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Zbl 1180.65130
Ma, Wen-Xiu; Chen, Min
Direct search for exact solutions to the nonlinear Schrödinger equation. (English)
[J] Appl. Math. Comput. 215, No. 8, 2835-2842 (2009). ISSN 0096-3003

Summary: A five-dimensional symmetry algebra consisting of Lie point symmetries is firstly computed for the nonlinear Schrödinger equation, which, together with a reflection invariance, generates two five-parameter solution groups. Three ansätze of transformations are secondly analyzed and used to construct exact solutions to the nonlinear Schrödinger equation. Various examples of exact solutions with constant, trigonometric function type, exponential function type and rational function amplitude are given upon careful analysis. A bifurcation phenomenon in the nonlinear Schrödinger equation is clearly exhibited during the solution process.

MSC 2000:: *65M70 Spectral, collocation and related methods (IVP of PDE)
35Q55 NLS-like (nonlinear Schroedinger) equations
35B32 Bifurcation (PDE)
35Q51 Solitons

Keywords: nonlinear Schrödinger equation; symmetry algebra; soliton solution; periodic solution; rational solution; numerical examples; bifurcation

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