On Some Canonical Classes of Cubic-Quintic Nonlinear Schrödinger Equations

Abstract

In this paper we bring into attention variable coefficient cubic-quintic nonlinear Schrödinger equations which admit Lie symmetry algebras of dimension four. Within this family, we obtain the reductions of canonical equations of nonequivalent classes to ordinary differential equations using tools of Lie theory. Painlevé integrability of these reduced equations is investigated. Exact solutions through truncated Painlevé expansions are achieved in some cases. One of these solutions, a conformal-group invariant one, exhibits blow-up behaviour in finite time in $L_p$, $L_{\infty }$ norm and in distributional sense.