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 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.

A numerical study of the diverging probability density function of flat-top solitons in an extended Korteweg–de Vries equation

“…Thus, we conduct Monte Carlo simulation to achieve the PDF of a parameter characterizing the soliton amplitude and verify its loglognormal divergence. This finding in turn, concludes that the loglognormal divergence of the amplitude PDF found in [19,20] is not restricted to the solitary waves of NLS type equations. We also note that theoretical analysis of the perturbed KdV type equations is an extremely challenging task due to the substantial effects of radiation.…”

Fast two-beam collisions in a linear optical medium with weak cubic loss in spatial dimension higher than 1

“…The optical medium's cubic loss typically arises due to two-photon absorption (2PA) [7,[14][15][16]. Propagation of optical pulses and optical beams in the presence of cubic loss has been studied in many earlier works, both in weakly perturbed linear media [8,10,[17][18][19][20], and in nonlinear media [21][22][23][24][25][26][27][28][29][30][31][32]. The subject attracted renewed attention in recent years due to the importance of 2PA in silicon nanowaveguides, which are expected to play a major role in many applications in optoelectronic devices [7,14,15,33,34].…”

Gaussian-like and flat-top solitons of atoms with spatially modulated repulsive interactions

“…Flat-top solitons-the solitary waves with constant intensity-are a new type of bright solitons that have been less understood in physics. Flat-top solitons appear as localized coherent structures in several areas of physics from nonlinear optics [51,52] to fluid physics [53] and plasma physics [54]. In terms of nonlinear optics, specifically, Flattop solitons exist in cubic-quintic nonlinear media [55,56], cubic-nonlinear media with non-Hermitian potentials [57], and nonlinear metamaterials [58].…”