On the asymptotic stability of the sine-Gordon kink in the energy space

Abstract

We consider the sine-Gordon (SG) equation in 1+1 dimensions. The kink is a static, non symmetric exact solution to SG, stable in the energy space H 1 × L 2 . It is well-known that the linearized operator around the kink has a simple kernel and no internal modes. However, it possesses an odd resonance at the bottom of the continuum spectrum, deeply related to the existence of the (in)famous wobbling kink, an explicit periodic-in-time solution of SG around the kink that contradicts the asymptotic stability of the kink in the energy space.In this paper we further investigate the influence of resonances in the asymptotic stability question. We also discuss the relationship between breathers, wobbling kinks and resonances in the SG setting. By gathering Bäcklund transformations (BT) as in [24,52] and Virial estimates around odd perturbations of the vacuum solution, in the spirit of [32], we first identify the manifold of initial data around zero under which BTs are related to the wobbling kink solution. It turns out that (even) small breathers are deeply related to odd perturbations around the kink, including the wobbling kink itself. As a consequence of this result and [32], using BTs we can construct a smooth manifold of initial data close to the kink, for which there is asymptotic stability in the energy space. The initial data has spatial symmetry of the form (kink + odd, even), non resonant in principle, and not preserved by the flow. This asymptotic stability property holds despite the existence of wobbling kinks in SG. We also show that wobbling kinks are orbitally stable under odd data, and clarify some interesting connections between SG and φ 4 at the level of linear Bäcklund transformations. Date: March 23, 2020. * M.A. Alejo was partially supported by CNPq grant no. 305205/2016-1. * * C. M. work was funded in part by Chilean research grants FONDECYT 1191412, project France-Chile ECOS-Sud C18E06 and CMM Conicyt PIA AFB170001. * * * J. M. P. was partially supported by Chilean research grants FONDECYT 1191412 and project France-Chile ECOS-Sud C18E06.

On modified scattering for 1D quadratic Klein-Gordon equations with non-generic potentials

“…) 1 The sine-Gordon model is completely integrable and the study of its dynamics is therefore amenable to inverse scattering techniques, see, e.g., the recent work [4].…”

Orbital stability and instability of periodic wave solutions for the $ϕ^4$-model

“…About orbital stability of explicit solutions to equations (1.1) and (1.2), there exists a vast literature regarding the aperiodic case. We refer the reader to [17] for a classical and rather general result about the orbital stability of Kink solutions, and to [25] for the first result regarding asymptotic stability of Kink solutions for equation (1.1) (see also [3] for a recent work in this direction). We also refer to [8] for an study of the asymptotic stability properties of these solutions in dimension 3.…”

Review on the stability of the Peregrine and related breathers

“…Some deeper connections between the stability of breathers and the nonzero background (modulational instability) are highly expected, but it seems that no proof of this fact is in the literature. Maybe Bäcklund transformations, in the spirit of [10,13,49], could help to give preliminary answers, and rigorous IST methods such as the ones in [24,25] may help to solve this question. Finally, the dichotomy blow up/global well-posedness, and ill-posedness for large data in NLS (1.1) with nonzero background, are interesting mathematical open problems to be treated elsewhere.…”

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