Integrals of nonlinear equations of evolution and solitary waves

Abstract

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.