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A semi-implicit moving mesh method for the focusing nonlinear Schrödinger equation. (English) Zbl 1010.35098
Summary: 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 solutions behavior for small dispersion parameters. Convincing evidence is presented of striking regular space-time patterns for both analytic and non-analytic initial data.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
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