  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.
