Onset of transverse instabilities of confined dark solitons

TL;DR

This paper studies how confined dark solitons in a 2D nonlinear Schrödinger equation become unstable due to transverse perturbations, revealing critical confinement conditions and connections to vortex formation in superfluid dynamics.

Contribution

It provides a spectral analysis of transverse instabilities of dark solitons in confined geometries, identifying critical confinement widths and linking bifurcations to vortex excitations.

Findings

Single-lobed solitons are stable under sufficient transverse confinement.

Transverse modulational instability occurs at a specific critical confinement width.

Bifurcation analysis links soliton instability to vortex and multi-vortex excitations.

Abstract

We investigate propagating dark soliton solutions of the two-dimensional defocusing nonlinear Schr\"odinger / Gross-Pitaevskii (NLS/GP) equation that are transversely confined to propagate in an infinitely long channel. Families of single, vortex, and multi-lobed solitons are computed using a spectrally-accurate numerical scheme. The multi-lobed solitons are unstable to small transverse perturbations. However, the single-lobed solitons are stable if they are sufficiently confined along the transverse direction, which explains their effective one-dimensional dynamics. The emergence of a transverse modulational instability is characterized in terms of a spectral bifurcation. The critical confinement width for this bifurcation is found to coincide with the existence of a propagating vortex solution and the onset of a "snaking" instability in the dark soliton dynamics that, in turn, give rise to vortex or multi-vortex excitations. These results shed light on the superfluidic hydrodynamics of dispersive shock waves in Bose-Einstein condensates and nonlinear optics.