Generating function of the Whitham-KdV hierarchy and effective solution of the Cauchy problem

Generating functions for a complete collection of symmetries of the multiphased averaged KdV equation are constructed. The isospectral generating function has a potential form with one of the canonical basis holomorphic differentials as a potential and possesses some remarkable properties at double points of the hyperelliptic Riemann surface. A new representation for the characteristic speeds of the Whitham-KdV hierarchy is obtained. A global solution to the Whitham system is constructed in an effective form for the case of smooth decreasing initial data with a finite number of inflection points. The large time asymptotics of this solution implies the single-phase limiting behaviour of the oscillations to correlate with the asymptotic predictions of the Lax-Levermore theory.