Bing-Ying Lu

Soliton resolution and asymptotic stability for the sine-Gordon equation

In this talk, 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.