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Information × Registration Number 0221U107193, 0121U111968 , R & D reports Title Nonstandard nonlocal and peakon integrable equations: asymptotics and inverse scattering transform popup.stage_title Head Filipkovska Mariia S., Кандидат фізико-математичних наук Registration Date 29-12-2021 Organization B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine popup.description2  The method of the inverse scattering transform (IST) and the associated Riemann-Hilbert (RH) formalism for initial-boundary value problems (IBVPs) for the system of the Maxwell-Bloch (MB) equations with an arbitrary inhomogeneous broadening were developed. The matrix RH problem which generates the solution of the corresponding IBVP for the MB equations was constructed. Also, we have studied the asymptotic behavior of the solution in certain sectors of the space-time plane. The obtained results supplement and develop the theory of nonlinear integrable equations and can be used to study the phenomena of the propagation of electromagnetic wave in a resonant medium with distributed two-level atoms. We investigated the asymptotic stage of modulation instability of nonlocal integrable equations. To this end we considered the initial value problem on the line with symmetric nonzero boundary conditions in the form of the stationary waves for the nonlocal nonlinear Schrödinger equation. This equation serves as a model example of nonlocal integrable equations. We adapted the inverse scattering transform method in the form of the RH problem to the considered Cauchy problem. It was shown that the associated RH problem has important distinctive features comparing with the local case. In particular, corresponding spectral functions has different symmetries. We have developed the RH problem formalism to the initial value problem for the modified Camassa-Holm equation on the line with non-zero boundary conditions, in the case when the solution is assumed to approach two different constants at different sides of the line. In particular, we have described the direct problem (Jost solutions, scattering relation, symmetries, behaviour at the branch points, discrete spectrum), formulated an inverse problem as RH problem and derived the parametric representation of the solution of this Cauchy problem in terms of the solution of an associated RH problem. Product Description popup.authors popup.nrat_date 2022-03-09 Close