Electron. J. Differential Equations, Vol. 2017 (2017), No. 56, pp. 1-26.

On the variational structure of breather solutions II: periodic mKdV equation

Miguel A. Alejo, Claudio Munoz, Jose M. Palacios

We study the periodic modified KdV equation, where a periodic in space and time breather solution is known from the work of Kevrekidis et al. [19]. We show that these breathers satisfy a suitable elliptic equation, and we also discuss via numerics its spectral stability. We also identify a source of nonlinear instability for the case described in [19], and we conjecture that, even if spectral stability is satisfied, nonlinear stability/instability depends only on the sign of a suitable discriminant function, a condition that is trivially satisfied in the case of non-periodic (in space) mKdV breathers. Finally, we present a new class of breather solution for mKdV, believed to exist from geometric considerations, and which is periodic in time and space, but has nonzero mean, unlike standard breathers.

Submitted January 21, 2017. Published February 22, 2017.
Math Subject Classifications: 35Q51, 35Q53, 37K10, 37K40.
Key Words: Modified KdV; sine-Gordon equation; periodic mKdV; integrability; breather; stability.

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Miguel A. Alejo
Departamento de Matemática
Universidade Federal de Santa Catarina, Brasil
email: miguel.alejo@ufsc.br
Claudio Muñnoz
CNRS and Departamento de Ingeniería Matemática DIM
Universidad de Chile, Chile
email: cmunoz@dim.uchile.cl
José M. Palacios
Departamento de Ingeniería Matemática DIM
Universidad de Chile, Chile
email: jpalacios@dim.uchile.cl

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