Symmetry classification of variable coefficient cubic-quintic nonlinear Schrödinger equations

Abstract

A Lie-algebraic classification of the variable coefficient cubic-quintic nonlinear Schrödinger equations involving 5 arbitrary functions of space and time is performed under the action of equivalence transformations. It is shown that the symmetry group can be at most four-dimensional in the case of genuine cubic-quintic nonlinearity. It may be five-dimensional (isomorphic to the Galilei similitude algebra \documentclass[12pt]{minimal}\begin{document}$\operatorname{\mathfrak {gs}}(1)$\end{document}gs(1)) when the equation is of cubic type, and six-dimensional (isomorphic to the Schrödinger algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sch}(1)$\end{document}sch(1)) when it is of quintic type.