Shepelsky D. Method of the Riemann-Hilbert problem in the theory of inverse problems and integrable equations

The objects are the inverse problems in the theory of propagation of electromagnetic waves in media with complex microstructure and initial boundary value problems for nonlinear integrable equations. The aim is to develop the Riemann-Hilbert method for solving inverse problems and to develop the inverse scattering method for studying initial boundary value problems. A systematic analytic approach has been developed that allowed to prove the uniqueness theorems of the parameter reconstruction from the scattering data in a wide class of models of complex media. The characterization of the boundary values of solutions of nonlinear equations (MKdV, NLS, sine-Gordon, Camassa-Holm) is obtained. Detailed asymptotic formulas are obtained for solutions of initial-boundary value problems for these equations. The results can be used in the theory of integration of nonlinear equations, theory of inverse problems, in the development of methods for solving the parameter synthesis problems.