Two infinite families of resonant solutions for the Gross-Pitaevskii equation

TL;DR

This paper constructs explicit resonant solutions for the 2D Gross-Pitaevskii equation in a harmonic trap, revealing modulated oscillations and vortex dynamics, advancing understanding of Bose-Einstein condensate behavior.

Contribution

It introduces two families of explicit analytic solutions for the resonant system of the Gross-Pitaevskii equation, focusing on fixed angular momentum modes and Landau levels.

Findings

Solutions describe modulated dark ring oscillations

Solutions depict vortex and antivortex precession

Results connect to broader nonlinear Hamiltonian systems

Abstract

We consider the two-dimensional Gross-Pitaevskii equation describing a Bose-Einstein condensate in an isotropic harmonic trap. In the small coupling regime, this equation is accurately approximated over long times by the corresponding nonlinear resonant system whose structure is determined by the fully resonant spectrum of the linearized problem. We focus on two types of consistent truncations of this resonant system: first, to sets of modes of fixed angular momentum, and second, to excited Landau levels. Each of these truncations admits a set of explicit analytic solutions with initial conditions parametrized by three complex numbers. Viewed in position space, the fixed angular momentum solutions describe modulated oscillations of dark rings, while the excited Landau level solutions describe modulated precession of small arrays of vortices and antivortices. We place our findings in the context of similar results for other spatially confined nonlinear Hamiltonian systems in recent literature.