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The inverse scattering transform is developed to solve the Maxwell-Blochsystem of equations that describes two-level systems with inhomogeneousbroadening, in the case of optical pulses that do not vanish at infinity in thefuture. The direct problem, which is formulated in terms of a suitably-defineduniformization variable, combines features of the formalism with decaying aswell as non-decaying fields. The inverse problem is formulated in terms of a$2\times 2$ matrix Riemann-Hilbert problem. A novel aspect of the problem isthat no reflectionless solutions can exist, and solitons are always accompaniedby radiation. At the same time, it is also shown that, when the medium isinitially in the ground state, the radiative components of the solutions decayupon propagation into the medium, giving rise to an asymptoticallyreflectionless states. Like what happens when the optical pulse decays rapidlyin the distant past and the distant future, a medium that is initially exciteddecays to the stable ground state as $t\to \infty$ and for sufficiently largepropagation distances. Finally, the asymptotic state of the medium and certainfeatures of the optical pulse inside the medium are considered, and theemergence of a transition region upon propagation in the medium is brieflydiscussed.

On Maxwell-Bloch systems with inhomogeneous broadening and one-sided nonzero background