptic” spectral problem and determine the generating operators for the equation of asymmetric chiral 0 (3) — field.
Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 91–104
New types of reduction of the Kadomtsev-Petviashvili (KP) hierarchy are considered on the basis of Sato's approach. As a result, we obtain a new multicomponent nonlinear integrable system. Bi-Hamiltonian structures for the new equations are presented.