Planar and Radial Kinks in Nonlinear Klein-Gordon Models: Existence, Stability and Dynamics

Abstract

We consider effectively one-dimensional planar and radial kinks in two-dimensional nonlinear Klein-Gordon models and focus on the sine-Gordon model and the ${\varphi }^4$ variants thereof. We adapt an adiabatic invariant formulation recently developed for nonlinear Schr{ö}dinger equations, and we study the transverse stability of these kinks. This enables us to characterize one-dimensional planar kinks as solitonic filaments, whose stationary states and corresponding spectral stability can be characterized not only in the homogeneous case, but also in the presence of external potentials. Beyond that, the full nonlinear (transverse) dynamics of such filaments are described using the reduced, one-dimensional, adiabatic invariant formulation. For radial kinks, this approach confirms their azimuthal stability. It also predicts the possibility of creating stationary and stable ring-like kinks. In all cases we corroborate the results of our methodology with full numerics on the original sine-Gordon and ${\varphi }^4$ models.

arXiv comment14 pages, 15 figures

References in corpus (15)

• Watching dark solitons decay into vortex rings in a Bose-Einstein condensate
• Dark solitons in atomic Bose-Einstein condensates: from theory to experiments
• Ring Dark Solitons and Vortex Necklaces in Bose-Einstein Condensates
• Landau dynamics of a grey soliton in a trapped condensate
• Observation of Hybrid Soliton Vortex-Ring Structures in Bose-Einstein Condensates
• Controlling the transverse instability of dark solitons and nucleation of vortices by a potential barrier
• A General Theory of Oscillon Dynamics
• Vortices and Ring Solitons in Bose-Einstein Condensates
• Ring dark and anti-dark solitons in nonlocal media
• Dark spherical shell solitons in three-dimensional Bose-Einstein condensates: Existence, stability and dynamics
• An Adiabatic Invariant Approach to Transverse Instability: Landau Dynamics of Soliton Filaments
• Radial sine-Gordon kinks as sources of fast breathers
• Onset of transverse instabilities of confined dark solitons
• Exact Soliton-like Solutions of the Radial Gross-Pitaevskii Equation
• Adiabatic invariant analysis of dark and dark-bright soliton stripes in two-dimensional Bose-Einstein condensates

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• Kink-inhomogeneity interaction in the sine-Gordon model
• Vortex and soliton dynamics in particle-hole symmetric superfluids
• Radial kinks in a Schwarzschild-like geometry
• Radial kinks in the boson stars