206 research outputs found
Attraction properties of the Ginzburg-Landau manifold
We consider solutions of weakly unstable PDE on an unbounded spatial domain. It has been shown earlier by the first author that the set of modulated solutions (called "Ginzburg-Landau manifold") is attracting. We seek to understand "how big" is the domain of attraction. Starting with general initial conditions of order " for the Fourier-transformed version of the given PDE we find that on the time-scale T " ; 2 (that is long in the terms of the original "physical" time t, but shorter than the natural time for the Ginzburg-Landau) the corresponding solutions evolve to the scaling of the clustered modes-distribution peaked at the integer multiples of the critical wave number, with the amplitudes sensitively dependent on such that for arbitrary close to zero after the time T " ; 2 solutions get on the Ginzburg-Landau manifold
Nonlinear wavelength selection in surface faceting under electromigration
We report on the control of the faceting of crystal surfaces by means of
surface electromigration. When electromigration reinforces the faceting
instability, we find perpetual coarsening with a wavelength increasing as
. For strongly stabilizing electromigration, the surface is stable.
For weakly stabilizing electromigration, a cellular pattern is obtained, with a
nonlinearly selected wavelength. The selection mechanism is not caused by an
instability of steady-states, as suggested by previous works in the literature.
Instead, the dynamics is found to exhibit coarsening {\it before} reaching a
continuous family of stable non-equilibrium steady-states.Comment: 5 pages, 4 figures, submitte
Transverse flow in thin superhydrophobic channels
We provide some general theoretical results to guide the optimization of
transverse hydrodynamic phenomena in superhydrophobic channels. Our focus is on
the canonical micro- and nanofluidic geometry of a parallel-plate channel with
an arbitrary two-component (low-slip and high-slip) coarse texture, varying on
scales larger than the channel thickness. By analyzing rigorous bounds on the
permeability, over all possible patterns, we optimize the area fractions, slip
lengths, geometry and orientation of the surface texture to maximize transverse
flow. In the case of two aligned striped surfaces, very strong transverse flows
are possible. Optimized superhydrophobic surfaces may find applications in
passive microfluidic mixing and amplification of transverse electrokinetic
phenomena.Comment: 4 page
Dynamics of stripe patterns in type-I superconductors subject to a rotating field
The evolution of stripe patterns in type-I superconductors subject to a
rotating in-plane magnetic field is investigated magneto-optically. The
experimental results reveal a very rich and interesting behavior of the
patterns. For small rotation angles, a small parallel displacement of the main
part of the stripes and a co-rotation of their very ends is observed. For
larger angles, small sideward protrusions develop, which then generate a zigzag
instability, ultimately leading to a breaking of stripes into smaller segments.
The short segments then start to co-rotate with the applied field although they
lag behind by approximately . Very interestingly, if the rotation is
continued, also reconnection of segments into longer stripes takes place. These
observations demonstrate the importance of pinning in type-I superconductors.Comment: To appear in Phys. Rev.
State selection in the noisy stabilized Kuramoto-Sivashinsky equation
In this work, we study the 1D stabilized Kuramoto Sivashinsky equation with
additive uncorrelated stochastic noise. The Eckhaus stable band of the
deterministic equation collapses to a narrow region near the center of the
band. This is consistent with the behavior of the phase diffusion constants of
these states. Some connections to the phenomenon of state selection in driven
out of equilibrium systems are made.Comment: 8 pages, In version 3 we corrected minor/typo error
Second-order gravitational self-force
Using a rigorous method of matched asymptotic expansions, I derive the
equation of motion of a small, compact body in an external vacuum spacetime
through second order in the body's mass (neglecting effects of internal
structure). The motion is found to be geodesic in a certain locally defined
regular geometry satisfying Einstein's equation at second order. I outline a
method of numerically obtaining both the metric of that regular geometry and
the complete second-order metric perturbation produced by the body.Comment: 5 pages, added clarifications in response to referee comments,
accepted for publication in PR
The Nikolaevskiy equation with dispersion
The Nikolaevskiy equation was originally proposed as a model for seismic
waves and is also a model for a wide variety of systems incorporating a
neutral, Goldstone mode, including electroconvection and reaction-diffusion
systems. It is known to exhibit chaotic dynamics at the onset of pattern
formation, at least when the dispersive terms in the equation are suppressed,
as is commonly the practice in previous analyses. In this paper, the effects of
reinstating the dispersive terms are examined. It is shown that such terms can
stabilise some of the spatially periodic traveling waves; this allows us to
study the loss of stability and transition to chaos of the waves. The secondary
stability diagram (Busse balloon) for the traveling waves can be remarkably
complicated.Comment: 24 pages; accepted for publication in Phys. Rev.
The Formation and Coarsening of the Concertina Pattern
The concertina is a magnetization pattern in elongated thin-film elements of
a soft material. It is a ubiquitous domain pattern that occurs in the process
of magnetization reversal in direction of the long axis of the small element.
Van den Berg argued that this pattern grows out of the flux closure domains as
the external field is reduced. Based on experimental observations and theory,
we argue that in sufficiently elongated thin-film elements, the concertina
pattern rather bifurcates from an oscillatory buckling mode. Using a reduced
model derived by asymptotic analysis and investigated by numerical simulation,
we quantitatively predict the average period of the concertina pattern and
qualitatively predict its hysteresis. In particular, we argue that the
experimentally observed coarsening of the concertina pattern is due to
secondary bifurcations related to an Eckhaus instability. We also link the
concertina pattern to the magnetization ripple and discuss the effect of a weak
(crystalline or induced) anisotropy
The self-consistent gravitational self-force
I review the problem of motion for small bodies in General Relativity, with
an emphasis on developing a self-consistent treatment of the gravitational
self-force. An analysis of the various derivations extant in the literature
leads me to formulate an asymptotic expansion in which the metric is expanded
while a representative worldline is held fixed; I discuss the utility of this
expansion for both exact point particles and asymptotically small bodies,
contrasting it with a regular expansion in which both the metric and the
worldline are expanded. Based on these preliminary analyses, I present a
general method of deriving self-consistent equations of motion for arbitrarily
structured (sufficiently compact) small bodies. My method utilizes two
expansions: an inner expansion that keeps the size of the body fixed, and an
outer expansion that lets the body shrink while holding its worldline fixed. By
imposing the Lorenz gauge, I express the global solution to the Einstein
equation in the outer expansion in terms of an integral over a worldtube of
small radius surrounding the body. Appropriate boundary data on the tube are
determined from a local-in-space expansion in a buffer region where both the
inner and outer expansions are valid. This buffer-region expansion also results
in an expression for the self-force in terms of irreducible pieces of the
metric perturbation on the worldline. Based on the global solution, these
pieces of the perturbation can be written in terms of a tail integral over the
body's past history. This approach can be applied at any order to obtain a
self-consistent approximation that is valid on long timescales, both near and
far from the small body. I conclude by discussing possible extensions of my
method and comparing it to alternative approaches.Comment: 44 pages, 4 figure
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