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