A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation
We present a new and general approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, when evaluating the long-time behavior of nonlinear wave equations solvable by the inverse scattering method. We restrict ourselves to the modified Korteweg-de Vries (MKdV) equation \[ y_ t-6y^ 2y_ x+y_{xxx}=0, \qquad -\infty<x<\infty, \qquad t\geq 0, \qquad y(x,t=0)=y_ 0(x), \] but the method extends naturally to the general class of wave equations solvable by the inverse scattering method, such as the KdV, nonlinear Schrödinger (NLS) and Boussinesq equations, and also to ``integrable'' ordinary differential equations such as the Painlevé transcendents.
long-time behavior of nonlinear wave equations
scientific article