New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems. (English) Zbl 0873.65111

Deift, P.; Vanakides, S.; Zhou, X.

Int. Math. Res. Not. 1997, No. 6, 285-299 (1997).

The purpose is to analyze the behaviour of the solution of the Korteweg-de Vries (KdV) equation \(u_t- 6uu_x+ \varepsilon^2 u_{xxx} =0\), \(u(x,0) =u_0(x)\), \(\varepsilon>0\) when \(\varepsilon\downarrow 0\). The approach to this small dispersion problem is based on the Riemann-Hilbert (RH) formulation of the inverse scattering transform, involving a steepest descent method for oscillatory RH problems.
A systematic procedure for determining the contour of the main contribution to the solution of the RH problem is introduced. The initial value problem of the modulation equations is reduced to the solution of a set of algebraic equations constrained by algebraic inequalities.

Reviewer: G.Jumarie (Montréal)

MSC:

65Z05	Applications to the sciences
35Q53	KdV equations (Korteweg-de Vries equations)
35Q15	Riemann-Hilbert problems in context of PDEs
81U40	Inverse scattering problems in quantum theory

Keywords:

Riemann-Hilbert problem; Korteweg-de Vries equation; inverse scattering; steepest descent method