Contour deformations for the collisionless shock region (|x/(2t)| = q_o)

Develop a detailed contour-deformation framework for the oscillatory Riemann–Hilbert problem associated with the inverse scattering transform for the defocusing nonlinear Schrödinger equation with symmetric nonzero boundary conditions in the collisionless shock region defined by |x/(2t)| = q_o, in order to carry out the Deift–Zhou nonlinear steepest descent analysis of the long-time asymptotics in this regime.

Background

The paper implements a numerical inverse scattering transform for the defocusing nonlinear Schrödinger equation with nonzero boundary conditions by reformulating the inverse problem as a Riemann–Hilbert problem in the uniformization variable z. The authors provide explicit contour deformations and factorizations tailored to two regimes of the phase parameter ξ = x/(2t): the solitonic region (|ξ| < q_o) and the solitonless region (|ξ| > q_o), each exhibiting distinct stationary phase structures.

At the boundary case |ξ| = q_o, referred to as the collisionless shock region, the stationary points coalesce and the standard steepest-descent contour deformations used in the other regimes are not directly applicable. The authors explicitly note that a rigorous and detailed analysis of the required contour deformations in this region has not yet been carried out in the long-time asymptotics literature, identifying it as an open problem.