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

TL;DR

This paper introduces a novel method using matrix Riemann-Hilbert problems and spectral analysis to solve the mixed problem for Maxwell-Bloch equations, extending solution types beyond previous inverse scattering approaches.

Contribution

The authors develop a new approach employing matrix Riemann-Hilbert problems for the Maxwell-Bloch equations, enabling solutions for the mixed problem in the quarter plane and generalizing previous methods.

Findings

Proves unique solvability of the new Riemann-Hilbert problem.

Generates solutions on the whole line and for periodic boundary conditions.

Extends solution framework beyond inverse scattering transform methods.

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.