Registration Number
0122U200505,
(
0223U002758
)
R & D request
Title
Analytical and asymptotic methods in theory of integrable nonlinear differential equations and mathematical control theory
Head
Shepelskyi Dmytro H., д.ф.-м.н.
Registration Date
08-07-2022
Organization
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
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The project aims at developing methods of qualitative analysis of initial and initial boundary value problems for new and classical integrable nonlinear partial differential equations, and methods for studying controllability problems for classical linear equations with variable coefficients in unbounded domains. Particularly, we are going to study asymptotically finite-gap solutions of the Korteweg-de Vries (KdV) and the Toda lattice equations. For these problems, we propose to develop and appropriately adapt the classical method of the inverse scattering problem (based on the transformation operators and Marchenko equations for spectral equations from the respective Lax pairs) as well as the method based on the formalism of the Riemann-Hilbert problem. In turn, the developed methods will allow us to study effectively various asymptotic properties of the respective solutions, which can be interpreted as generalized shock and rarefaction waves for respective nonlinear dispersive systems. Also, the developed methods will allow us to study controllability properties for the heat equation in unbounded domains.
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2024-12-09
