| We develop a rigorous nonlinear stationary phase method to study asymptotical behaviors of oscillatory Riemann-Hilbert problems arising in the AKNS hierarchy of integrable nonlinear partial differential equations, where the oscillating phase is not assumed to be analytic and has a finite number of stationary phase points of arbitrary orders. The main idea is to localize the given Riemann-Hilbert problem to small neighborhoods of stationary points, where the phase function could then be well-approximated by suitable analytic functions and thus allows for a steepest descent argument. The method developed in this thesis is based on previous work of Varzugin and may have applications to other oscillatory Riemann-Hilbert settings, and potential adaptations are discussed. |
