On nonlinear nonlocal equations in wave theory. I: The Whitham equation (Q1173549)

The article is devoted to the discussion of the theory of nonlinear nonlocal equations. The survey of physical problems generating nonlinear nonlocal equations is given. In this item, which is the first part of the whole article, the equation of J. Whitham is considered. The principles of preserving, breaking of solutions in finite time, smoothing in the course of time of initial discontinuous perturbations, a global with respect to time existence of the solution, the existence of generalised solutions, and the properties of solutions of the Cauchy problem for the periodic equation of J. Whitham are considered here.