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

Alejo, Miguel; Muñoz, Claudio

doi:10.2140/apde.2015.8.629

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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

DOI: 10.2140/apde.2015.8.629

Abstract

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 $H^{1}$ level of regularity, uniformly away from each blow-up point. These new properties are used to prove that mKdV breathers are $H^{1}$ -stable, improving our previous result [Comm. Math. Phys. 324:1 (2013) 233–262], where we only proved $H^{2}$ -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 $L^{2} (ℝ)$ perturbations, provided a corresponding local well-posedness theory is available.

Keywords

mKdV equation, Bäcklund transformation, solitons, breather, stability

Mathematical Subject Classification 2010

Primary: 35Q51, 35Q53

Secondary: 37K10, 37K40

Milestones

Received: 13 February 2014

Revised: 4 December 2014

Accepted: 9 February 2015

Published: 3 June 2015

Authors

Miguel A. Alejo
Instituto Nacional de Matemática Pura e Aplicada 22081-010 Rio de Janeiro, RJ Brazil

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

Vol. 8, No. 3, 2015