5,068 research outputs found
Continuation for thin film hydrodynamics and related scalar problems
This chapter illustrates how to apply continuation techniques in the analysis
of a particular class of nonlinear kinetic equations that describe the time
evolution through transport equations for a single scalar field like a
densities or interface profiles of various types. We first systematically
introduce these equations as gradient dynamics combining mass-conserving and
nonmass-conserving fluxes followed by a discussion of nonvariational amendmends
and a brief introduction to their analysis by numerical continuation. The
approach is first applied to a number of common examples of variational
equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including
certain thin-film equations for partially wetting liquids on homogeneous and
heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal
equations. Second we consider nonvariational examples as the
Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard
equations and thin-film equations describing stationary sliding drops and a
transversal front instability in a dip-coating. Through the different examples
we illustrate how to employ the numerical tools provided by the packages
auto07p and pde2path to determine steady, stationary and time-periodic
solutions in one and two dimensions and the resulting bifurcation diagrams. The
incorporation of boundary conditions and integral side conditions is also
discussed as well as problem-specific implementation issues
Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. I: General presentation and periodic solutions
We present experimental results on hydrothermal traveling-waves dynamics in
long and narrow 1D channels. The onset of primary traveling-wave patterns is
briefly presented for different fluid heights and for annular or bounded
channels, i.e., within periodic or non-periodic boundary conditions. For
periodic boundary conditions, by increasing the control parameter or changing
the discrete mean-wavenumber of the waves, we produce modulated waves patterns.
These patterns range from stable periodic phase-solutions, due to supercritical
Eckhaus instability, to spatio-temporal defect-chaos involving traveling holes
and/or counter-propagating-waves competition, i.e., traveling sources and
sinks. The transition from non-linearly saturated Eckhaus modulations to
transient pattern-breaks by traveling holes and spatio-temporal defects is
documented. Our observations are presented in the framework of coupled complex
Ginzburg-Landau equations with additional fourth and fifth order terms which
account for the reflection symmetry breaking at high wave-amplitude far from
onset. The second part of this paper (nlin.PS/0208030) extends this study to
spatially non-periodic patterns observed in both annular and bounded channel.Comment: 45 pages, 21 figures (elsart.cls + AMS extensions). Accepted in
Physica D. See also companion paper "Nonlinear dynamics of waves and
modulated waves in 1D thermocapillary flows. II: Convective/absolute
transitions" (nlin.PS/0208030). A version with high resolution figures is
available on N.G. web pag
Chaos assisted tunnelling with cold atoms
In the context of quantum chaos, both theory and numerical analysis predict
large fluctuations of the tunnelling transition probabilities when irregular
dynamics is present at the classical level. We consider here the
non-dissipative quantum evolution of cold atoms trapped in a time-dependent
modulated periodic potential generated by two laser beams. We give some precise
guidelines for the observation of chaos assisted tunnelling between invariant
phase space structures paired by time-reversal symmetry.Comment: submitted to Phys. Rev. E ; 16 pages, 13 figures; figures of better
quality can be found at http://www.phys.univ-tours.fr/~mouchet
Evolution of wave packets in quasi-1D and 1D random media: diffusion versus localization
We study numerically the evolution of wavepackets in quasi one-dimensional
random systems described by a tight-binding Hamiltonian with long-range random
interactions. Results are presented for the scaling properties of the width of
packets in three time regimes: ballistic, diffusive and localized. Particular
attention is given to the fluctuations of packet widths in both the diffusive
and localized regime. Scaling properties of the steady-state distribution are
also analyzed and compared with theoretical expression borrowed from
one-dimensional Anderson theory. Analogies and differences with the kicked
rotator model and the one-dimensional localization are discussed.Comment: 32 pages, LaTex, 11 PostScript figure
Nonlinear Dynamics of Particles Excited by an Electric Curtain
The use of the electric curtain (EC) has been proposed for manipulation and
control of particles in various applications. The EC studied in this paper is
called the 2-phase EC, which consists of a series of long parallel electrodes
embedded in a thin dielectric surface. The EC is driven by an oscillating
electric potential of a sinusoidal form where the phase difference of the
electric potential between neighboring electrodes is 180 degrees. We
investigate the one- and two-dimensional nonlinear dynamics of a particle in an
EC field. The form of the dimensionless equations of motion is codimension two,
where the dimensionless control parameters are the interaction amplitude ()
and damping coefficient (). Our focus on the one-dimensional EC is
primarily on a case of fixed and relatively small , which is
characteristic of typical experimental conditions. We study the nonlinear
behaviors of the one-dimensional EC through the analysis of bifurcations of
fixed points. We analyze these bifurcations by using Floquet theory to
determine the stability of the limit cycles associated with the fixed points in
the Poincar\'e sections. Some of the bifurcations lead to chaotic trajectories
where we then determine the strength of chaos in phase space by calculating the
largest Lyapunov exponent. In the study of the two-dimensional EC we
independently look at bifurcation diagrams of variations in with fixed
and variations in with fixed . Under certain values of
and , we find that no stable trajectories above the surface exists;
such chaotic trajectories are described by a chaotic attractor, for which the
the largest Lyapunov exponent is found. We show the well-known stable
oscillations between two electrodes come into existence for variations in
and the transitions between several distinct regimes of stable motion for
variations in
Dynamical tunnelling with ultracold atoms in magnetic microtraps
The study of dynamical tunnelling in a periodically driven anharmonic
potential probes the quantum-classical transition via the experimental control
of the effective Planck's constant for the system. In this paper we consider
the prospects for observing dynamical tunnelling with ultracold atoms in
magnetic microtraps on atom chips. We outline the driven anharmonic potentials
that are possible using standard magnetic traps, and find the Floquet spectrum
for one of these as a function of the potential strength, modulation, and
effective Planck's constant. We develop an integrable approximation to the
non-integrable Hamiltonian and find that it can explain the behaviour of the
tunnelling rate as a function of the effective Planck's constant in the regular
region of parameter space. In the chaotic region we compare our results with
the predictions of models that describe chaos-assisted tunnelling. Finally we
examine the practicality of performing these experiments in the laboratory with
Bose-Einstein condensates.Comment: V1: 12 pages, 10 figures. V2: 14 pages, 12 figures, significantly
updated in response to referee report. Some figures are lower quality to
reduce file sizes, please contact submitter for high quality versions. V3:
Introduction rewritten, but mostly unchanged; updated to published versio
Signatures of chaotic tunnelling
Recent experiments with cold atoms provide a significant step toward a better
understanding of tunnelling when irregular dynamics is present at the classical
level. In this paper, we lay out numerical studies which shed light on the
previous experiments, help to clarify the underlying physics and have the
ambition to be guidelines for future experiments.Comment: 11 pages, 9 figures, submitted to Phys. Rev. E. Figures of better
quality can be found at http://www.phys.univ-tours.fr/~mouchet
An introduction to the Ginzburg-Landau theory of phase transitions and nonequilibrium patterns
This paper presents an introduction to phase transitions and critical
phenomena on the one hand, and nonequilibrium patterns on the other, using the
Ginzburg-Landau theory as a unified language. In the first part, mean-field
theory is presented, for both statics and dynamics, and its validity tested
self-consistently. As is well known, the mean-field approximation breaks down
below four spatial dimensions, where it can be replaced by a scaling
phenomenology. The Ginzburg-Landau formalism can then be used to justify the
phenomenological theory using the renormalization group, which elucidates the
physical and mathematical mechanism for universality. In the second part of the
paper it is shown how near pattern forming linear instabilities of dynamical
systems, a formally similar Ginzburg-Landau theory can be derived for
nonequilibrium macroscopic phenomena. The real and complex Ginzburg-Landau
equations thus obtained yield nontrivial solutions of the original dynamical
system, valid near the linear instability. Examples of such solutions are plane
waves, defects such as dislocations or spirals, and states of temporal or
spatiotemporal (extensive) chaos
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