  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.
