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We consider multisoliton solutions of the matrix KdV equation. We obtain the formulas for changing phases and amplitudes during the interaction of two solitons and prove that no multiparticle effects appear during the multisoliton interaction. We find the conditions ensuring the symmetry of the corresponding solutions of the matrix KdV equation if they are constructed by the matrix Darboux transformation applied to the Schrodinger operator with zero potential.

A Schrodinger operator with a matrix-valued rational potential U(z) is said to have trivial monodromy if all the solutions of the corresponding Schrodinger equations are single-valued in the complex plane for any . A local criterion of this property in terms of the Laurent coefficients of the potential U near its singularities, which are assumed to be regular, is found. It is proved that any such operator with a potential vanishing at infinity can be obtained by a matrix analogue of the Darboux transformation from the Schrodinger operator . This generalizes the well known Duistermaat-Grunbaum result to the matrix case and gives the explicit description of the Schrodinger operators with trivial monodromy in this case.

We consider the problem of steady-state oscillations of a plane body with an inclusion. A solution algorithm is constructed. Computational experiments are described, which establish the applicability of the proposed procedure for analyzing the effect of defects inside the body on wave fields excited by ultrasonic sources.

We consider the problem of determining the stress-strain state of an elastoplastic layer under impulse heating. The theory of small elastoplastic strains with linear hardening is used. A boundary-value problem is obtained for the equations of thermoelasticity whose coefficients at any time are functionals of strain history. A method is developed for solving this problem, based on discretization by space and time variables and application of an appropriate difference scheme. This scheme constructs a recursive evolution process for the state column at the nodes of the space grid. Numerical implementation of the method has demonstrated its reliability and efficiency.

The variational problem of contact equilibrium of a punch and an elastic body is considered. An equivalent formulation of the problem is given in variational inequality form. Existence and uniqueness of the solution is investigated in a particular case. A penalty method is proposed for approximate solution of the problem.

New examples of the Yang-Baxter maps (or set-theoretical solutions to the quantum Yang-Baxter equation) on the Grassmannians, arising from the theory of the matrix KdV equation are discussed. The Lax pairs for these maps are produced using the relations with the inverse scattering problem for the matrix Schrodinger operator

The Schrodinger operators with matrix rational potential, which are D-integrable, i.e., can be intertwined with the pure Laplacian, are investigated. Corresponding potentials are uniquely determined by their singular data which are a configuration of the hyperplanes in Cn with prescribed matrices. We describe some algebraic conditions (matrix locus equations) on these data, which are sufficient for D-integrability. As the examples some matrix generalizations of the Calogero–Moser operators are considered.