THE INITIAL-VALUE PROBLEM FOR THE WHITHAM AVERAGED SYSTEM

We study the initial value problem for the Whitham averaged system whi ch is important in determining the KdV zero dispersion limit. We use t he hodograph method to show that, for a generic non-trivial monotone i nitial data, the Whitham averaged system has a solution within a regio n in the x-t plane for all time bigger than a large time. Furthermore, the Whitham solution matches the Burgers solution on the boundaries o f the region. For hump-like initial data, the hodograph method is modi fied to solve the non-monotone (in x) solutions of the Whitham average d system. In this way, we show that, for a hump-like initial data, the Whitham averaged system has a solution within a cusp for a short time after the increasing and decreasing parts of the initial data begin t o interact. On the cusp, the Whitham and Burgers solutions are matched .