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Integrals of nonlinear equations of evolution and solitary waves
Author(s) -
Lax Peter D.
Publication year - 1968
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160210503
Subject(s) - 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 , paleontology , biochemistry , chemistry , repressor , biology , transcription factor , gene , spanning tree
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.