The $\phi^4$ kink on a wormhole spacetime

TL;DR

This paper investigates the stability and spectral properties of a modified $ ext{phi}^4$ kink solution on a 3+1 dimensional wormhole spacetime, providing analytical and numerical insights into its linear and nonlinear stability.

Contribution

It proves the linear stability of the modified kink on a wormhole spacetime and compares its spectral properties to the standard $ ext{phi}^4$ kink on flat space.

Findings

The modified kink is linearly stable on the wormhole spacetime.

Spectral analysis shows differences in discrete modes compared to flat space.

Numerical evidence suggests asymptotic stability for certain parameters.

Abstract

The soliton resolution conjecture states that solutions to solitonic equations with generic initial data should, after some non--linear behaviour, eventually resolve into a finite number of solitons plus a radiative term. This conjecture is intimately tied to soliton stability, which has been investigated for a number of solitonic equations, including that of $\phi^4$ theory on $\mathbb{R}^{1,1}$. We study a modification of this theory on a $3+1$ dimensional wormhole spacetime which has a spherical throat of radius $a$, with a focus on the stability properties of the modified kink. In particular, we prove that the modified kink is linearly stable, and compare its discrete spectrum to that of the $\phi^4$ kink on $\mathbb{R}^{1,1}$. We also study the resonant coupling between the discrete modes and the continuous spectrum for small but non--linear perturbations. Some numerical and analytical evidence for asymptotic stability is presented for the range of $a$ where the kink has exactly one discrete mode.