58 research outputs found
About coherent structures in random shell models for passive scalar advection
A study of anomalous scaling in models of passive scalar advection in terms
of singular coherent structures is proposed. The stochastic dynamical system
considered is a shell model reformulation of Kraichnan model. We extend the
method introduced in \cite{DDG99} to the calculation of self-similar instantons
and we show how such objects, being the most singular events, are appropriate
to capture asymptotic scaling properties of the scalar field. Preliminary
results concerning the statistical weight of fluctuations around these optimal
configurations are also presented.Comment: 4 pages, 2 postscript figures, submitted to PR
Surmounting collectively oscillating bottlenecks
We study the collective escape dynamics of a chain of coupled, weakly damped
nonlinear oscillators from a metastable state over a barrier when driven by a
thermal heat bath in combination with a weak, globally acting periodic
perturbation. Optimal parameter choices are identified that lead to a drastic
enhancement of escape rates as compared to a pure noise-assisted situation. We
elucidate the speed-up of escape in the driven Langevin dynamics by showing
that the time-periodic external field in combination with the thermal
fluctuations triggers an instability mechanism of the stationary homogeneous
lattice state of the system. Perturbations of the latter provided by incoherent
thermal fluctuations grow because of a parametric resonance, leading to the
formation of spatially localized modes (LMs). Remarkably, the LMs persist in
spite of continuously impacting thermal noise. The average escape time assumes
a distinct minimum by either tuning the coupling strength and/or the driving
frequency. This weak ac-driven assisted escape in turn implies a giant speed of
the activation rate of such thermally driven coupled nonlinear oscillator
chains
Anomalous and dimensional scaling in anisotropic turbulence
We present a numerical study of anisotropic statistical fluctuations in
homogeneous turbulent flows. We give an argument to predict the dimensional
scaling exponents, (p+j)/3, for the projections of p-th order structure
function in the j-th sector of the rotational group. We show that measured
exponents are anomalous, showing a clear deviation from the dimensional
prediction. Dimensional scaling is subleading and it is recovered only after a
random reshuffling of all velocity phases, in the stationary ensemble. This
supports the idea that anomalous scaling is the result of a genuine inertial
evolution, independent of large-scale behavior.Comment: 4 pages, 3 figure
Supersonic Discrete Kink-Solitons and Sinusoidal Patterns with "Magic" wavenumber in Anharmonic Lattices
The sharp pulse method is applied to Fermi-Pasta-Ulam (FPU) and Lennard-Jones
(LJ) anharmonic lattices. Numerical simulations reveal the presence of high
energy strongly localized ``discrete'' kink-solitons (DK), which move with
supersonic velocities that are proportional to kink amplitudes. For small
amplitudes, the DK's of the FPU lattice reduce to the well-known ``continuous''
kink-soliton solutions of the modified Korteweg-de Vries equation. For high
amplitudes, we obtain a consistent description of these DK's in terms of
approximate solutions of the lattice equations that are obtained by restricting
to a bounded support in space exact solutions with sinusoidal pattern
characterized by the ``magic'' wavenumber . Relative displacement
patterns, velocity versus amplitude, dispersion relation and exponential tails
found in numerical simulations are shown to agree very well with analytical
predictions, for both FPU and LJ lattices.Comment: Europhysics Letters (in print
Fluctuation-response relation in turbulent systems
We address the problem of measuring time-properties of Response Functions
(Green functions) in Gaussian models (Orszag-McLaughin) and strongly
non-Gaussian models (shell models for turbulence). We introduce the concept of
{\it halving time statistics} to have a statistically stable tool to quantify
the time decay of Response Functions and Generalized Response Functions of high
order. We show numerically that in shell models for three dimensional
turbulence Response Functions are inertial range quantities. This is a strong
indication that the invariant measure describing the shell-velocity
fluctuations is characterized by short range interactions between neighboring
shells
Oscillatory Instabilities of Standing Waves in One-Dimensional Nonlinear Lattices
In one-dimensional anharmonic lattices, we construct nonlinear standing waves
(SWs) reducing to harmonic SWs at small amplitude. For SWs with spatial
periodicity incommensurate with the lattice period, a transition by breaking of
analyticity versus wave amplitude is observed. As a consequence of the
discreteness, oscillatory linear instabilities, persisting for arbitrarily
small amplitude in infinite lattices, appear for all wave numbers Q not equal
to zero or \pi. Incommensurate analytic SWs with |Q|>\pi/2 may however appear
as 'quasi-stable', as their instability growth rate is of higher order.Comment: 4 pages, 6 figures, to appear in Phys. Rev. Let
Pattern formation and localization in the forced-damped FPU lattice
We study spatial pattern formation and energy localization in the dynamics of
an anharmonic chain with quadratic and quartic intersite potential subject to
an optical, sinusoidally oscillating field and a weak damping. The
zone-boundary mode is stable and locked to the driving field below a critical
forcing that we determine analytically using an approximate model which
describes mode interactions. Above such a forcing, a standing modulated wave
forms for driving frequencies below the band-edge, while a ``multibreather''
state develops at higher frequencies. Of the former, we give an explicit
approximate analytical expression which compares well with numerical data. At
higher forcing space-time chaotic patterns are observed.Comment: submitted to Phys.Rev.
On modulational instability and energy localization in anharmonic lattices at finite energy density
The localization of vibrational energy, induced by the modulational
instability of the Brillouin-zone-boundary mode in a chain of classical
anharmonic oscillators with finite initial energy density, is studied within a
continuum theory. We describe the initial localization stage as a gas of
envelope solitons and explain their merging, eventually leading to a single
localized object containing a macroscopic fraction of the total energy of the
lattice. The initial-energy-density dependences of all characteristic time
scales of the soliton formation and merging are described analytically. Spatial
power spectra are computed and used for the quantitative explanation of the
numerical results.Comment: 12 pages, 7 figure
Hamiltonian Hopf bifurcations in the discrete nonlinear Schr\"odinger trimer: oscillatory instabilities, quasiperiodic solutions and a 'new' type of self-trapping transition
Oscillatory instabilities in Hamiltonian anharmonic lattices are known to
appear through Hamiltonian Hopf bifurcations of certain time-periodic solutions
of multibreather type. Here, we analyze the basic mechanisms for this scenario
by considering the simplest possible model system of this kind where they
appear: the three-site discrete nonlinear Schr\"odinger model with periodic
boundary conditions. The stationary solution having equal amplitude and
opposite phases on two sites and zero amplitude on the third is known to be
unstable for an interval of intermediate amplitudes. We numerically analyze the
nature of the two bifurcations leading to this instability and find them to be
of two different types. Close to the lower-amplitude threshold stable
two-frequency quasiperiodic solutions exist surrounding the unstable stationary
solution, and the dynamics remains trapped around the latter so that in
particular the amplitude of the originally unexcited site remains small. By
contrast, close to the higher-amplitude threshold all two-frequency
quasiperiodic solutions are detached from the unstable stationary solution, and
the resulting dynamics is of 'population-inversion' type involving also the
originally unexcited site.Comment: 25 pages, 11 figures, to be published in J. Phys. A: Math. Gen.
Revised and shortened version with few clarifying remarks adde
- …