Variational Approach to the Snake Instability of a Bose-Einstein Condensate Soliton

TL;DR

This paper develops a variational method to analyze the snake instability of dark solitons in Bose-Einstein condensates, predicting stability conditions and growth rates that align with numerical simulations.

Contribution

It introduces a novel variational ansatz capturing transverse bending and vortex formation, providing analytical insights into soliton stability in anisotropic traps.

Findings

Identifies critical trap anisotropy to suppress snake instability.

Derives equations of motion for soliton dynamics.

Matches analytical predictions with numerical simulations.

Abstract

Solitons are striking manifestations of nonlinearity, encountered in diverse physical systems such as water waves, nonlinear optics, and Bose-Einstein condensates (BECs). In BECs, dark solitons emerge as exact stationary solutions of the one-dimensional Gross-Pitaevskii equation. While they can be long-lived in elongated traps, their stability is compromised in higher dimensions due to the snake instability, which leads to the decay of the soliton into vortex structures among other excitations. We investigate the dynamics of a dark soliton in a Bose-Einstein condensate confined in an anisotropic harmonic trap. Using a variational ansatz that incorporates both the transverse bending of the soliton plane and the emergence of vortices along the nodal line, we derive equations of motion governing the soliton's evolution. This approach allows us to identify stable oscillation modes as well as the growth rates of the unstable perturbations. In particular, we determine the critical trap anisotropy required to suppress the snake instability. Our analytical predictions are in good agreement with full numerical simulations of the Gross-Pitaevskii equation.