Non-Monte Carlo formulations and computational techniques for the stochastic non-linear Schrِdinger equation

Stochastic ordinary and partial differential equations (SOPDEs) in various forms arise and are successfully utilized in the modeling of a variety of physical and engineered systems such as telecommunication systems, electronic circuits, cosmological systems, financial systems, meteorological and climate systems. While the theory of stochastic partial and especially ordinary differential equations is more or less well understood, there has been much less work on practical formulations and computational approaches to solving these equations. In this paper, we concentrate on the stochastic non-linear Schrِdinger equation (SNLSE) that arises in the analysis of wave propagation phenomena, mainly motivated by its predominant role as a modeling tool in the design of optically amplified long distance fiber telecommunication systems. We present novel formulations and computational methods for the stochastic characterization of the solution of the SNLSE. Our formulations and techniques are not aimed at computing individual realizations, i.e., sample paths, for the solution of the SNLSE ل la Monte Carlo. Instead, starting with the SNLSE, we derive new systems of differential equations and develop associated computational techniques. The numerical solutions of these new equations directly produce the ensemble-averaged stochastic characterization desired for the solution of the SNLSE, in a non-Monte Carlo manner without having to compute many realizations needed for ensemble-averaging.