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

Mashkin, Timur

J. Math. Anal. Appl. 486, No. 2, Article ID 123904, 37 p. (2020).

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

Keywords:

soliton; symplectic decomposition; modulation equation; Lyapunov function

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