Variable coefficient nonlinear Schrödinger equations with four-dimensional symmetry groups and analysis of their solutions

Analytical solutions of variable coefficient nonlinear Schrödinger equations having four-dimensional symmetry groups, which are in fact the next closest to the integrable ones occurring only when the Lie symmetry group is five-dimensional, are obtained using two different tools. The first tool is to use one-dimensional subgroups of the full symmetry group to generate solutions from those of the reduced ordinary differential equations, namely, group invariant solutions. The other is by truncation in their Painlevé expansions.