Long-time asymptotics and stability for the sine-Gordon equation

Abstract

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.

arXiv admin note: text overlap with arXiv:1907.07115

References in corpus (3)

• Asymptotic stability of Landau solutions to Navier-Stokes system
• Long-time Asymptotics for the NLS equation via dbar methods
• On the asymptotic stability of the sine-Gordon kink in the energy space

Cited by in corpus (4)

• Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities
• Long-time Asymptotic Behavior of the coupled dispersive AB system in Low Regularity Spaces
• Long-time dynamics of small solutions to 1$d$ cubic nonlinear Schrödinger equations with a trapping potential
• A $\dbar$-steepest descent method for oscillatory Riemann-Hilbert problems