## Stability of the solitary manifold of the perturbed sine-Gordon equation.(English)Zbl 1442.35065

Summary: 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 $$\varepsilon = 0$$, i.e., $$\partial_\varepsilon^l F(0, \cdot) = 0$$ for $$0 \leq l \leq k$$. 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 $$1 /(\widetilde{C} \varepsilon^{\frac{ k + 1}{2}})$$ 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.

### MSC:

 35C08 Soliton solutions 35L71 Second-order semilinear hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs
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