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arXiv · 0910.2533

A nonlinear stationary phase method for oscillatory Riemann-Hilbert problems

Yen Do
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Abstract

We study the asymptotic behavior of oscillatory Riemann-Hilbert problems arising in the AKNS hierarchy of integrable nonlinear PDE's. Our method is based on the Deift-Zhou nonlinear steepest descent method in which the given Riemann-Hilbert problem localizes to small neighborhoods of stationary phase points. In their original work, Deift and Zhou only considered analytic phase functions. Subsequently Varzugin extended the Deift-Zhou method to a certain restricted class of non-analytic phase functions. In this paper, we extend Varzugin's method to a substantially more general class of non-analytic phase functions. In our work real variable methods play a key role.

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Yen Do. 2010-08-12. A nonlinear stationary phase method for oscillatory Riemann-Hilbert problems. https://arxiv.org/abs/0910.2533

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