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Similarity solutions of the generalized Korteweg–de Vries equation

Published online by Cambridge University Press:  01 September 1999

J. L. BONA  and
F. B. WEISSLER
J. L. BONA
Affiliation:
Department of Mathematics and Texas Institute for Computational and Applied Mathematics, The University of Texas at Austin, Austin, TX 78712, USA
F. B. WEISSLER
Affiliation:
Laboratoire Analyse, Geométrie et Applications, UMA CNRS 7539, Institut Galilée-Université Paris XIII, F-93430, Villetaneuse, France
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Abstract

Numerical simulations and group invariance considerations point to the existence of similarity solutions of the form

formula here

of the generalized Korteweg–de Vries equation

formula here

Here, x∗, t∗ and c are real parameters, x and t are real variables with t≠t∗, p is a positive integer and interest is focussed on the case where p[ges ]4 for which solutions of the initial-value problem for (**) are not known to be always globally defined. It is shown that smooth solutions of (**) of the form appearing in (*) do indeed exist. Some detailed properties of the function ψ appearing in (*) are also obtained.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 127 , Issue 2 , September 1999 , pp. 323 - 351
Copyright
The Cambridge Philosophical Society 1999

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