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.

Riemann-Hilbert approach to the time-dependent generalized sine kernel

“…RHP with multiple saddle-points have been considered in [47]. This work was later extended to the case of less regular functions and higer order saddle-points in [17].…”

Long-time behavior of solutions to the derivative nonlinear Schrödinger equation for soliton-free initial data

“…A formal analysis of general oscillatory RHP with Schwartz class scattering data is presented in Varzugin [21]. More recently, Do [10] developed a version of the Deift-Zhou steepest descent method that emphasizes real-variable methods and extends to a much larger class of RHPs. A key step in the nonlinear steepest descent method consists in deforming the contour associated to the RHP in a way adapted to the structure of the phase function that defines the oscillatory dependence on parameters (for our case, see (1.7) for the jump matrix, (1.8) for the phase function, and Figure 4.1 for the deformation).…”

Inverse Scattering in 60 Minutes

“…To appreciate what the analysis of such a Riemann-Hilbert problem entails, we consider an illuminating "toy model" problem that appears in the paper of Do [10]. In what follows it will be useful to recall: The phase function (4.1b) has a single stationary point at ξ "´x{4t.…”