GENERATING FUNCTION OF THE WHITHAM-KDV HIERARCHY AND EFFECTIVE SOLUTION OF THE CAUCHY-PROBLEM

Generating functions for a complete collection of symmetries of the mu ltiphased averaged KdV equation are constructed. The isospectral gener ating function has a potential form with one of the canonical basis ho lomorphic differentials as a potential and possesses some remarkable p roperties at double points of the hyperelliptic Riemann surface. A new representation for the characteristic speeds of the Whitham-KdV hiera rchy is obtained. A global solution to the Whitham system is construct ed 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 o scillations to correlate with the asymptotic predictions of the Lax-Le vermore theory.