Lie Symmetries of (1+1)-Dimensional Cubic Schr\"odinger Equation with Potential

TL;DR

This paper classifies all potentials in a (1+1)-dimensional cubic Schrödinger equation that admit non-trivial Lie symmetries, extending previous classifications by using algebraic and compatibility methods.

Contribution

It provides a complete group classification of cubic Schrödinger equations with arbitrary potentials, identifying all cases with enhanced Lie symmetries.

Findings

Identified all inequivalent potentials with non-trivial Lie symmetries.

Extended and amended earlier classifications in the literature.

Constructed symmetry algebras for the classified equations.

Abstract

We perform the complete group classification in the class of cubic Schr\"odinger equations of the form $i\psi_t+\psi_{xx}+\psi^2\psi^*+V(t,x)\psi=0$ where $V$ is an arbitrary complex-valued potential depending on $t$ and $x$. We construct all possible inequivalent potentials for which these equations have non-trivial Lie symmetries using algebraic and compatibility methods simultaneously. Our classification essentially amends earlier works on the subject.