The Gardner equation and the $L^2$-stability of the $N$-soliton solution of the Korteweg-de Vries equation
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- by Miguel A. Alejo, Claudio Muñoz and Luis Vega PDF
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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.References
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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
- MR Author ID: 975414
- 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
- MR Author ID: 806855
- 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
- MR Author ID: 237776
- Email: luis.vega@ehu.es
- Received by editor(s): December 24, 2010
- Received by editor(s) in revised form: January 21, 2011
- Published electronically: June 22, 2012
- © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 365 (2013), 195-212
- MSC (2010): Primary 35Q51, 35Q53; Secondary 37K10, 37K40
- DOI: https://doi.org/10.1090/S0002-9947-2012-05548-6
- MathSciNet review: 2984057