Invariant manifold of modified solitons for the perturbed sine-Gordon equation

Abstract

We study the perturbed sine-Gordon equation θ tt − θ xx + sin θ = F(ɛ, x), where we assume that the perturbation F is analytic in ɛ and that its derivatives with respect to ɛ satisfy certain bounds at ɛ = 0. We construct implicitly an, adjusted to the perturbation F, manifold which is invariant in the following sense: the initial value problem for the perturbed sine-Gordon equation with an appropriate initial state on the manifold has a unique solution which follows a trajectory on the manifold. The trajectory is precisely described by two parameters which satisfy specific ODEs. In our approach, the invariant manifold is generated as a limit of a sequence whose elements are created one after another by successive distortion of the classical solitary manifold. For the proof, we firstly modify the iteration scheme introduced in Mashkin (2020 Stability of the solitary manifold of the perturbed sine-Gordon equation J. Math. Anal. Appl. 486 123904). By using the modified iteration scheme we build the mentioned sequence. Thereafter we show convergence of the sequence to establish the result.