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2015 Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers
Miguel Alejo, Claudio Muñoz
Anal. PDE 8(3): 629-674 (2015). 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 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.

Citation

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Miguel Alejo. Claudio Muñoz. "Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers." Anal. PDE 8 (3) 629 - 674, 2015. https://doi.org/10.2140/apde.2015.8.629

Information

Received: 13 February 2014; Revised: 4 December 2014; Accepted: 9 February 2015; Published: 2015
First available in Project Euclid: 16 November 2017

zbMATH: 1323.35156
MathSciNet: MR3353827
Digital Object Identifier: 10.2140/apde.2015.8.629

Subjects:
Primary: 35Q51, 35Q53
Secondary: 37K10, 37K40

Rights: Copyright © 2015 Mathematical Sciences Publishers

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