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Numerical simulations of the Cubic Schroedinger equations for Periodic Perturbations

Rob Knapp
Organization: Wolfram Research, Inc.
Department: Kernel Technology

1999 International Mathematica Symposium

In the course of investigating of 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 which 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 will show you the results I have obtained and conclude with some observations about the effective width.

I will structure this presentation roughly as follows. First, I will give a brief description of the numerical method. Next, I will present a leading order perturbation analysis for a slowly varying potential. Then, I will show how well this theory bears out in numerical simulations with a slowly varying potential. The results start to 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 will try to bridge the entire range of length scales with some conclusions and questions.

*Applied Mathematics > Numerical Methods
*Mathematics > Calculus and Analysis > Differential Equations
*Science > Physics > Quantum Physics

Schroedinger equation, soliton, periodic potentials, near-integrable