Diverging probability-density functions for flat-top solitary waves

We investigate the statistics of flat-top solitary wave parameters in the presence of weak multiplicative dissipative disorder. We consider first propagation of solitary waves of the cubic-quintic nonlinear Schrödinger equation (CQNLSE) in the presence of disorder in the cubic nonlinear gain. We show by a perturbative analytic calculation and by Monte Carlo simulations that the probability-density function (PDF) of the amplitude eta exhibits loglognormal divergence near the maximum possible amplitude eta(m), a behavior that is similar to the one observed earlier for disorder in the linear gain [A. Peleg, Phys. Rev. E 72, 027203 (2005)]. We relate the loglognormal divergence of the amplitude PDF to the superexponential approach of eta to eta(m) in the corresponding deterministic model with linear/nonlinear gain. Furthermore, for solitary waves of the derivative CQNLSE with weak disorder in the linear gain both the amplitude and the group velocity beta become random. We therefore study analytically and by Monte Carlo simulations the PDF of the parameter p, where p = eta/(1-epsilon(s)beta/2) and epsilon(s) is the self-steepening coefficient. Our analytic calculations and numerical simulations show that the PDF of p is loglognormally divergent near the maximum p value.