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arXiv · 2011.14205

A $\dbar$-steepest descent method for oscillatory Riemann-Hilbert problems

Fudong Wang·Wen-Xiu Ma
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Abstract

We study the asymptotic behavior of Riemann-Hilbert problems (RHP) arising in the AKNS hierarchy of integrable equations. Our analysis is based on the $\dbar$-steepest descent method. We consider RHPs arising from the inverse scattering transform of the AKNS hierarchy with $H^{1,1}(\R)$ initial data. The analysis will be divided into three regions: fast decay region, oscillating region and self-similarity region (the Painlev\'e region). The resulting formulas can be directly applied to study the long-time asymptotic of the solutions of integrable equations such as NLS, mKdV and their higher-order generalizations.

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BibTeXRIS

Fudong Wang, Wen-Xiu Ma. 2020-11-28. A $\dbar$-steepest descent method for oscillatory Riemann-Hilbert problems. https://arxiv.org/abs/2011.14205

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