  In this paper, we study the long-time dynamics and stability properties of
the sine-Gordon equation $$f_{tt}-f_{xx}+\sin f=0.$$ Firstly, we use the
nonlinear steepest descent for Riemann-Hilbert problems to compute the
long-time asymptotics of the solutions to the sine-Gordon equation whose
initial condition belongs to some weighted Sobolev spaces. Secondly, we study
the asymptotic stability of the sine-Gordon equation. It is known that the
obstruction to the asymptotic stability of the sine-Gordon equation in the
energy space is the existence of small breathers which is also closely related
to the emergence of wobbling kinks. Combining the long-time asymptotics and a
refined approximation argument, we analyze the asymptotic stability properties
of the sine-Gordon equation in weighted energy spaces. Our stability analysis
gives a criterion for the weight which is sharp up to the endpoint so that the
asymptotic stability holds.
