Diverging probability-density functions for flat-top solitary waves
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 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 η exhibits loglognormal divergence near the maximum possible amplitude ηm , 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 η to ηm 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 β become random. We therefore study analytically and by Monte Carlo simulations the PDF of the parameter p , where p=η/(1-ɛsβ/2) and ɛs 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.
- Publication:
-
Physical Review E
- Pub Date:
- August 2009
- DOI:
- 10.1103/PhysRevE.80.026602
- arXiv:
- arXiv:0906.3001
- Bibcode:
- 2009PhRvE..80b6602P
- Keywords:
-
- 05.45.Yv;
- 05.40.-a;
- 47.54.-r;
- 42.65.Tg;
- Solitons;
- Fluctuation phenomena random processes noise and Brownian motion;
- Pattern selection;
- pattern formation;
- Optical solitons;
- nonlinear guided waves;
- Nonlinear Sciences - Pattern Formation and Solitons
- E-Print:
- 9 pages, 6 figures. Submitted to Phys. Rev. E