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

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.