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

Abstract

We perform the complete group classification in the class of nonlinear Schrödinger 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.

10 pages

References in corpus (4)

• Group classification of heat conductivity equations with a nonlinear source
• Critical nonlinear Schroedinger equations with and without harmonic potential
• Gauge Transformations in Quantum Mechanics and the Unification of Nonlinear Schrödinger Equations
• Lie Symmetries of (1+1)-Dimensional Cubic Schrödinger Equation with Potential

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• Dynamical symmetries of semi-linear Schrödinger and diffusion equations
• Equivalence groupoid and group classification of a class of variable-coefficient Burgers equations
• Symmetries of Schroedinger equation with scalar and vector potentials
• Similarity Inner Solutions for the Pulsar Equation
• Equivalence groupoids and group classification of multidimensional nonlinear Schrödinger equations
• Symmetry classification of variable coefficient cubic-quintic nonlinear Schrödinger equations
• Solution of the Master Equation for Quantum Brownian Motion given by the Schrödinger Equation