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

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