Matrix Riemann–Hilbert Problems and Maxwell–Bloch Equations without Spectral Broadening

The Maxwell–Bloch equations have been intensively studied by many authors. The main results are based on the inverse scattering transform and the Marchenko integral equations. However this method is not acceptable for mixed problems. In the paper, we develop a method allowing to linearize mixed problems. It is based on simultaneous spectral analysis of both Lax equations and the matrix Riemann–Hilbert problems. We consider the case of infinitely narrow spectral line, i.e., without spectrum broadening. The proposed matrix Riemann–Hilbert problem can be used for studying temporal/spatial asymptotics of the solutions of Maxwell–Bloch equations by using a nonlinear method of steepest descent.