Sine-Gordon on a wormhole

TL;DR

This paper studies the Sine-Gordon equation on a wormhole spacetime, demonstrating the existence and stability of topological solitons called kinks, and analyzing their long-term behavior and relaxation dynamics.

Contribution

It establishes the existence and uniqueness of stable kinks in each topological sector on a wormhole, and provides numerical and analytical insights into their asymptotic relaxation.

Findings

Existence of unique stable kinks in each topological sector.

Numerical evidence that kinks are global attractors.

Asymptotic analysis of relaxation dynamics for the 1-kink.

Abstract

In an attempt to understand the soliton resolution conjecture, we consider the Sine-Gordon equation on a spherically symmetric wormhole spacetime. We show that within each topological sector (indexed by a positive integer degree $n$) there exists a unique linearly stable soliton, which we call the $n$-kink. We give numerical evidence that the $n$-kink is a global attractor in the evolution of any smooth, finite energy solutions of degree $n$. When the radius of the wormhole throat $a$ is large enough, the convergence to the $n$-kink is shown to be governed by internal modes that slowly decay due to the resonant transfer of energy to radiation. We compute the exact asymptotics of this relaxation process for the $1$-kink using the Soffer-Weinstein weakly nonlinear perturbation theory.