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.

arXiv comment35 pages, no figures

References in corpus (5)

• Global dynamics above the ground state for the nonlinear Klein-Gordon equation without a radial assumption
• Quadratic Klein-Gordon equations with a potential in one dimension
• Dynamics near the threshold for blowup in the one-dimensional focusing nonlinear Klein-Gordon equation
• The $1$d nonlinear Schrödinger equation with a weighted $L^1$ potential
• On $1$d quadratic Klein-Gordon equations with a potential and symmetries