Long-time asymptotics of the modified KdV equation in weighted Sobolev spaces

Abstract

AbstractThe long-time behaviour of solutions to the defocussing modified Korteweg-de Vries (MKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method of Deift and Zhou and its reformulation by Dieng and McLaughlin through$\overline {\partial }$-derivatives. To extend the asymptotics to solutions with initial data in lower-regularity spaces, we apply a global approximation via PDE techniques.