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Dynamics of
complex-valued modified KdV solitons with applications to the
stability of breathers
Miguel A. Alejo and Claudio Muñoz |
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Vol. 8 (2015), No. 3, 629–674
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Abstract |
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We study the long-time dynamics of complex-valued modified Korteweg–de Vries (mKdV) solitons, which are distinguished because they blow up in finite time. We establish stability properties at the level of regularity, uniformly away from each blow-up point. These new properties are used to prove that mKdV breathers are -stable, improving our previous result [Comm. Math. Phys. 324:1 (2013) 233–262], where we only proved -stability. The main new ingredient of the proof is the use of a Bäcklund transformation which relates the behavior of breathers, complex-valued solitons and small real-valued solutions of the mKdV equation. We also prove that negative energy breathers are asymptotically stable. Since we do not use any method relying on the inverse scattering transform, our proof works even under perturbations, provided a corresponding local well-posedness theory is available. |
Keywords
mKdV equation, Bäcklund transformation, solitons, breather,
stability
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Mathematical Subject Classification 2010
Primary: 35Q51, 35Q53
Secondary: 37K10, 37K40
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Milestones
Received: 13 February 2014
Revised: 4 December 2014
Accepted: 9 February 2015
Published: 3 June 2015
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Authors |
| Miguel A. Alejo | |
| Instituto Nacional de Matemática
Pura e Aplicada 22081-010 Rio de Janeiro, RJ Brazil |
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| Claudio Muñoz | |
| CNRS and Laboratoire de
Mathématiques d’Orsay UMR 8628 Université Paris-Sud Bât. 425 Faculté des Sciences d’Orsay F-91405 Orsay France |
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