Importance Sampling for Dispersion-Managed Solitons

Abstract

Abstract. The dispersion-managed nonlinear Schrödinger (DMNLS) equation governs the long-term dynamics of systems which are subject to large and rapid dispersion variations. We present a method to study large, noise-induced amplitude and phase perturbations of dispersion-managed solitons. The method is based on the use of importance sampling to bias Monte Carlo simulations toward regions of state space where rare events of interest-large phase or amplitude variations-are most likely to occur. Implementing the method thus involves solving two separate problems: finding the most likely noise realizations that produce a small change in the soliton parameters, and finding the most likely way that these small changes should be distributed in order to create a large, sought-after amplitude or phase change. Both steps are formulated and solved in terms of a variational problem. In addition, the first step makes use of the results of perturbation theory for dispersion-managed systems recently developed by the authors. We demonstrate this method by reconstructing the probability density function of amplitude and phase deviations of noise-perturbed dispersion-managed solitons and comparing the results to those of the original, unaveraged system.

Universal Behavior of Modulationally Unstable Media

“…The remaining three models (i.e., the DMNLS equation, the thermal media system and the AL system) were integrated using a pseudo-spectral Fourier method in space and an integrating-factor fourth-order Runge-Kutta algorithm in time (e.g., as in [41,47]). For the DMNLS equation ( 5), a computationally efficient algorithm (discussed in detail in [41,47]) was also used to evaluate the double integral. The same ICs as for the NLS equations were considered, and periodic boundary conditions were used for all models.…”

A Path‐Based Method for Simulating Large Deviations and Rare Events in Nonlinear Lightwave Systems

“…We also note that the method as currently implemented may have difficulty if more than one type of pulse deformation or large deviation occurs with roughly equal probability, leading to possible bifurcations of the most probable error mode . In such situations, the iterative procedure described here would need to be modified to incorporate branch switching techniques from bifurcation theory to detect and follow such changes .…”

“…IS has also been used to determine the phase distributions of nonsoliton pulses using a root‐mean‐square approximation . Studies further extended the method to dispersion‐managed (DM) systems by taking advantage of a path‐averaged governing equation and a singular perturbation technique . More recently, a method applicable to arbitrarily shaped pulses was developed ; in this case, the most probable noise configurations were found using a combination of the singular value decomposition (SVD) and the cross‐entropy (CE) method.…”