Exact Soliton-like Solutions of the Radial Gross-Pitaevskii Equation

TL;DR

This paper derives exact ring-shaped soliton solutions for the radial Gross-Pitaevskii equation using similarity transformations, revealing stationary, oscillating, and bouncing behaviors with potential applications in Bose-Einstein condensates.

Contribution

It introduces a method to construct explicit solutions of the radial Gross-Pitaevskii equation, including novel oscillating and bouncing soliton-like states.

Findings

Exact stationary dark and bright ring solutions identified.

Solutions include oscillating and bouncing states linked to Painlevé transcendents.

Potential can be chosen to be time-independent for these solutions.

Abstract

We construct exact ring soliton-like solutions of the cylindrically symmetric (i.e., radial) Gross- Pitaevskii equation with a potential, using the similarity transformation method. Depending on the choice of the allowed free functions, the solutions can take the form of stationary dark or bright rings whose time dependence is in the phase dynamics only, or oscillating and bouncing solutions, related to the second Painlev\'e transcendent. In each case the potential can be chosen to be time-independent.