Weakly Nonlinear Analysis of Vortex Formation in a Dissipative Variant of the Gross-Pitaevskii Equation

TL;DR

This paper investigates how vortices form in a dissipative two-dimensional Gross-Pitaevskii equation under rotation, using weakly nonlinear analysis and numerical simulations to understand symmetry breaking and the role of dissipation.

Contribution

It derives a one-dimensional amplitude equation to describe vortex formation near the instability threshold in a dissipative Gross-Pitaevskii model.

Findings

Vortex formation is initiated by a modulational instability.

Dissipation influences the symmetry breaking process.

Numerical results confirm the weakly nonlinear analysis.

Abstract

For a dissipative variant of the two-dimensional Gross-Pitaevskii equation with a parabolic trap under rotation, we study a symmetry breaking process that leads to the formation of vortices. The first symmetry breaking leads to the formation of many small vortices distributed uniformly near the Thomas-Fermi radius. The instability occurs as a result of a linear instability of a vortex-free steady state as the rotation is increased above a critical threshold. We focus on the second subsequent symmetry breaking, which occurs in the weakly nonlinear regime. At slightly above threshold, we derive a one-dimensional amplitude equation that describes the slow evolution of the envelope of the initial instability. We show that the mechanism responsible for initiating vortex formation is a modulational instability of the amplitude equation. We also illustrate the role of dissipation in the symmetry breaking process. All analyses are confirmed by detailed numerical computations.