Adjust nonlinear steepest descent to the singular RHP#0

Develop an adaptation of the Deift–Zhou nonlinear steepest descent method for the original Riemann–Hilbert problem RHP#0 arising from the KdV scattering data with Wigner–von Neumann–type initial profiles, where the reflection coefficient R(k) may satisfy |R(k)| ≥ 1 and the factor R(k)/(1 − |R(k)|^2) is not approximable by analytic functions in the L1 norm, so that long-time asymptotics of the corresponding KdV solutions can be analyzed within this framework.

Background

The paper reformulates the time-evolved scattering identity for the KdV equation as a vector Riemann–Hilbert problem (RHP#0) tailored to Wigner–von Neumann–type initial data that induce a resonance. In contrast to classical short-range settings, the associated jump matrix involves a reflection coefficient R(k) for which |R(k)| may not be strictly less than 1, and the standard nonlinear steepest descent approach relies on approximating R(k)/(1 − |R(k)|^2) by analytic functions in L1, a step that fails here.

The authors note that while RHP#0 is well-posed and contour deformation is feasible, extending the nonlinear steepest descent method to this singular setting remains unresolved. An adaptation would enable rigorous derivation of long-time asymptotics in the resonance regime produced by Wigner–von Neumann resonances.