Direct search for exact solutions to the nonlinear Schrödinger equation |
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| Authors: | Wen-Xiu Ma Min Chen |
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| Institution: | a Department of Mathematics, Zhejiang Normal University, Jinhua 321004, PR China
b Department of Applied Mathematics, Donghua University, Shanghai 200051, PR China |
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| Abstract: | 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. |
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| Keywords: | The nonlinear Schrö dinger equation Symmetry algebra Soliton solution Periodic solution Rational solution |
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