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Review of long time asymptotics and collision of solitons for the quartic generalized Korteweg—de Vries equation

Published online by Cambridge University Press:  04 April 2011

Yvan Martel  and
Frank Merle
Yvan Martel
Affiliation:
Laboratoire de Mathématiques de Versailles, UMR CNRS 8100, Université de Versailles St Quentin and Institut Universitaire de France, 45 Avenue des États-Unis, 78035 Versailles, Franceyvan.martel@math.uvsq.fr
Frank Merle
Affiliation:
Laboratoire de Mathématiques, UMR CNRS 8088, Université de Cergy-Pontoise and Institut des Hautes Études Scientifiques, 2 Avenue Adolphe Chauvin, 95302 Cergy-Pontoise, Francefrank.merle@math.u-cergy.fr
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Abstract

We review recent nonlinear partial differential equation techniques developed to address questions concerning solitons for the quartic generalized Korteweg—de Vries equation (gKdV) and other generalizations of the KdV equation. We draw a comparison between results obtained in this way and some elements of the classical integrability theory for the original KdV equation, which serve as a reference for soliton and multi-soliton problems. First, known results on stability and asymptotic stability of solitons for gKdV equations are reviewed from several different sources. Second, we consider the problem of the interaction of two solitons for the quartic gKdV equation. We focus on recent results and techniques from a previous paper by the present authors concerning the interaction of two almost-equal solitons.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 141 , Issue 2 , April 2011 , pp. 287 - 317
Copyright
Copyright © Royal Society of Edinburgh 2011

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