Numerical Simulations of the Cubic Schrödinger Equations for Periodic Perturbations

In the course of investigating the effect of a random potential on pulses with the cubic nonlinear Schrödinger equation, I became intrigued by what one should define as the effective width of a pulse or soliton, particularly in cases that are near-integrable. In an attempt to answer that question, I did some numerical simulations with periodic potentials and came across an interesting range of behavior depending on how the length scales of potential and pulse compare. Some of the results are explainable by simple perturbation methods, and some remain more mysterious. I show the results obtained and conclude with some observations about the effective width. I first give a brief description of the numerical method. Next, I present a leading order perturbation analysis for a slowly varying potential. Then, I show how well this theory bears out in numerical simulations with a slowly varying potential. The results get more interesting when the assumption that the potential is slowly varying breaks down. Beyond this, some simulations with a rapidly varying potential show some mildly surprising behavior. Finally, I try to bridge the entire range of length scales with some conclusions and questions.