Asymptotic stability near the soliton for quartic Klein-Gordon in 1D

TL;DR

This paper proves optimal decay and local decay for solutions near the unstable soliton of the 1D focusing Klein-Gordon equation, confirming numerical results and providing new space-time stability insights.

Contribution

It establishes decay estimates for perturbations near the soliton in 1D Klein-Gordon, extending prior work and confirming numerical findings for nonlinearities with p ≥ 4.

Findings

Proves optimal decay for even perturbations near the soliton.

Confirms numerical results for nonlinearities p ≥ 4.

Provides new space-time stability information for localized perturbations.

Abstract

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.