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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The Gardner equation and the $ L^2$-stability of the $ N$-soliton solution of the Korteweg-de Vries equation


Authors: Miguel A. Alejo, Claudio Muñoz and Luis Vega
Journal: Trans. Amer. Math. Soc. 365 (2013), 195-212
MSC (2010): Primary 35Q51, 35Q53; Secondary 37K10, 37K40
Published electronically: June 22, 2012
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Abstract: Multi-soliton solutions of the Korteweg-de Vries equation (KdV) are shown to be globally $ L^2$-stable, and asymptotically stable in the sense of Martel and Merle. The proof is surprisingly simple and combines the Gardner transform, which links the Gardner and KdV equations, together with the Martel-Merle-Tsai and Martel-Merle recent results on stability and asymptotic stability in the energy space, applied this time to the Gardner equation. As a by-product, the results of Maddocks-Sachs, and Merle-Vega are improved in several directions.


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Additional Information

Miguel A. Alejo
Affiliation: Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Bilbao, España
Address at time of publication: Department of Mathematical Sciences, University of Copenhagen, 2100 Copenhagen, Denmark
Email: miguelangel.alejo@ehu.es, miguel.alejo@math.ku.dk

Claudio Muñoz
Affiliation: Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Bilbao, España
Address at time of publication: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: Claudio.Munoz@math.uvsq.fr, cmunoz@math.uchicago.edu

Luis Vega
Affiliation: Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Bilbao, España
Email: luis.vega@ehu.es

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05548-6
PII: S 0002-9947(2012)05548-6
Keywords: KdV equation, Gardner equation, integrability, multi-soliton, stability, asymptotic stability, Miura transform
Received by editor(s): December 24, 2010
Received by editor(s) in revised form: January 21, 2011
Published electronically: June 22, 2012
Article copyright: © Copyright 2012 American Mathematical Society