Periodic solutions of the KdV equation

Abstract In this paper we construct a large family of special solutions of the KdV equation which are periodic in x and almost periodic in t . These solutions lie on N ‐dimensional tori; very likely they are dense among all solutions. The special solutions are characterized variationally; they minimize F N ( u ), subject to the constraints F j ( u ) = A j , j = −1,…, N − 1; here F j denote the remarkable sequence of conserved functionals discovered by Kruskal and Zabusky. The above minimum problem was originally suggested by them. In exploring the manifold of solutions of this minimum problem we make essential use of Gardner's discovery that these functionals are in involution with respect to a suitable Poisson bracket. Gardner, Greene, Kruskal and Miura have shown that the eigenvalues of the Schrödinger operator are conserved functionals if the potential is a function of t and satisfies the KdV equation. In Section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the Schrödinger operator and the F j . Another result in Section 6 is a slight sharpening of an earlier result of the author and J. Moser: for the special solutions constructed above all but 2 N + 1 eigenvalues of the Schrödinger operator are double. The simplest class of special solutions, N = 1, are cnoidal waves. In an appendix, M. Hyman describes the results of computing numerically the next simplest case, N = 2. These calculations show that the shape of these solutions recurs exactly after a finite time, in a shifted position. The theory verifies this fact.