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March  2002, 1(1): 1-18. doi: 10.3934/cpaa.2002.1.1

A semi-implicit moving mesh method for the focusing nonlinear Schrödinger equation

Hector D. Ceniceros 1,

1. 

Department of Mathematics, University of California, Santa Barbara California, 93106, United States

Received  March 2001 Published  December 2001

An efficient adaptive moving mesh method for investigation of the semi-classical limit of the focusing nonlinear schrödinger equation is presented. The method employs a dynamic mesh to resolve the sea of solitons observed for small dispersion parameters. A second order semi-implicit discretization is used in conjunction with a dynamic mesh generator to achieve a cost-efficient, accurate, and stable adaptive scheme. This method is used to investigate with highly resolved numerics the solution's behavior for small dispersion parameters. Convincing evidence is presented of striking regular space-time patterns for both analytic and non-analytic inital data.
Keywords: Semi-classical limit, dynamically adaptive mesh, modulational instability.
Mathematics Subject Classification: 35Q55, 65M5.
Citation: Hector D. Ceniceros. A semi-implicit moving mesh method for the focusing nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2002, 1 (1) : 1-18. doi: 10.3934/cpaa.2002.1.1
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