Asymptotic behaviour of solutions of the Korteweg-de Vries equation

A rigorous proof that any solution of the Korteweg-de Vries equation with smooth initial data decaying sufficiently fast at infínity tends as t → ±∞ to a pure N-soliton solution at a spatially uniform rate of ∥t∥ −1 3 is provided. It is also proved that the solitonless solutions have a spatially uni...

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Bibliographic Details
Published in:Annals of physics 1985, Vol.162 (1), p.132-154
Main Authors: Scharf, G., Wreszinski, W.F.
Format: Article
Language:English
Subjects:
Exact sciences and technology
Function theory, analysis
Mathematical methods in physics
Physics
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Summary:A rigorous proof that any solution of the Korteweg-de Vries equation with smooth initial data decaying sufficiently fast at infínity tends as t → ±∞ to a pure N-soliton solution at a spatially uniform rate of ∥t∥ −1 3 is provided. It is also proved that the solitonless solutions have a spatially uniform decay rate of ∥t∥ −2 3 , i.e., faster than the solutions of the corresponding linear equation. Some possible implications for scattering theory are discussed.
ISSN:0003-4916
1096-035X
DOI:10.1016/0003-4916(85)90231-3