Symmetry classification of variable coefficient cubic-quintic nonlinear Schr\"{o}dinger equations

TL;DR

This paper classifies the symmetry groups of variable coefficient cubic-quintic nonlinear Schrödinger equations, revealing how their symmetries depend on the type of nonlinearity and providing a detailed algebraic structure analysis.

Contribution

It provides a Lie-algebraic classification of these equations, identifying the maximum symmetry dimensions and their isomorphisms for different nonlinearities.

Findings

Maximum symmetry group is four-dimensional for genuine cubic-quintic nonlinearity.

Six-dimensional symmetry group for pure quintic nonlinearity, isomorphic to Schrödinger algebra.

Five-dimensional symmetry group for pure cubic nonlinearity, isomorphic to Galilei similitude algebra.

Abstract

A Lie-algebraic classification of the variable coefficient cubic-quintic nonlinear Schr\"odinger equations involving 5 arbitrary functions of space and time is performed under the action of equivalence transformations. It is shown that their symmetry group can be at most four-dimensional in the genuine cubic-quintic nonlinearity. It is only five-dimensional (isomorphic to the Galilei similitude algebra gs(1)) when the equations are of cubic type, and six-dimensional (isomorphic to the Schr\"odinger algebra sch(1)) when they are of quintic type.