Stability of the solitary manifold of the perturbed sine-Gordon equation

Abstract

We study the perturbed sine-Gordon equation θtt−θxx+sin⁡θ=F(ε,x), where F is of differentiability class Cn in ε and the first k derivatives vanish at ε=0, i.e., ∂εlF(0,⋅)=0 for 0≤l≤k. We construct implicitly a virtual solitary manifold by deformation of the classical solitary manifold in n iteration steps. Our main result establishes that the initial value problem with an appropriate initial state εn-close to the virtual solitary manifold has a unique solution, which follows up to time 1/(C˜εk+12) and errors of order εn a trajectory on the virtual solitary manifold. The trajectory on the virtual solitary manifold is described by two parameters, which satisfy a system of ODEs. In contrast to previous works our stability result yields arbitrarily high accuracy as long as the perturbation F is sufficiently often differentiable.