Almost Periodic Solutions of the KdV Equation

Abstract

In this talk we discuss the almost periodic behavior in time of space periodic solutions of the KdV equation \[ u_t + uu_x + u_{xxx} = 0.\] We present a new proof, based on a recursion relation of Lenart, for the existence of an infinite sequence of conserved functionals $F_n (u)$ of form$\int {P_n (u)dx} $, $P_n $ a polynomial in u and its derivatives; the existence of such functionals is due to Kruskal, Zabusky, Miura and Gardner. We review and extend the following result of the speaker: the functions u minimizing $F_{N + 1} (u)$ subject to the constraints $F_j (u) = A_j $,$j = 0, \cdots ,N,$ form N-dimensional tori which are invariant under the KdV flow. The extension consists of showing that for certain ranges of the constraining parameters $A_j $ the functional $F_{N + 1} (u)$ has minimax stationary points; these too form invariant N-tori. The Hamiltonian structure of the KdV equation, discovered by Gardner and also by Faddeev and Zakharov, which is used in these studies, is described briefly. In an ...