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

TL;DR

This paper investigates the modulational stability of the Gross-Pitaevskii equation with quadratic external potential, using a transformation to analyze stability and suggesting experiments for soliton formation in Bose-Einstein condensates.

Contribution

It introduces a lens-type transformation to analyze stability of the Gross-Pitaevskii equation with time-varying potential, linking theoretical analysis with experimental implications.

Findings

Stability depends on specific time-varying potentials.

Numerical analysis confirms stability regimes.

Proposes experimental setups for soliton generation.

Abstract

The modulational stability of the nonlinear Schr{\"o}dinger (NLS) equation is examined in the case with a quadratic external potential. This study is motivated by recent experimental studies in the context of matter waves in Bose-Einstein condensates (BECs). The theoretical analysis invokes a lens-type transformation that converts the Gross-Pitaevskii into a regular NLS equation with an additional growth term. This analysis suggests the particular interest of a specific time-varying potential ((t+t*)^{-2}). We examine both this potential, as well as the time independent one numerically and conclude by suggesting experiments for the production of solitonic wave-trains in BEC.