A nonlinear stationary phase method for oscillatory Riemann-Hilbert problems
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.
Cite
@article{arxiv.0910.2533,
title = {A nonlinear stationary phase method for oscillatory Riemann-Hilbert problems},
author = {Yen Do},
journal= {arXiv preprint arXiv:0910.2533},
year = {2010}
}
Comments
78 pages, referee's corrections and suggestions incorporated, to appear in IMRN