Work

The project belongs to the fundamental science (mathematics) and aims at the investigation of the long time asymptotics of solutions of initial value problems for integrable systems described by new nonstandard nonlocal and peakon nonlinear partial differential equations. Nonlocal equations can be used for modeling the PT (parity-time) symmetric systems, which are known to be state of the art area in modern theoretical physics, whereas peakon equations are useful in modeling waves in shallow water. Particularly, we will deal with solutions of the nonlocal nonlinear Schrödinger equation and modified Camassa-Holm equation satisfying nonzero asymmetric boundary conditions. Our project aims at both the development of the Riemann-Hilbert formalism adapted to nonlocal and peakon problems and usage of this formalism in the asymptotic analysis. The method that we develop will not only lead to the characterization of new asymptotic regimes for particular equations – the nonlocal nonlinear Schrödinger equation and modified Camassa-Holm equation, but also will be of use for studying other nonlocal and peakon systems. Also we are going to develop the inverse scattering transform method and the associated formalism of the Riemann-Hilbert problems for initial-boundary value problems for the system of Maxwell-Bloch equations with spectral broadening. The asymptotic analysis of initial-boundary value problems for the system of the Maxwell-Bloch equations will be carried out and the asymptotics of solutions of these problems in different sectors of the space-time plane will be obtained. The obtained results will make a significant contribution to the theory of initial-boundary value problems for nonlinear integrable equations and will be important for solving practical problems, since they can be used to study the phenomena of propagation of electromagnetic waves, in particular, problems about self-induced transparency, superfluorescence, and a quantum laser attenuator.