Dark solitons, dispersive shock waves, and transverse 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.

34 pages, 15 figures

References in corpus (8)

• On Dispersive and Classical Shock Waves in Bose-Einstein Condensates and Gas Dynamics
• Observation of a gradient catastrophe generating solitons
• Oblique dark solitons in supersonic flow of a Bose-Einstein condensate
• Stabilization of Solitons Generated by a Supersonic Flow of Bose-Einstein Condensate Past an Obstacle
• The theory of optical dispersive shock waves in photorefractive media
• Wave patterns generated by a supersonic moving body in a binary Bose-Einstein condensate
• Suppression of transverse instabilities of dark solitons and their dispersive shock waves
• Condition for convective instability of dark solitons

Cited by in corpus (9)

• Dispersive shock waves and modulation theory
• Dynamics and stability of dark solitons in exciton-polariton condensates
• Oblique spatial dispersive shock waves in nonlinear Schrödinger flows
• Oblique breathers generated by a flow of two-component Bose-Einstein condensate past a polarized obstacle
• Onset of transverse instabilities of confined dark solitons
• Oblique solitons in the flow of polariton condensate past an obstacle
• Time-dependent spectral renormalization method
• Dynamics of shock waves in a superfluid unitary Fermi gas
• Wave patterns generated by a flow of two-component Bose-Einstein condensate with spin-orbit interaction past a localized obstacle