Integrals of nonlinear equations of evolution and solitary waves

1968/09/01 by Peter D. Lax · 22 citations
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Nonlinear Photonic Systems #Advanced Mathematical Physics Problems #Mathematics #Eigenvalues and eigenvectors #Superposition principle #Nonlinear system #Korteweg–de Vries equation #Operator (biology) #Conjecture #Mathematical analysis #Series (stratigraphy) #Kruskal's algorithm #Mathematical physics #Pure mathematics #Physics #Quantum mechanics #Discrete mathematics

paper · doi:10.1002/cpa.3160210503

openalex publication_date 1968/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves
• Korteweg-de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion
• Method for Solving the Korteweg-deVries Equation
• Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States

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