Diverging probability density functions for flat-top solitary waves

TL;DR

This paper analyzes the statistical behavior of flat-top solitary waves under weak multiplicative disorder, revealing a loglognormal divergence in their amplitude and related parameters near maximum values, using analytical and simulation methods.

Contribution

It introduces a detailed analysis of the probability density functions of solitary wave parameters under disorder, highlighting a novel loglognormal divergence behavior.

Findings

Amplitude PDF exhibits loglognormal divergence near maximum amplitude.

Parameter p's PDF also diverges loglognormally near its maximum.

Disorder causes the wave parameters to become statistically unpredictable near their limits.

Abstract

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 Schr\"odinger 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 et al., Phys. Rev. E {\bf 72}, 027203 (2005)]. We relate the loglognormal divergence of the amplitude PDF to the super-exponential 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 $p=\eta/(1-\varepsilon_s\beta/2)$ and $\varepsilon_s$ 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.