  We consider the perturbed sine-Gordon equation $\theta_{tt}-\theta_{xx}+\sin
\theta=
  \varepsilon^2 f(\varepsilon x)$, where the external perturbation
$\varepsilon^2 f(\varepsilon x)$ corresponds to a small, slowly varying
electric field. We show that the initial value problem with an appropriate
initial state close enough to the solitary manifold has a unique solution,
which follows up to time $1/{\varepsilon}$ and errors of order $\varepsilon^{
3/4}$ a trajectory on the solitary manifold. The trajectory on the solitary
manifold is described by ODEs, which agree approximately up to errors of order
$\varepsilon^3$ with Hamilton equations for the restricted to the solitary
manifold sine-Gordon Hamiltonian.
