In the first part of this talk, I will illustrate how to compute the long-time asymptotics of the sine-Gordon equation using its integrable structure and nonlinear steepest descent. Then I will discuss the asymptotic stability of solitons of the sine-Gordon equation in weighted energy space. It is known that the obstruction to the asymptotic stability for 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. Our stability analysis gives a criterion for the weight which is sharp up to the endpoint so that the asymptotic stability holds.

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