Miguel A. Alejo, Claudio Munoz, Jose M. Palacios
Abstract:
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 |
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Claudio Muñnoz CNRS and Departamento de Ingeniería Matemática DIM Universidad de Chile, Chile email: cmunoz@dim.uchile.cl |
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José M. Palacios Departamento de Ingeniería Matemática DIM Universidad de Chile, Chile email: jpalacios@dim.uchile.cl |
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