Random Perturbations in the Nonlocal Nonlinear Schr¨odinger Equation

Abstract

We discuss the nonlinear nonlocal Schr¨odinger equation a˙ected by random perturbations both in propagational as well as transverse directions. Firstly, we revise the stability properties of fundamental bright solitons in such systems. Both numerical simulations and analytical estimates show that the stability of fundamental bright solitons in the presence of random perturbations increases dramatically with the nonlocality-induced finite correlation length of the noise in the transverse plane. In fact, solitons are practically insensitive to noise when the correlation length of the noise becomes comparable to the extent of the wave packet. Fundamental soliton stability can be characterized by two di˙erent criteria based on the evolution of the Hamiltonian of the soliton and its power. Moreover, a simplified mean field approach is used to calculate the power loss analytically in the physically relevant case of weakly correlated noise. Secondly, we discuss how these criteria and concepts carry over to higher-order soliton solutions. It turns out that while basic results hold, noise induced random phase shifts may trigger additional instability mechanisms which are not relevant in the case of fundamental solitons.