Soliton resolution for the focusing modified KdV equation

Chen, Gong; Liu, Jiaqi

doi:10.1016/j.anihpc.2021.02.008

Chen, Gong ¹ ; Liu, Jiaqi ²

¹ a Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
² b School of mathematical sciences, University of Chinese academy of sciences, Beijing, China

Annales de l'I.H.P. Analyse non linéaire, novembre – décembre 2021, Tome 38 (2021) no. 6, pp. 2005-2071.

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Résumé

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 $\overline{\partial}$ -derivatives. From the view of stationary points, we give precise asymptotic formulas along trajectory $x = v t$ 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.

Reçu le : 2020-05-03
Révisé le : 2021-02-05
Accepté le : 2021-02-15

DOI : 10.1016/j.anihpc.2021.02.008

Mots-clés : Soliton resolution, Breather stability, Long time asymptotics, Riemann-Hilbert problems

Affiliations des auteurs :

Chen, Gong ¹ ; Liu, Jiaqi ²

¹ a Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
² b School of mathematical sciences, University of Chinese academy of sciences, Beijing, China

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Chen, Gong; Liu, Jiaqi. Soliton resolution for the focusing modified KdV equation. Annales de l'I.H.P. Analyse non linéaire, novembre – décembre 2021, Tome 38 (2021) no. 6, pp. 2005-2071. doi : 10.1016/j.anihpc.2021.02.008. http://www.numdam.org/articles/10.1016/j.anihpc.2021.02.008/

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