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

TL;DR

This paper investigates the existence, stability, and dynamics of planar and radial kinks in two-dimensional nonlinear Klein-Gordon models, using an adiabatic invariant approach validated by numerical simulations.

Contribution

It adapts an adiabatic invariant method to Klein-Gordon models to analyze kink stability and dynamics, including external potentials and ring-like configurations.

Findings

Planar kinks are characterized as solitonic filaments with spectral stability.

Radial kinks are azimuthally stable and can form stable ring-like structures.

The methodology's predictions are confirmed through numerical simulations.

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 $\phi^4$ variants thereof. We adapt an adiabatic invariant formulation recently developed for nonlinear Schr{\"o}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 $\phi^4$ models.