Periodic Orbits of Gross Pitaevskii in the Disc with Vortices Following Point Vortex Flow

TL;DR

This paper proves the existence of periodic vortex solutions to the Gross-Pitaevskii equations, where vortices follow point vortex system orbits, using constrained minimization and topological minimax methods.

Contribution

It introduces a novel approach combining constrained minimization and topological minimax techniques to establish vortex solutions following point vortex dynamics.

Findings

Existence of non-constant periodic vortex solutions for small fixed epsilon.

Vortices follow periodic orbits of the point vortex system.

Application of combined minimization techniques in a rotational ansatz.

Abstract

We prove the existence of non-constant time periodic vortex solutions to the Gross-Pitaevskii equations for small but \textit{fixed} $\varepsilon > 0.$ The vortices of these solutions follow periodic orbits to the point vortex system of ordinary differential equations \textit{for all time}. The construction uses two approaches-- constrained minimization techniques adapted from \cite{GS} and topological minimax techniques adapted from \cite{LinMinMax}, applied to a formulation of the problem within a rotational ansatz.