  We consider the nonlinear focusing Klein-Gordon equation in $1 + 1$
dimensions and the global space-time dynamics of solutions near the unstable
soliton. Our main result is a proof of optimal decay, and local decay, for even
perturbations of the static soliton originating from well-prepared initial data
belonging to a subset of the stable manifold constructed in Bates-Jones
(Dynamics reported, 1989) and Kowalczyk-Martel-Mu\~noz (J. Eur. Math. Soc.,
2021). Our results complement those of Kowalczyk-Martel-Mu\~noz (J. Eur. Math.
Soc., 2021) and confirm numerical results of Bizon-Chmaj-Szpak (J. Math. Phys.,
2011) when considering nonlinearities $u^p$ with $p \geq 4$. In particular, we
provide new information both local and global in space about asymptotically
stable perturbations of the soliton under localization assumptions on the data.
