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.
Highlights
In the past two decades, the study of coherent structures in the form of dark solitons has been a theme pervading a wide range of areas within physics
There has been a host of additional systems including notably a wide variety of experiments in atomic Bose-Einstein condensates (BECs) summarized, e.g., in Refs. [5,6], and realizations in electromagnetic [7], hydrodynamic [8], acoustic [9], plasma [10], and exciton-polariton [11] systems among others
In order to obtain a theory with a definitive Lagrangian framework enabling the systematic derivation of Euler-Lagrange equations of motion, we develop an alternative variational approach (VA)
Summary
In the past two decades, the study of coherent structures in the form of dark solitons has been a theme pervading a wide range of areas within physics. There has been a surge of further interest in the subject [28,29,30] fueled by an adiabatic invariant (AI) based insight enabling the derivation of effective equations for the dark soliton stripes [in two dimensions (2D)] and planes (in 3D), and the ability of this methodology to tackle ring (in 2D, and in 3D in the form of spherical) solitons [31] This methodology was seen to have the right long wavelength limit (a fundamental prerequisite for such a theory). While the above AI methodology captures the correct long wavelength limit, it does not a priori capture the restabilization of perturbations of large wave numbers (above a certain kc) As it stands, the theory is developed solely for the evolutionary dynamics of the center of the dark solitonic stripes (or planes), but does not arise as a coupled theory for the evolution of the center and the width (or velocity) of the structure. V we summarize our results and give possible avenues for future research
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