On the modulational stability of Gross-Pittaevskii type equations in 1+1 dimensions

TL;DR

This paper investigates the modulational stability of the nonlinear Schrödinger equation with external potentials, motivated by Bose-Einstein condensate experiments, using transformations to analyze stability in different potential scenarios.

Contribution

It provides a detailed analysis of modulational stability for NLS equations with linear and quadratic potentials, applying transformations to facilitate the study.

Findings

Linear potential case analyzed via Tappert transformation.

Quadratic potential case examined using lens-type transformation.

Results inform stability conditions in Bose-Einstein condensate contexts.

Abstract

The modulational stability of the nonlinear Schr{\"o}dinger (NLS) equation is examined in the cases with linear and quadratic external potential. This study is motivated by recent experimental studies in the context of matter waves in Bose-Einstein condensates. The linear case can be examined by means of the Tappert transformation and can be mapped to the NLS in the appropriate (constant acceleration) frame. The quadratic case can be examined by using a lens-type transformation that converts it into a regular NLS with an additional linear growth term.