337 research outputs found
7th International Conference on Nonlinear Vibrations, Localization and Energy Transfer: Extended Abstracts
International audienceThe purpose of our conference is more than ever to promote exchange and discussions between scientists from all around the world about the latest research developments in the area of nonlinear vibrations, with a particular emphasis on the concept of nonlinear normal modes and targeted energytransfer
Asymptotic profiles for a travelling front solution of a biological equation
We are interested in the existence of depolarization waves in the human
brain. These waves propagate in the grey matter and are absorbed in the white
matter. We consider a two-dimensional model u_t=\Delta u + f(u) \1_{|y|\leq R}
- \alpha u \1_{|y|>R}, with a bistable nonlinearity taking effect only on
the domain \Rm\times [-R,R], which represents the grey matter layer. We study
the existence, the stability and the energy of non-trivial asymptotic profiles
of possible travelling fronts. For this purpose, we present dynamical systems
technics and graphic criteria based on Sturm-Liouville theory and apply them to
the above equation. This yields three different behaviours of the solution
after stimulation, depending of the thickness of the grey matter. This may
partly explain the difficulties to observe depolarization waves in the human
brain and the failure of several therapeutic trials
Simulations of the Nonlinear Helmholtz Equation: Arrest of Beam Collapse, Nonparaxial Solitons, and Counter-Propagating Beams
We solve the (2+1)D nonlinear Helmholtz equation (NLH) for input beams that
collapse in the simpler NLS model. Thereby, we provide the first ever numerical
evidence that nonparaxiality and backscattering can arrest the collapse. We
also solve the (1+1)D NLH and show that solitons with radius of only half the
wavelength can propagate over forty diffraction lengths with no distortions. In
both cases we calculate the backscattered field, which has not been done
previously. Finally, we compute the dynamics of counter-propagating solitons
using the NLH model, which is more comprehensive than the previously used
coupled NLS model.Comment: 6 pages, 6 figures, Lette
Localized Dispersive States in Nonlinear Coupled Mode Equations for Light Propagation in Fiber Bragg Gratings.
Dispersion effects induce new instabilities and dynamics in the weakly nonlinear description of light propagation in fiber Bragg gratings. A new family of dispersive localized pulses that propagate with the group velocity is numerically found, and its stability is also analyzed. The unavoidable different asymptotic order of transport and dispersion effects plays a crucial role in the determination of these localized states. These results are also interesting from the point of view of general pattern formation since this asymptotic imbalance is a generic situation in any transport-dominated (i.e., nonzero group velocity) spatially extended system
On the driven Frenkel-Kontorova model: I. Uniform sliding states and dynamical domains of different particle densities
The dynamical behavior of a harmonic chain in a spatially periodic potential
(Frenkel-Kontorova model, discrete sine-Gordon equation) under the influence of
an external force and a velocity proportional damping is investigated. We do
this at zero temperature for long chains in a regime where inertia and damping
as well as the nearest-neighbor interaction and the potential are of the same
order. There are two types of regular sliding states: Uniform sliding states,
which are periodic solutions where all particles perform the same motion
shifted in time, and nonuniform sliding states, which are quasi-periodic
solutions where the system forms patterns of domains of different uniform
sliding states. We discuss the properties of this kind of pattern formation and
derive equations of motion for the slowly varying average particle density and
velocity. To observe these dynamical domains we suggest experiments with a
discrete ring of at least fifty Josephson junctions.Comment: Written in RevTeX, 9 figures in PostScrip
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