A steepest descent method for oscillatory Riemann-Hilbert problems

Abstract

In this announcement we present a general and new approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, in evaluating the long-time behavior of nonlinear wave equations solvable by the inverse scattering method. We will restrict ourselves here exclusively to the modified Korteweg de Vries (MKdV) equation, yt -6y2yx + yxxx = 0, -oo < x < oo, t > 0, y(x, t = 0)=y0(x), but it will be clear immediately to the reader with some experience in the field, that the method extends naturally and easily to the general class of wave equations solvable by the inverse scattering method, such as the KdV, nonlinear Schrdinger (NLS), and Boussinesq equations, etc., and also to "integrable" ordinary differential equations such as the Painlev transcendents. As described, for example, in [IN] or [BC], the inverse scattering method for the MKdV equation leads to a Riemann-Hilbert factorization problem for a 2x2 matrix valued function m = m(-; x, t) analytic in C\R, m+(z) = m-(z)vXJ, zgR,| m(z) - / asz->oo, J m(z) = lim m(z ie; x, t), 10 VXit(z) = e-^'zi+x2)a}v^zy(4tz'+xz)ai ^ ffj _ 1 ^> \ and _ If yo(x) is in Schwartz space, then so is r(z) and r(z) = -r(-z), sup \r(z)\ < 1. zeR From the inverse point of view, given v(z), one considers a singular integral