The competition between non-linearity, dispersion and randomness in signal propagation

The phenomenon of self-induced transparency (SIT) is reinterpreted in the context of competition between randomness, non-linearity and dispersion, and furthermore the problem is recast to show that it is isomorphic to a problem of the non-linear Schroedinger (NLS) type with a random potential in which the randomness is manifested spatially. It is shown that, under mild assumptions, the SIT result continues to hold when we replace the uniform medium of inhomogeneously broadened two-level atoms by a series of intervals in each of which the frequency mismatch is randomly chosen from some distribution. The exact solution of this problem confirms and revelas the reason for the fact that non-linearity can help improve the transparency of the medium. Also, the small amplitude, almost monochromatic limit of SIT is taken and results in a complex envelope equation which turns out to be an exactly integrable combination of NLS and a modified SIT equation. Finally, some generalizations are made to describe a broad class of integrable systems which combine randomness, non-linearity and dispersion.