On the nonlinear stability of mKdV breathers
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On the nonlinear stability of mKdV breathers. / Alejo Plana, Miguel Angel; Muñoz, Claudio.
I: Journal of Physics A: Mathematical and Theoretical, Bind 45, Nr. 43, 430001, 2012.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - On the nonlinear stability of mKdV breathers
AU - Alejo Plana, Miguel Angel
AU - Muñoz, Claudio
PY - 2012
Y1 - 2012
N2 - Breather modes of the mKdV equation on the real line are known to be elastic under collisions with other breathers and solitons. This fact indicates very strong stability properties of breathers. In this communication we describe a rigorous, mathematical proof of the stability of breathers under a class of small perturbations. Our proof involves the existence of a nonlinear equation satisfied by all breather profiles, and a new Lyapunov functional which controls the dynamics of small perturbations and instability modes. In order to construct such a functional, we work in a subspace of the energy one. However, our proof introduces new ideas in order to attack the corresponding stability problem in the energy space. Some remarks about the sine-Gordon case are also considered.
AB - Breather modes of the mKdV equation on the real line are known to be elastic under collisions with other breathers and solitons. This fact indicates very strong stability properties of breathers. In this communication we describe a rigorous, mathematical proof of the stability of breathers under a class of small perturbations. Our proof involves the existence of a nonlinear equation satisfied by all breather profiles, and a new Lyapunov functional which controls the dynamics of small perturbations and instability modes. In order to construct such a functional, we work in a subspace of the energy one. However, our proof introduces new ideas in order to attack the corresponding stability problem in the energy space. Some remarks about the sine-Gordon case are also considered.
U2 - 10.1088/1751-8113/45/43/432001
DO - 10.1088/1751-8113/45/43/432001
M3 - Journal article
VL - 45
JO - Journal of Physics A: Mathematical and Theoretical
JF - Journal of Physics A: Mathematical and Theoretical
SN - 1751-8113
IS - 43
M1 - 430001
ER -
ID: 49604340