Integrals of Nonlinear Equations of Evolution and Solitary Waves (Classic Reprint)

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Excerpt from 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 t 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.
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