Stability of the Solitary Manifold of the Perturbed Sine-Gordon Equation

Abstract

We study the perturbed sine-Gordon equation ${\theta }_{tt}-{\theta }_{xx}+\sin \theta =F(\epsilon ,x)$, where $F$ is of differentiability class $C^n$ in $\epsilon$ and the first $k$ derivatives vanish at $0$, i.e., ${\partial }_{\epsilon }^{l}F(0,\cdot )=0$ for $0\le l\le k$. 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 ${\epsilon }^n$-close to the virtual solitary manifold has a unique solution which follows up to time $1/(\tilde{C}{\epsilon }^{\frac{k+1}{2}})$ and errors of order ${\epsilon }^n$ 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.