  In this paper we obtain a global characterization of the dynamics of even
solutions to the one-dimensional nonlinear Klein-Gordon (NLKG) equation on the
line with focusing nonlinearity |u|^{p-1}u, p>5, provided their energy exceeds
that of the ground state only sightly. The method is the same as in the
three-dimensional case arXiv:1005.4894, the major difference being in the
construction of the center-stable manifold. The difficulty there lies with the
weak dispersive decay of 1-dimensional NLKG. In order to address this specific
issue, we establish local dispersive estimates for the perturbed linear
Klein-Gordon equation, similar to those of Mizumachi arXiv:math/0605031. The
essential ingredient for the latter class of estimates is the absence of a
threshold resonance of the linearized operator.
