Vol. 8, No. 3, 2015

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Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers

Miguel A. Alejo and Claudio Muñoz

Vol. 8 (2015), No. 3, 629–674

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 H1 level of regularity, uniformly away from each blow-up point. These new properties are used to prove that mKdV breathers are H1-stable, improving our previous result [Comm. Math. Phys. 324:1 (2013) 233–262], where we only proved H2-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 L2() perturbations, provided a corresponding local well-posedness theory is available.

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mKdV equation, Bäcklund transformation, solitons, breather, stability
Mathematical Subject Classification 2010
Primary: 35Q51, 35Q53
Secondary: 37K10, 37K40
Received: 13 February 2014
Revised: 4 December 2014
Accepted: 9 February 2015
Published: 3 June 2015
Miguel A. Alejo
Instituto Nacional de Matemática Pura e Aplicada
22081-010 Rio de Janeiro, RJ
Claudio Muñoz
CNRS and Laboratoire de Mathématiques d’Orsay UMR 8628
Université Paris-Sud
Bât. 425 Faculté des Sciences d’Orsay
F-91405 Orsay