6 research outputs found
Nonlinear stability of mKdV breathers
Breather solutions of the modified Korteweg-de Vries equation are shown to be
globally stable in a natural H^2 topology. Our proof introduces a new Lyapunov
functional, at the H^2 level, which allows to describe the dynamics of small
perturbations, including oscillations induced by the periodicity of the
solution, as well as a direct control of the corresponding instability modes.
In particular, degenerate directions are controlled using low-regularity
conservation laws.Comment: 24 pp., submitte
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 recognized because they blow-up in finite time. We
establish stability properties at the H^1 level of regularity, uniformly away
from each blow-up point. These new properties are used to prove that mKdV
breathers are H^1 stable, improving our previous result, where we only proved
H^2 stability. The main new ingredient of the proof is the use of a B\"acklund
transformation which links 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 Transformation, our proof works even
under rough perturbations, provided a corresponding local well-posedness theory
is available.Comment: 45 pages, we thank Yvan Martel for pointing us a gap in the previous
version of this pape
On the variational structure of breather solutions
In this paper we give a systematic and simple account that put in evidence
that many breather solutions of integrable equations satisfy suitable
variational elliptic equations, which also implies that the stability problem
reduces in some sense to the study of the spectrum of explicit linear
systems (\emph{spectral stability}), and the understanding of how bad
directions (if any) can be controlled using low regularity conservation laws.
We exemplify this idea in the case of the modified Korteweg-de Vries (mKdV),
Gardner, and sine-Gordon (SG) equations. Then we perform numerical simulations
that confirm, at the level of the spectral problem, our previous rigorous
results, where we showed that mKdV breathers are and stable,
respectively. In a second step, we also discuss the Gardner and the Sine-Gordon
cases, where the spectral study of a fourth-order linear matrix system is the
key element to show stability. Using numerical methods, we confirm that all
spectral assumptions leading to the stability of SG breathers
are numerically satisfied, even in the ultra-relativistic, singular regime. In
a second part, we study the periodic mKdV case, where a periodic breather is
known from the work of Kevrekidis et al. We rigorously show that these
breathers satisfy a suitable elliptic equation, and we also show numerical
spectral stability. However, we also identify the source of nonlinear
instability in the case described in Kevrekidis et al. 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.Comment: 55 pages; This paper is an improved version of our previous paper
1309.0625 and hence we replace i