Complete linearization of a mixed problem to the Maxwell-Bloch equations by matrix Riemann-Hilbert problem

Abstract

Considered in this paper the Maxwell-Bloch (MB) equations became known after Lamb [1-4]. In [5] Ablowitz, Kaup and Newell proposed the inverse scattering transform (IST) to the Maxwell-Bloch equations for studying a physical phenomenon known as the self-induced transparency. A description of general solutions to the MB equations and their classification was done in [6] by Gabitov, Zakharov and Mikhailov. In particular, they gave an approximate solution of the mixed problem to the MB equations in the domain $x,t\in(0,L)\times (0,\infty)$ and, on this bases, a description of the phenomenon of superfluorescence. It was emphasized in [6] that the IST method is non-adopted for the mixed problem. Authors of the mentioned papers have developed the IST method in the form of the Marchenko integral equations. We propose another approach for solving the mixed problem to the MB equations in the quarter plane. We use matrix Riemann-Hilbert (RH) problems and simultaneous spectral analysis of the both Lax operators. First, we introduce appropriate compatible solutions of the corresponding Ablowitz-Kaup-Newell-Segur (AKNS) equations and then we suggest such a matrix RH problem which corresponds to the mixed problem for MB equations. Second, we generalize this matrix RH problem, prove a unique solvability of the new RH problem and show that the RH problem (after a specialization of jump matrix) generates the MB equations. As a result we obtain solutions defined on the whole line and studied in [5] and [6], solutions to the mixed problem studied below in this paper and solutions with a periodic (finite-gap) boundary conditions. The kind of solution is defined by the specialization of conjugation contour and jump matrix.