Soliton dynamics for the 1D NLKG equation with symmetry and in the absence of internal modes

We consider the dynamics of even solutions one-dimensional nonlinear Klein-Gordon equation $\partial_t^2 \phi - \partial_x^2 + |\phi|^{2\alpha} =0$ for $\alpha>1$, in vicinity unstable soliton $Q$. Our main result is that stability energy space $H^1(\mathbb R)\times L^2(\mathbb R)$ implies asymptotic a local norm. In particular, there exists Lipschitz graph initial data leading to stable and asymptotically trajectories. The condition $\alpha>1$ corresponds cases where linearized operator around $Q$ has no resonance internal mode. Recall case $\alpha>2$ treated Krieger-Nakanishi-Schlag using Strichartz other dispersive estimates. Since these tools are not available low power nonlinearities, our approach based on virial type estimates particular structure observed Chang-Gustafson-Nakanishi-Tsai.