The competition between nonlinearity, dispersion and randomness in signal propagation

The phenomenon of self-induced transparency (SIT) is reinterpreted in the c ontext of competition between randomness, nonlinearity and dispersion, and furthermore the problem is recast to show that it is isomorphic to a proble m of the nonlinear Schroedinger (NLS) type with a random potential in which the randomness is manifested spatially. It is shown that, under mild assum ptions, the SIT result continues to hold when we replace the uniform medium of inhomogeneously broadened two-level atoms by a series of intervals in e ach of which the frequency mismatch is randomly chosen from some distributi on. The exact solution of this problem confirms and reveals the reason for the fact that nonlinearity 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 int egrable combination of NLS and a modified SIT equation. Finally, some gener alizations are made to describe a broad class of integrable systems which c ombine randomness, nonlinearity and dispersion.