Symmetry classification of variable coefficient cubic-quintic nonlinear Schr?dinger equations

摘要

A Lie-algebraic classification of the variable coefficient cubic-quintic nonlinear Schr?dinger 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 gs(1)) when the equation is of cubic type, and six-dimensional (isomorphic to the Schr?dinger algebra sch(1)) when it is of quintic type.