On Some Canonical Classes of Cubic-Quintic Nonlinear Schr\"odinger Equations

TL;DR

This paper studies variable coefficient cubic-quintic nonlinear Schrödinger equations with four-dimensional Lie symmetry algebras, deriving reductions to ODEs, analyzing integrability, and finding exact solutions including blow-up phenomena.

Contribution

It classifies certain variable coefficient cubic-quintic NLS equations by symmetry, reduces them to ODEs, and constructs explicit solutions demonstrating blow-up behavior.

Findings

Reduced equations exhibit Painlevé integrability in some cases

Exact solutions including blow-up solutions are obtained

Symmetry analysis classifies equations with four-dimensional Lie algebras

Abstract

In this paper we bring into attention variable coefficient cubic-quintic nonlinear Schr\"odinger 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. Painlev\'e integrability of these reduced equations is investigated. Exact solutions through truncated Painlev\'e 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.