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

Abstract

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 ODEs (Ordinary Differential Equations), namely group invariant solutions. The other is by truncation in their Painlevé expansions.

On some canonical classes of cubic–quintic nonlinear Schrödinger equations

“…By this way one can also obtain Bäcklund transformations relating solutions of the equation. We refer the interested reader to [13,14,15] and [16], among many other works on Schrödinger and other type equations. After this overview of the methods we shall make use of, we can turn back to have a look at the material we obtained in the previous section.…”

“…of the symmetry algebra L 1 . For the case q 1 = q 2 = 0, inspired by [14], we obtained interesting exact solutions in [15], giving rise to blow-up analysis studied in [16].…”

“…Even the leading order α cannot be determined. However, we keep on tracking the route we followed in the absence of the quintic term technically (see [15] for details), we write (3.16) with its complex conjugate v as the system…”

“…The system (3.17) has no leading order. Those powers −1 ∓ iδ are proposed in regard to cubic case studied in [15]. When we substitute (3.18) in (3.17), terms with coefficients Φ −5∓iδ ,Φ −3∓iδ ,Φ −2∓iδ ,Φ −1∓iδ appear in the system.…”

Symmetry classification of variable coefficient cubic-quintic nonlinear Schrödinger equations

“…An in-depth analysis of the constant coefficient version of (1.1) with k = 0 in 3+1-dimensions was done in a series of papers [4,5,6] where the authors studied Lie point symmetries and gave a complete subalgebra classification of symmetry algebras, reductions and a comprehensive analysis of the explicit (group-invariant) solutions. Recently we looked at the solutions of the cubic version admitting only 4-dimensional Lie point symmetries [7]. Other symmetry classification results relevant to several one and multidimensional versions of the nonlinear Schrödinger equations involving arbitrary functions depending not only on space-time variables but also on dependent variables and its space derivatives can be found for example in [8,9,10].…”