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Direct search for exact solutions to the nonlinear Schrödinger equation. (English) Zbl 1180.65130
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:
65M70Spectral, collocation and related methods (IVP of PDE)
35Q55NLS-like (nonlinear Schrödinger) equations
35B32Bifurcation (PDE)
35Q51Soliton-like equations
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