Soliton resolution for the 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 $x=\text{v}t$ for any fixed $\text{v}$. To extend the asymptotics to solutions with initial data in low regularity spaces, we apply a global approximation via PDE techniques. As byproducts of our long-time asymptotics, we also obtain the asymptotic stability of nonlinear structures involving solitons and breathers.

70 pages. arXiv admin note: text overlap with arXiv:1903.03855. This is a revision of previous versions