Variable coefficient nonlinear Schr\"odinger equations with four-dimensional symmetry groups and analysis of their solutions

TL;DR

This paper derives analytical solutions for variable coefficient nonlinear Schrödinger equations with four-dimensional symmetry groups, using symmetry reduction and Painlevé truncation methods, advancing understanding of near-integrable cases.

Contribution

It introduces two methods—symmetry reduction and Painlevé truncation—to find solutions for equations with four-dimensional symmetry groups, close to integrability.

Findings

Solutions obtained via symmetry reduction and Painlevé truncation.

Characterization of equations with four-dimensional symmetry groups.

Enhanced understanding of near-integrable nonlinear Schrödinger equations.

Abstract

Analytical solutions of variable coefficient nonlinear Schr\"odinger 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 ODEs (Ordinary Differential Equations), namely group invariant solutions. The other is by truncation in their Painlev\'e expansions.