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 gs(1)) when the equation is of cubic type, and six-dimensional (isomorphic to the Schrödinger algebra sch(1)) when it is of quintic type. C 2013 American Institute of Physics.

On some canonical classes of cubic–quintic nonlinear Schrödinger equations

“…Equations (1.1) with coefficients from Table 1 are representatives of class of equations which admit non-isomorphic four-dimensional symmetry algebras. If we are to say that the symmetry algebras do not extend to five-or six-dimensional ones at all, we have to put the conditions (g, h) = (0, 0) for L 1 , (g, h 2 ) = (0, 0) or (q, h 2 ) = (0, 0) for L 2 , (g, h 2 ) = (0, 1 4 ) or (q, h 2 ) = (0, {0, 1 2 }) for L 3 and (g, h 2 ) = (0, 0) for L 4 [1]. Let us note that L 1 is non-solvable, L 2 is nilpotent, L 3 and L 4 are solvable algebras.…”

“…In this paper we aim at studying a class of cubic-quintic nonlinear Schrödinger (CQNLS) equations, given in the form iu t + u xx + g(x, t)|u| 2 u + q(x, t)|u| 4 u + h(x, t)u = 0 (1.1) in which the complex coefficients g, q and h will be assumed to have some specific forms so that the equation under consideration admits a four-dimensional Lie symmetry algebra. The motivation for this study comes from the recent work [1] on a general class of cubic-quintic nonlinear Schrödinger equations given as iu t + f (x, t)u xx + k(x, t) u x + g(x, t)|u| 2 u + q(x, t)|u| 4 u + h(x, t)u = 0.…”

“…In [1] we transformed (1.2) to (1.1) by point transformations. Therefore (1.1) appears as a canonical equation when classifying the family (1.2) with respect to Lie symmetries According to these results, when a variable coefficient CQNLS equation has a 5-or 6-dimensional Lie symmetry algebra, it can be transformed to well-known constant coefficient cubic or quintic NLS equations, respectively.…”

“…The motivation for this study comes from the recent work [1] on a general class of cubic-quintic nonlinear Schrödinger equations given as iu t + f (x, t)u xx + k(x, t) u x + g(x, t)|u| 2 u + q(x, t)|u| 4 u + h(x, t)u = 0.…”

“…In [1] we transformed (1.2) to (1.1) by point transformations. Therefore (1.1) appears as a canonical equation when classifying the family (1.2) with respect to Lie symmetries Table 1: Four dimensional symmetry algebras and the coefficients in (1.1) No Algebra…”