  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.
