On the nonlinear stability of mKdV breathers

Miguel Angel Alejo Plana; Claudio Muñoz

doi:10.1088/1751-8113/45/43/432001

On the nonlinear stability of mKdV breathers

Miguel Angel Alejo Plana, Claudio Muñoz

Institut for Matematiske Fag

7 Citationer (Scopus)

Abstract

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.

Originalsprog	Engelsk
Artikelnummer	430001
Tidsskrift	Journal of Physics A: Mathematical and Theoretical
Vol/bind	45
Udgave nummer	43
ISSN	1751-8113
DOI	https://doi.org/10.1088/1751-8113/45/43/432001
Status	Udgivet - 2 nov. 2012

Adgang til dokumentet

10.1088/1751-8113/45/43/432001

Citationsformater

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AU - Alejo Plana, Miguel Angel

AU - Muñoz, Claudio

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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.

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DO - 10.1088/1751-8113/45/43/432001

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