178 research outputs found
The effective equation method
In this chapter we present a general method of constructing the effective
equation which describes the behaviour of small-amplitude solutions for a
nonlinear PDE in finite volume, provided that the linear part of the equation
is a hamiltonian system with a pure imaginary discrete spectrum. The effective
equation is obtained by retaining only the resonant terms of the nonlinearity
(which may be hamiltonian, or may be not); the assertion that it describes the
limiting behaviour of small-amplitude solutions is a rigorous mathematical
theorem. In particular, the method applies to the three-- and four--wave
systems. We demonstrate that different possible types of energy transport are
covered by this method, depending on whether the set of resonances splits into
finite clusters (this happens, e.g. in case of the Charney-Hasegawa-Mima
equation), or is connected (this happens, e.g. in the case of the NLS equation
if the space-dimension is at least two). For equations of the first type the
energy transition to high frequencies does not hold, while for equations of the
second type it may take place. In the case of the NLS equation we use next some
heuristic approximation from the arsenal of wave turbulence to show that under
the iterated limit "the volume goes to infinity", taken after the limit "the
amplitude of oscillations goes to zero", the energy spectrum of solutions for
the effective equation is described by a Zakharov-type kinetic equation.
Evoking the Zakharov ansatz we show that stationary in time and homogeneous in
space solutions for the latter equation have a power law form. Our method
applies to various weakly nonlinear wave systems, appearing in plasma,
meteorology and oceanology
Mean flow and spiral defect chaos in Rayleigh-Benard convection
We describe a numerical procedure to construct a modified velocity field that
does not have any mean flow. Using this procedure, we present two results.
Firstly, we show that, in the absence of mean flow, spiral defect chaos
collapses to a stationary pattern comprising textures of stripes with angular
bends. The quenched patterns are characterized by mean wavenumbers that
approach those uniquely selected by focus-type singularities, which, in the
absence of mean flow, lie at the zig-zag instability boundary. The quenched
patterns also have larger correlation lengths and are comprised of rolls with
less curvature. Secondly, we describe how mean flow can contribute to the
commonly observed phenomenon of rolls terminating perpendicularly into lateral
walls. We show that, in the absence of mean flow, rolls begin to terminate into
lateral walls at an oblique angle. This obliqueness increases with Rayleigh
number.Comment: 14 pages, 19 figure
Pattern Formation of Ion Channels with State Dependent Electrophoretic Charges and Diffusion Constants in Fluid Membranes
A model of mobile, charged ion channels in a fluid membrane is studied. The
channels may switch between an open and a closed state according to a simple
two-state kinetics with constant rates. The effective electrophoretic charge
and the diffusion constant of the channels may be different in the closed and
in the open state. The system is modeled by densities of channel species,
obeying simple equations of electro-diffusion. The lateral transmembrane
voltage profile is determined from a cable-type equation. Bifurcations from the
homogeneous, stationary state appear as hard-mode, soft-mode or hard-mode
oscillatory transitions within physiologically reasonable ranges of model
parameters. We study the dynamics beyond linear stability analysis and derive
non-linear evolution equations near the transitions to stationary patterns.Comment: 10 pages, 7 figures, will be submitted to Phys. Rev.
Statistics and Characteristics of Spatio-Temporally Rare Intense Events in Complex Ginzburg-Landau Models
We study the statistics and characteristics of rare intense events in two
types of two dimensional Complex Ginzburg-Landau (CGL) equation based models.
Our numerical simulations show finite amplitude collapse-like solutions which
approach the infinite amplitude solutions of the nonlinear Schr\"{o}dinger
(NLS) equation in an appropriate parameter regime. We also determine the
probability distribution function (PDF) of the amplitude of the CGL solutions,
which is found to be approximately described by a stretched exponential
distribution, , where . This
non-Gaussian PDF is explained by the nonlinear characteristics of individual
bursts combined with the statistics of bursts. Our results suggest a general
picture in which an incoherent background of weakly interacting waves,
occasionally, `by chance', initiates intense, coherent, self-reinforcing,
highly nonlinear events.Comment: 7 pages, 9 figure
Cross-Newell equations for hexagons and triangles
The Cross-Newell equations for hexagons and triangles are derived for general
real gradient systems, and are found to be in flux-divergence form. Specific
examples of complex governing equations that give rise to hexagons and
triangles and which have Lyapunov functionals are also considered, and explicit
forms of the Cross-Newell equations are found in these cases. The general
nongradient case is also discussed; in contrast with the gradient case, the
equations are not flux-divergent. In all cases, the phase stability boundaries
and modes of instability for general distorted hexagons and triangles can be
recovered from the Cross-Newell equations.Comment: 24 pages, 1 figur
Amplitude equations near pattern forming instabilities for strongly driven ferromagnets
A transversally driven isotropic ferromagnet being under the influence of a
static external and an uniaxial internal anisotropy field is studied. We
consider the dissipative Landau-Lifshitz equation as the fundamental equation
of motion and treat it in ~dimensions. The stability of the spatially
homogeneous magnetizations against inhomogeneous perturbations is analyzed.
Subsequently the dynamics above threshold is described via amplitude equations
and the dependence of their coefficients on the physical parameters of the
system is determined explicitly. We find soft- and hard-mode instabilities,
transitions between sub- and supercritical behaviour, various bifurcations of
higher codimension, and present a series of explicit bifurcation diagrams. The
analysis of the codimension-2 point where the soft- and hard-mode instabilities
coincide leads to a system of two coupled Ginzburg-Landau equations.Comment: LATeX, 25 pages, submitted to Z.Phys.B figures available via
[email protected] in /pub/publications/frank/zpb_95
(postscript, plain or gziped
Spatiotemporally localized solitons in resonantly absorbing Bragg reflectors
We predict the existence of spatiotemporal solitons (``light bullets'') in
two-dimensional self-induced transparency media embedded in a Bragg grating.
The "bullets" are found in an approximate analytical form, their stability
being confirmed by direct simulations. These findings suggest new possibilities
for signal transmission control and self-trapping of light.Comment: RevTex, 3 pages, 2 figures, to be published in PR
Parametrically Excited Surface Waves: Two-Frequency Forcing, Normal Form Symmetries, and Pattern Selection
Motivated by experimental observations of exotic standing wave patterns in
the two-frequency Faraday experiment, we investigate the role of normal form
symmetries in the pattern selection problem. With forcing frequency components
in ratio m/n, where m and n are co-prime integers, there is the possibility
that both harmonic and subharmonic waves may lose stability simultaneously,
each with a different wavenumber. We focus on this situation and compare the
case where the harmonic waves have a longer wavelength than the subharmonic
waves with the case where the harmonic waves have a shorter wavelength. We show
that in the former case a normal form transformation can be used to remove all
quadratic terms from the amplitude equations governing the relevant resonant
triad interactions. Thus the role of resonant triads in the pattern selection
problem is greatly diminished in this situation. We verify our general results
within the example of one-dimensional surface wave solutions of the
Zhang-Vinals model of the two-frequency Faraday problem. In one-dimension, a
1:2 spatial resonance takes the place of a resonant triad in our investigation.
We find that when the bifurcating modes are in this spatial resonance, it
dramatically effects the bifurcation to subharmonic waves in the case of
forcing frequencies are in ratio 1/2; this is consistent with the results of
Zhang and Vinals. In sharp contrast, we find that when the forcing frequencies
are in ratio 2/3, the bifurcation to (sub)harmonic waves is insensitive to the
presence of another spatially-resonant bifurcating mode.Comment: 22 pages, 6 figures, late
Frozen spatial chaos induced by boundaries
We show that rather simple but non-trivial boundary conditions could induce
the appearance of spatial chaos (that is stationary, stable, but spatially
disordered configurations) in extended dynamical systems with very simple
dynamics. We exemplify the phenomenon with a nonlinear reaction-diffusion
equation in a two-dimensional undulated domain. Concepts from the theory of
dynamical systems, and a transverse-single-mode approximation are used to
describe the spatially chaotic structures.Comment: 9 pages, 6 figures, submitted for publication; for related work visit
http://www.imedea.uib.es/~victo
Phase Dynamics of Nearly Stationary Patterns in Activator-Inhibitor Systems
The slow dynamics of nearly stationary patterns in a FitzHugh-Nagumo model
are studied using a phase dynamics approach. A Cross-Newell phase equation
describing slow and weak modulations of periodic stationary solutions is
derived. The derivation applies to the bistable, excitable, and the Turing
unstable regimes. In the bistable case stability thresholds are obtained for
the Eckhaus and the zigzag instabilities and for the transition to traveling
waves. Neutral stability curves demonstrate the destabilization of stationary
planar patterns at low wavenumbers to zigzag and traveling modes. Numerical
solutions of the model system support the theoretical findings
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