A Nonlinear Stationary Phase Method for Oscillatory Riemann–Hilbert Problems

We study the asymptotic behavior of oscillatory Riemann–Hilbert problems (RHPs) arising in the Ablowitz-Kaup-Newell-Segur hierarchy of integrable nonlinear partial differential equations. Our method is based on the Deift–Zhou nonlinear steepest descent method in which the given RHP 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 nonanalytic phase functions. In this paper, we extend Varzugin’s method to a substantially more general class of nonanalytic phase functions. In our work, real variable methods play a key role.