Almost Periodic Solutions of the KdV Equation

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 Appendix, M. Hyman describes numerical studies of the stability of some invariant 2-tori for the KdV flow; the numerical evidence points to stability.

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