  We calculate the Lyapunov exponent for the non-Hermitian Zakharov-Shabat
eigenvalue problem corresponding to the attractive non-linear Schroedinger
equation with a Gaussian random pulse as initial value function. Using an
extension of the Thouless formula to non-Hermitian random operators, we
calculate the corresponding average density of states. We analyze two cases,
one with circularly symmetric complex Gaussian pulses and the other with real
Gaussian pulses. We discuss the implications in the context of the information
transmission through non-linear optical fibers.
