Analysis & PDE

  • Anal. PDE
  • Volume 8, Number 3 (2015), 629-674.

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

Miguel Alejo and Claudio Muñoz

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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 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.

Article information

Anal. PDE, Volume 8, Number 3 (2015), 629-674.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q51: Soliton-like equations [See also 37K40] 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.) 37K40: Soliton theory, asymptotic behavior of solutions

mKdV equation Bäcklund transformation solitons breather stability


Alejo, Miguel; Muñoz, Claudio. Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers. Anal. PDE 8 (2015), no. 3, 629--674. doi:10.2140/apde.2015.8.629.

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