The inverse problem for the Gross - Pitaevskii equation

TL;DR

This paper introduces two novel methods for generating exact solutions to the stationary Gross-Pitaevskii equation, including an inverse problem approach, with applications demonstrated in 1D and 2D cases involving localized structures.

Contribution

It presents a new inverse problem method for constructing potentials that yield desired solutions to the GPE, expanding solution generation techniques.

Findings

Methods produce a variety of localized solutions, including vortices.

Solutions are tested for stability via direct simulations.

Applicable to both attractive and repulsive nonlinearities.

Abstract

Two different methods are proposed for the generation of wide classes of exact solutions to the stationary Gross - Pitaevskii equation (GPE). The first method, suggested by the work by Kondrat'ev and Miller (1966), applies to one-dimensional (1D) GPE. It is based on the similarity between the GPE and the integrable Gardner equation, all solutions of the latter equation (both stationary and nonstationary ones) generating exact solutions to the GPE, with the potential function proportional to the corresponding solutions. The second method is based on the "inverse problem" for the GPE, i.e. construction of a potential function which provides a desirable solution to the equation. Systematic results are presented for 1D and 2D cases. Both methods are illustrated by a variety of localized solutions, including solitary vortices, for both attractive and repulsive nonlinearity in the GPE. The stability of the 1D solutions is tested by direct simulations of the time-dependent GPE.