ASYMPTOTIC STABILITY NEAR THE SOLITON FOR QUARTIC KLEIN–GORDON EQUATION IN 1D

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 center-stable manifold constructed by Bates and Jones (1989) and Kowalczyk, Martel and Muñoz (2022). Our results complement those of Kowalczyk, Martel and Muñoz and confirm numerical results of Bizoń, Chmaj and Szpak (2011) when considering nonlinearities u^p with p ≥ 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.