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
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.
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
Permanent link to this document
Digital Object Identifier
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
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. https://projecteuclid.org/euclid.apde/1510843096