Asymptotic stability of the sineGordon kink under odd perturbations
Abstract
We establish the asymptotic stability of the sineGordon kink under odd perturbations that are sufficiently small in a weighted Sobolev norm. Our approach is perturbative and does not rely on the complete integrability of the sineGordon model. Key elements of our proof are a specific factorization property of the linearized operator around the sineGordon kink, a remarkable nonresonance property exhibited by the quadratic nonlinearity in the KleinGordon equation for the perturbation, and a variable coefficient quadratic normal form introduced in [53]. We emphasize that the restriction to odd perturbations does not bypass the effects of the odd threshold resonance of the linearized operator. Our techniques have applications to soliton stability questions for several wellknown nonintegrable models, for instance, to the asymptotic stability problem for the kink of the $\phi^4$ model as well as to the conditional asymptotic stability problem for the solitons of the focusing quadratic and cubic KleinGordon equations in one space dimension.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 DOI:
 10.48550/arXiv.2106.09605
 arXiv:
 arXiv:2106.09605
 Bibcode:
 2021arXiv210609605L
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics
 EPrint:
 72 pages, 3 figures. Minor Revisions. To appear in Duke Math. J