Integrals of nonlinear equations of evolution and solitary waves†
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.
Abstract
In Section 1 we present a general principle for associating nonlinear equations evolutions with linear operators so that the eigenvalues of the linear operator integrals of the nonlinear equation. A striking instance of such a procedure discovery by Gardner, Miura and Kruskal that the eigenvalues of the Schrödinger 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 |I| large behave as the superposition of two solitary waves travelling at different speeds. The main tool used is the first of remarkable series of integrals discovered by Kruskal and Zabusky.
