# An Adiabatic Invariant Approach to Transverse Instability: Landau Dynamics of Soliton Filaments

## TL;DR

This paper extends the Landau dynamics approach to analyze the transverse stability and instability modes of lower-dimensional solitons embedded in higher-dimensional spaces, providing a theoretical framework and numerical validation.

## Contribution

It introduces an adiabatic invariant method to capture Kelvin modes of soliton filaments, advancing understanding of their transverse dynamics and stability in higher dimensions.

## Key findings

- The theory accurately predicts transverse instability modes.
- Numerical simulations confirm the stability analysis.
- Application to 2D and 3D soliton structures.

## Abstract

Assume a lower-dimensional solitonic structure embedded in a higher dimensional space, e.g., a 1D dark soliton embedded in 2D space, a ring dark soliton in 2D space, a spherical shell soliton in 3D space etc. By extending the Landau dynamics approach [Phys. Rev. Lett. {\bf 93}, 240403 (2004)], we show that it is possible to capture the transverse dynamical modes (the "Kelvin modes") of the undulation of this "soliton filament" within the higher dimensional space. These are the transverse stability/instability modes and are the ones potentially responsible for the breakup of the soliton into structures such as vortices, vortex rings etc. We present the theory and case examples in 2D and 3D, corroborating the results by numerical stability and dynamical computations.