Group classification of (1+1)-Dimensional Schr\"odinger Equations with Potentials and Power Nonlinearities

TL;DR

This paper performs a complete group classification of a class of nonlinear Schrödinger equations with arbitrary potentials and power nonlinearities, identifying symmetries and classifying inequivalent potentials.

Contribution

It introduces a systematic approach combining algebraic and compatibility methods for classifying symmetries in nonlinear Schrödinger equations with potentials.

Findings

Identified all inequivalent potentials with non-trivial Lie symmetries.

Developed a method applicable to various classes of differential equations in physics.

Provided a comprehensive classification for equations with power nonlinearities.

Abstract

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.