Asymptotic stability of the sine-Gordon kinks under perturbations in weighted Sobolev norms

We study the asymptotic stability of the sine-Gordon kinks under small perturbations in weighted Sobolev norms. Our main tool is the Bäcklund transform which reduces the study of the asymptotic stability of the kinks to the study of the asymptotic decay of solutions near zero. Our results consist of two parts. First, we prove an asymptotic stability result similar to the local results in Alejo et al. and Chen et al. Our assumptions are the same as those in the local result in Chen et al. In its proof, we apply a result obtained by the inverse scattering method on the local decay of the solutions with sufficiently small and localized initial data. Moreover, we derive an asymptotic formula for the perturbations, i.e., the difference between solutions and kinks. This result is similar to that in Lührmann and Schlag and the full asymptotic stability result in Chen et al. In its proof, we apply a result obtained by the method of testing by wave packets on the pointwise decay of the solutions with small and localized data.