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

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.