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

Abstract

Considered in this paper, the Maxwell–Bloch (MB) equations became known after Lamb (1967 Phys. Lett. A 25 181–2; 1971 Rev. Mod. Phys. 43 99–144; 1973 Phys. Rev. Lett. 31 196–199; 1974 Phys. Rev. A 9 422–30). Ablowitz, Kaup and Newell (1974 J. Math. Phys. 15 1852–58) proposed the inverse scattering transform (IST) to the MB 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 by Gabitov, Zakharov and Mikhailov (1985 Teor. Mat. Fiz. 63 11–31). In particular, they gave an approximate solution of the mixed problem to the MB equations in the domain x, t ∈ (0, L) × (0, ∞) and, on this basis, a description of the phenomenon of superfluorescence. It was emphasized in Gabitov et al (1985 Teor. Mat. Fiz. 63 11–31) 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 a simultaneous spectral analysis of both the Lax operators and matrix Riemann–Hilbert (RH) problems. Firstly, we introduce appropriate compatible solutions of the corresponding Ablowitz–Kaup–Newell–Segur equations and then we suggest such a matrix RH problem which corresponds to the mixed problem for MB equations. Secondly, 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 Ablowits (1974 J. Math. Phys. 15 1852–58) and Gabitov et al (1985 Teor. Mat. Fiz. 63 11–31), solutions to the mixed problem studied below in this paper and solutions with periodic (finite-gap) boundary conditions. The type of solution is defined by specialization of the conjugation contour and jump matrix. Suggested matrix RH problems will be useful for studying the long time/long distance () asymptotic behavior of solutions to the MB equations using the Deift–Zhou method of steepest decent.