Dark solitons, dispersive shock waves, and transverse instabilities

TL;DR

This paper analyzes the transverse instabilities of dark solitons and dispersive shock waves in the (2+1)-D nonlinear Schrödinger equation, providing analytical and numerical insights into their stability, transition regimes, and implications for controlling nonlinear waves.

Contribution

It offers a detailed analytical and numerical study of transverse instabilities in (2+1)-D NLS/Gross-Pitaevskii equations, especially in the shallow amplitude limit, including stability criteria and transition analysis.

Findings

Derived dispersion relation for shallow solitons beyond KP limit.

Identified maximum growth rate and wavenumber of instabilities.

Analyzed transition between convective and absolute instabilities.

Abstract

The nature of transverse instabilities to dark solitons and dispersive shock waves for the (2+1)-dimensional defocusing nonlinear Schrodinger equation / Gross-Pitaevskii (NLS / GP) equation is considered. Special attention is given to the small (shallow) amplitude regime, which limits to the Kadomtsev-Petviashvili (KP) equation. We study analytically and numerically the eigenvalues of the linearized NLS / GP equation. The dispersion relation for shallow solitons is obtained asymptotically beyond the KP limit. This yields 1) the maximum growth rate and associated wavenumber of unstable perturbations; and 2) the separatrix between convective and absolute instabilities. The latter result is used to study the transition between convective and absolute instabilities of oblique dispersive shock waves (DSWs). Stationary and nonstationary oblique DSWs are constructed analytically and investigated numerically by direct simulations of the NLS / GP equation. The instability properties of oblique DSWs are found to be directly related to those of the dark soliton. It is found that stationary and nonstationary oblique DSWs have the same jump conditions in the shallow and hypersonic regimes. These results have application to controlling nonlinear waves in dispersive media.