Reduced dynamics for one and two dark soliton stripes in the defocusing nonlinear Schr\"odinger equation: a variational approach

TL;DR

This paper develops a variational method to simplify the complex dynamics of dark soliton stripes in the 2D defocusing nonlinear Schrödinger equation, achieving good qualitative and quantitative agreement with numerical simulations.

Contribution

It introduces a reduced set of filament equations for dark soliton dynamics and extends the approach to interacting pairs, outperforming previous adiabatic invariant methods.

Findings

Good qualitative agreement with numerical results on transverse instability.

Significant quantitative improvements with a phenomenological amendment.

More accurate representation of full dynamics for various perturbation wavelengths.

Abstract

We study the dynamics and pairwise interactions of dark soliton stripes in the two-dimensional defocusing nonlinear Schr\"odinger equation. By employing a variational approach we reduce the dynamics for dark soliton stripes to a set of coupled one-dimensional "filament" equations of motion for the position and velocity of the stripe. The method yields good qualitative agreement with the numerical results as regards the transverse instability of the stripes. We propose a phenomenological amendment that also significantly improves the quantitative agreement of the method with the computations. Subsequently, the method is extended for a pair of symmetric dark soliton stripes that include the mutual interactions between the filaments. The reduced equations of motion are compared with a recently proposed adiabatic invariant method and its corresponding findings and are found to provide a more adequate representation of the original full dynamics for a wide range of cases encompassing perturbations with long and short wavelengths, and combinations thereof.