Complete linearization of a mixed problem to the Maxwell-Bloch equations by matrix Riemann-Hilbert problems (Q2851177)

Maxwell-Bloch (MB) equations are known as early as 1967 in the study of pulse propagation in atomic media by Lamb. \textit{M. J. Ablowitz}, \textit{D. Kaup} and \textit{A. C. Newell} [``Coherent pulse propagation, a dispersive, irreversible phenomenon'', J. Math. Phys. 15, 1852--1858 (1974)] proposed the inverse scattering transform (IST) to study the physical phenomenon of self-induced transparency (SIT). \textit{I. R. Gabitov}, \textit{V. E. Zakharov} and \textit{A. V. Mikhailov} [``Maxwell-Bloch equations and inverse scattering transform method'', Teor. Mat. Fiz. 63, No. 1, 11--31 (1985)] have shown that the IST method is non-adopted for the mixed problem. An alternative approach is presented in this paper to show that the IST is applicable for the mixed problem of the MB equations in the quarter plane using a simultaneous spectral analysis of both the Lax operators and the matrix Riemann-Hilbert (RH) problems, and hence the problem is completely linearizable. Among the generated derived solutions are: previously studied solutions by Abitov et al. [loc. cit.], Gabitov et al. [loc. cit.] for \(t\in \mathbb{R}\), \(x\in \mathbb{R}_+\), solutions to the mixed problem of the MB equations with decreasing or periodic input pulse for \(t,x\in \mathbb{R}_+\). Suggested matrix RH problems are useful to study the long time/large distance \((x\in \mathbb{R}_+)\) asymptotic behavior of solutions of the MB equations using the Deift-Zhou method at steepest descent.