  We study the asymptotic stability of the sine-Gordon kinks under small
perturbations in weighted Sobolev norms. Our main tool is the B\"acklund
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 present a different proof of the local asymptotic
stability result in arXiv:2009.04260. 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 prove an
$L^\infty$-type asymptotic stability result which is similar to that in
arXiv:2106.09605; the main difference is that we remove the assumptions on the
spatial symmetry of the perturbations. 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.
