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

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-Muñoz (J. Eur. Math. Soc., 2021). Our results complement those of Kowalczyk-Martel-Muñoz (J. Eur. Math. Soc., 2021) and confirm numerical results of Bizon-Chmaj-Szpak (J. Math. Phys., 2011) when considering nonlinearities $u^p$ with $p\ge 4$. 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.

35 pages, no figures

References in corpus (1)

• On $1$d quadratic Klein-Gordon equations with a potential and symmetries