Group classification of (1+1)-Dimensional Schrödinger Equations with Potentials and Power Nonlinearities

Details Group classification of (1+1)-Dimensional Schrödinger Equations with Potentials and Power Nonlinearities Group classification of (1+1)-Dimensional Schrödinger Equations with Potentials and Power Nonlinearities

2003-11-24

arxiv.org/abs/math-ph/0311039v2

J.Math.Phys. 45 (2004) 3049-3057

Physics

Mathematical Physics

10 pages

Scientific paper

10.1063/1.1765748

We perform the complete group classification in the class of nonlinear Schr\"odinger equations of the form $i\psi_t+\psi_{xx}+|\psi|^\gamma\psi+V(t,x)\psi=0$ where $V$ is an arbitrary complex-valued potential depending on $t$ and $x,$ $\gamma$ is a real non-zero constant. We construct all the possible inequivalent potentials for which these equations have non-trivial Lie symmetries using a combination of algebraic and compatibility methods. The proposed approach can be applied to solving group classification problems for a number of important classes of differential equations arising in mathematical physics.

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