Initial-Boundary Value Problem for the Maxwell–Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function

The initial-boundary value problem (IBVP) for the Maxwell–Bloch equations with an arbitrary inhomogeneous broadening and periodic boundary condition is studied. This IBVP describes the propagation of an electromagnetic wave generated by periodic pumping in a resonant medium with distributed two-level atoms. We extended the inverse scattering transform method in the form of the matrix Riemann–Hilbert problem for solving the considered IBVP. Using the system of Ablowitz–Kaup–Newell–Segur equations equivalent to the system of the Maxwell–Bloch (MB) equations, we construct the associated matrix Riemann–Hilbert (RH) problem. Theorems on the existence, uniqueness and smoothness properties of a solution of the constructed RH problem are proved, and it is shown that a solution of the considered IBVP is uniquely defined by the solution of the associated RH problem. It is proved that the RH problem provides the causality principle. The representation of a solution of the MB equations in terms of a solution of the associated RH problem are given. The significance of this method also lies in the fact that, having studied the asymptotic behavior of the constructed RH problem and equivalent ones, we can obtain formulas for the asymptotics of a solution of the corresponding IBVP for the MB equations.