205 research outputs found
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
Nonlinear Propagation of Light in One Dimensional Periodic Structures
We consider the nonlinear propagation of light in an optical fiber waveguide
as modeled by the anharmonic Maxwell-Lorentz equations (AMLE). The waveguide is
assumed to have an index of refraction which varies periodically along its
length. The wavelength of light is selected to be in resonance with the
periodic structure (Bragg resonance). The AMLE system considered incorporates
the effects non-instantaneous response of the medium to the electromagnetic
field (chromatic or material dispersion), the periodic structure (photonic band
dispersion) and nonlinearity. We present a detailed discussion of the role of
these effects individually and in concert. We derive the nonlinear coupled mode
equations (NLCME) which govern the envelope of the coupled backward and forward
components of the electromagnetic field. We prove the validity of the NLCME
description and give explicit estimates for the deviation of the approximation
given by NLCME from the {\it exact} dynamics, governed by AMLE. NLCME is known
to have gap soliton states. A consequence of our results is the existence of
very long-lived {\it gap soliton} states of AMLE. We present numerical
simulations which validate as well as illustrate the limits of the theory.
Finally, we verify that the assumptions of our model apply to the parameter
regimes explored in recent physical experiments in which gap solitons were
observed.Comment: To appear in The Journal of Nonlinear Science; 55 pages, 13 figure
Normal form for travelling kinks in discrete Klein-Gordon lattices
We study travelling kinks in the spatial discretizations of the nonlinear
Klein--Gordon equation, which include the discrete lattice and the
discrete sine--Gordon lattice. The differential advance-delay equation for
travelling kinks is reduced to the normal form, a scalar fourth-order
differential equation, near the quadruple zero eigenvalue. We show numerically
non-existence of monotonic kinks (heteroclinic orbits between adjacent
equilibrium points) in the fourth-order equation. Making generic assumptions on
the reduced fourth-order equation, we prove the persistence of bounded
solutions (heteroclinic connections between periodic solutions near adjacent
equilibrium points) in the full differential advanced-delay equation with the
technique of center manifold reduction. Existence and persistence of multiple
kinks in the discrete sine--Gordon equation are discussed in connection to
recent numerical results of \cite{ACR03} and results of our normal form
analysis
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