Soliton resolution for the modified KdV equation

Gong Chen; Jiaqi Liu

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arXiv · 1907.07115

Soliton resolution for the modified KdV equation

Gong Chen·Jiaqi Liu

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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 $x=\textrm{v}t$ for any fixed $ \textrm{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.

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Gong Chen, Jiaqi Liu. 2019-10-09. Soliton resolution for the modified KdV equation. https://arxiv.org/abs/1907.07115

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