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Information × Registration Number 0223U002758, 0122U200505 , R & D reports Title Analytical and asymptotic methods in theory of integrable nonlinear differential equations and mathematical control theory popup.stage_title Head Shepelskyi Dmytro H., д.ф.-м.н. Registration Date 05-03-2023 Organization B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine popup.description2 The work is aimed at developing methods of qualitative analysis of initial and initial boundary value problems with classic and non-standard boundary conditions for classic and new nonlinear integrable partial differential equations, as well as methods for studying controllability problems for linear equations with variable coefficients. In the theory of initial problems for integrable nonlinear equations, the asymptotic behavior of solutions of initial problems for the Korteweg-de Vries equation with step-like initial data associated with the so-called shock waves is described and substantiated. Two methods were developed for this problem: the classical method of the inverse scattering problem in the form of the Marchenko equations and the method of the Riemann-Hilbert problem. The soliton asymptotics of the solution are obtained and substantiated under the condition that the initial data approach their backgrounds with a speed characterized by a given finite moment. In the theory of nonlocal integrable nonlinear equations, the concept of global solutions of initial problems for the nonlocal nonlinear Schrödinger equation in the case when the solutions are singular in a certain set of points of the "space-time" plane is proposed and developed. The concept is based on the application of the representation of the solution in terms of the solution of an associated Riemann-Hilbert problem. The characterization of the set of singular points of solutions is obtained. In the control theory, the transformation operators for the Sturm-Liouville problem are adapted for their application to the heat equation with variable coefficients on the semi-axis governed by the Neumann condition. With their help, a one-to-one correspondence was established between the set of solutions of a controlled system with variable coefficients and the set of solutions of the corresponding controlled system with constant coefficients on the half-axis. Product Description popup.authors popup.nrat_date 2023-03-05 Close