A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. (English) Zbl 0771.35042

Deift, P.; Zhou, X.

Ann. Math. (2) 137, No. 2, 295-368 (1993).

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.

MSC:

35Q15	Riemann-Hilbert problems in context of PDEs
35Q53	KdV equations (Korteweg-de Vries equations)
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

long-time behavior of nonlinear wave equations; inverse scattering method