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

Abstract

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