Soliton resolution for the focusing modified KdV equation

Abstract

The soliton resolution for the focusing 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 and its reformulation through $\bar{\partial }$-derivatives. From the view of stationary points, we give precise asymptotic formulas along trajectory $x=\mathrm{v}t$ for any fixed v. To extend the asymptotics to solutions with initial data in low regularity spaces, we apply a global approximation via PDE techniques. As by-products of our long-time asymptotics, we also obtain the asymptotic stability of nonlinear structures involving solitons and breathers.