Please access the talk through the Research Seminars page ( https://researchseminars.org/seminar/MO_Analysis ); you will need to register for a free account. If you have any difficulty, please contact Samuel Walsh (email@example.com).
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 the sine-Gordon equation in weighted energy space. 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. Our stability analysis gives a criterion for the weight which is sharp up to the endpoint so that the asymptotic stability holds.