Numerical Explorations in the Non-Linear Schrodinger Equation

Authors: Yonah Moise and Yedidya Moise, B.A./M.A. in Mathematics
Faculty Advisor: Jeremy Schiff, Ph.D., Bar-Ilan University

Abstract:
The Nonlinear Schrodinger equation is a partial differential equation (PDE) whose principal application is to the propagation of a beam of light. Saturated nonlinearity acts as a limitation on the nonlinear component of the equation to prevent it from blowing up. After reproducing initial conditions from Gatz and Herrmann (1997), which are based on certain constraints, the split-step method was applied to step forward in time and approximate the solution to this PDE for any given time. We then constructed a Gaussian function of two dimensions (with a power equivalent to the power of the solution) and ran the split-step method on this function to study it as an approximation of the solution. The observed two internal modes in the behavior of the widths, as well as other observed behavior, provide a basis for analysis of this approximate solution