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Information × Registration Number 0121U111968, ( 0221U107193  0223U000174  ) R & D request Title Nonstandard nonlocal and peakon integrable equations: asymptotics and inverse scattering transform Head Filipkovska Mariia S., Кандидат фізико-математичних наук Registration Date 01-07-2021 Organization B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine popup.description1 The main aim of the project is qualitative analysis of initial and initial-boundary value problems for new nonstandard and classical nonlinear integrable PDEs. Considering these problems leads, on the one hand, to developing the existent methods used for studying such problems and, on the other hand, could provide new important results in the theory of integrable equations interesting both in physical and mathematical perspective. During the project we are going to consider the initial value problems for the nonlocal nonlinear Schrödinger (NNLS) equation and modified Camassa-Holm (mCH) equation (also known as the Fokas-Olver-Rosenau-Qiao equation) – new physically significant integrable models and initial-value problems for the classical system of the Maxwell-Bloch equations. Cauchy problems for the NNLS and mCH equations will be considered with nonzero asymmetric boundary conditions. Such problems are known to be a rich source of the significant nonlinear effects, such as modulation instability, asymptotic solitons, elliptic waves, to name but a few. For the initial-value problems for the NNLS equation and mCH equation we are going to adapt the inverse scattering transform (IST) method and investigate the asymptotics of solutions. Concerning the initial-boundary value problems for integrable equations it is known that, in general, such problems cannot be solved by the IST method without additional information about boundary conditions. But in the case of the Maxwell-Bloch equation it is possible, in principle, to adapt the IST method for the well-posed problem and obtain the large-time asymptotics of the solution. In the framework of our project we are going to develop the IST method for initial-boundary value problems for the system of the Maxwell-Bloch equations with spectral broadening and investigate the large-time asymptotics in different sectors in the spatial-time plane. popup.nrat_date 2024-12-09 Close