Motions in a Bose condensate: X. New results on stability of axisymmetric solitary waves of the Gross-Pitaevskii equation

TL;DR

This paper investigates the stability of axisymmetric solitary waves in the Gross-Pitaevskii equation, revealing that upper branch rarefaction waves are unstable but tend to evolve towards stable lower branch waves or dissipate.

Contribution

It provides new insights into the linear stability of solitary waves in Bose condensates using advanced spectral methods and numerical simulations.

Findings

Upper branch rarefaction waves are linearly unstable.

Lower branch and 2D solitary waves are stable.

Unstable waves tend to evolve towards stable states or dissipate.

Abstract

The stability of the axisymmetric solitary waves of the Gross-Pitaevskii (GP) equation is investigated. The Implicitly Restarted Arnoldi Method for banded matrices with shift-invert was used to solve the linearised spectral stability problem. The rarefaction solitary waves on the upper branch of the Jones-Roberts dispersion curve are shown to be unstable to axisymmetric infinitesimal perturbations, whereas the solitary waves on the lower branch and all two-dimensional solitary waves are linearly stable. The growth rates of the instabilities on the upper branch are so small that an arbitrarily specified initial perturbation of a rarefaction wave at first usually evolves towards the upper branch as it acoustically radiates away its excess energy. This is demonstrated through numerical integrations of the GP equation starting from an initial state consisting of an unstable rarefaction wave and random non-axisymmetric noise. The resulting solution evolves towards, and remains for a significant time in the vicinity of, an unperturbed unstable rarefaction wave. It is shown however that, ultimately (or for an initial state extremely close to the upper branch), the solution evolves onto the lower branch or is completely dissipated as sound.