Asymptotic stability of the sine-Gordon kink under odd perturbations

We establish the asymptotic stability of the sine-Gordon 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 sine-Gordon model. Key elements of our proof are a specific factorization property of the linearized operator around the sine-Gordon kink, a remarkable non-resonance property exhibited by the quadratic nonlinearity in the Klein-Gordon 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 well-known non-integrable 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 Klein-Gordon equations in one space dimension.