Quasiclassical asymptotics of the inverse scattering solutions of the KdV equation and the solution of Whitham's modulation equations

The initial value problem for the KdV equation is studied in the limit of weak dispertion. It may be considered as a model for nondissipative shock wave in plasmas. The perturbation theory in power series of the weak dispersion parameter leads to the Riemann simple wave equation describing nonlinear effects of wave front sharpening and “overturning”. The subsequent phase of the nondissipative shock wave evolution is described by Whitham's modulation equations.
Another approach used in this paper is based on the asymptotic study of the exact solution by the inverse scattering problem technique. The WKB formulas for the direct scattering problem solution and the exact multisolution of the inverse problem are considered. As a result a system of closed relations between $x$, $t$ and the modulation parameters is obtained, which presents an exact solution of the Whitham's equations.