Analytical and computational methods for the study of rare event probabilities in dispersive and dissipative waves

ANALYTICAL AND COMPUTATIONAL METHODS FOR THE STUDY OF RARE EVENT PROBABILITIES IN DISPERSIVE AND DISSIPATIVE WAVES by Daniel S. Cargill The main focus of this dissertation is the application of importance sampling (IS) to calculate the probabilities associated with rare events in nonlinear, large-dimensional lightwave systems that are driven by noise, including models for fiber-based optical communication system and mode-locked lasers. Throughout the last decade, IS has emerged as a valuable tool for improving the efficiency of simulating rare events in such systems. In particular, it has shown great success in simulating various sources of transmission impairments found in optical communication systems, with examples ranging from large polarization fluctuations resulting from randomly varying fiber birefringence to large pulse-width fluctuations resulting from imperfections in the optical fiber. In many cases, the application of IS is guided by a low-dimensional reduction of the system dynamics. Combining the low-dimensional reduction with Monte Carlo simulations of the original system has been shown to be an extremely effective scheme for computing, for example, the probability with which a pulse deviates significantly from its initial form due to a random forcing. In the context of nonlinear optics, this might represent a transmission error where the propagation model is the nonlinear Schrödinger equation (NLSE) with additive or multiplicative noise. A shortcoming of this method is that the efficiency of the IS technique depends strongly on the accuracy of the low-dimensional reduction used to guide the simulations. These low-dimensional reductions are often derived from a formal perturbation theory, referred to as soliton perturbation theory (SPT) for the case of soliton propagation under the forced NLSE. As demonstrated here, such reduction methods are often inadequate in their description of the pulse’s dynamics. In particular, the interaction between a propagating pulse and dispersive radiation leads to a radiation-induced drift in a pulse’s phase, which is largely unaccounted for in the reduced systems currently in use. The first part of this dissertation is devoted to understanding the interaction between a pulse and dispersive radiation, leading to the derivation of an improved reduced system based on a variational approach. Once this system is derived and verified numerically, it serves as the basis for an improved IS method that incorporates the dynamics of the radiation, which is subsequently extended to more realistic propagation models. Of particular interest is the case of the NLSE with a periodic modulation of the dispersion constant, referred to as dispersion management (DM), and a related model where this modulation is averaged to give an autonomous, nonlocal equation. Following the nomenclature commonly use in literature, the former (nonautonomous) equation will be referred to as the NLSE+DM and the latter (autonomous) equation as the DMNLSE. A complicating aspect of these more realistic models is that, unlike the NLSE, exact solutions only exist as numerical objects rather than as closed-form solutions, which introduces an addition source of error in the derivation of a reduced system for the pulse dynamics. In the second part of this dissertation, the IS method is extended to the calculation of phase-slip probabilities in mode-locked lasers (MLL). Realistic models for pulse propagation in MLL include the dissipative effects of gain and loss, in addition to nonlocal saturation effects. As a result most of the reduced systems derived for pulse dynamics are extremely complicated, which diminishes their applicability as guides for IS simulations. Therefore, a MLL operating in the soliton propagation regime is considered, where the effects of gain, loss and saturation are treated perturbatively. A simple reduced system for the pulse dynamics is derived for this MLL model, allowing the IS technique to be effectively applied. ANALYTICAL AND COMPUTATIONAL METHODS FOR THE STUDY OF RARE EVENT PROBABILITIES IN DISPERSIVE AND DISSIPATIVE WAVES by Daniel S. Cargill A Dissertation Submitted to the Faculty of New Jersey Institute of Technology and Rutgers, The State University of New Jersey – Newark in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematical Sciences Department of Mathematical Sciences, NJIT Department of Mathematics and Computer Science, Rutgers-Newark