Integrals of Nonlinear Equations of Evolution and Solitary Waves *

In Section 1 we present a general principle for associating nonlinear equations of evolutions with linear operators so that the eigenvalues of the linear operator are integrals of the nonlinear equation. A striking instance of such a procedure is the discovery by Gardner, Miura and Kruskal that the eigenvalues of the Schrodinger operator are integrals of the Korteweg-de Vries equation. In Section 2 we prove the simplest case of a conjecture of Kruskal and Zabusky concerning the existence of double wave solutions of the Korteweg-de Vries equation, i.e., of solutions which for It( large behave as the superposition of two solitary waves travelling at different speeds. The main tool used is the first of a remarkable series of integrals discovered by Kruskal and Zabusky. 91. In this paper we study the equation (1.1) U t + uu, + Uxxx = 0 introduced by Korteweg and de Vries in their approximate theory of water waves, [3]; we shall refer to it as the KdV equation. Subsequently the KdV equation was found to be relevant for the description of hydromagnetic waves, [2], and in the description of acoustic waves in an anharmonic crystal, 181. Equation (1.1) is a special instance of a nonlinear evolution equation of the form (1.2) Ut = K(u) . We shall study C" solutions of (1.1) defined for all x in ( co, co), which tend to zero as x 4 fa, together with all their x derivatives. It is easy to show that such solutions are uniquely determined by their initial values. Let v be another solution of (1.1) : (1.l)v vt + DUX + vzxx = 0 . Subtracting this from (1.1) and denoting u v by w, we obtain the linear equation Wt + uwx + wvx + wzxx = 0 * This research represents results obtained at the Courant Institute, New York University, under the sponsorship of the Atomic Energy Commission, contract AT(30-1)-1480. Reproduction in whole or in part is permitted for any purpose of the United States Government. 467