1,058 research outputs found
Kink Localization under Asymmetric Double-Well Potential
We study diffuse phase interfaces under asymmetric double-well potential
energies with degenerate minima and demonstrate that the limiting sharp
profile, for small interface energy cost, on a finite space interval is in
general not symmetric and its position depends exclusively on the second
derivatives of the potential energy at the two minima (phases). We discuss an
application of the general result to porous media in the regime of solid-fluid
segregation under an applied pressure and describe the interface between a
fluid-rich and a fluid-poor phase. Asymmetric double-well potential energies
are also relevant in a very different field of physics as that of Brownian
motors. An intriguing analogy between our result and the direction of the dc
soliton current in asymmetric substrate driven Brownian motors is pointed out
Book Reviews
Reviews of the following books: Uncollected Short Stories of Sarah Orne Jewett edited by Richard Cary; The Care of Historical Collections: A Conservation Handbook for the Non-Specialist by Per E. Guldbeck; Cushing’s Island: Two Memoirs by Robert and Agnes Hale; The History of Fort Halifax by Carleton E. Fishe
Finite to infinite steady state solutions, bifurcations of an integro-differential equation
We consider a bistable integral equation which governs the stationary
solutions of a convolution model of solid--solid phase transitions on a circle.
We study the bifurcations of the set of the stationary solutions as the
diffusion coefficient is varied to examine the transition from an infinite
number of steady states to three for the continuum limit of the
semi--discretised system. We show how the symmetry of the problem is
responsible for the generation and stabilisation of equilibria and comment on
the puzzling connection between continuity and stability that exists in this
problem
Deviations from the local field approximation in negative streamer heads
Negative streamer ionization fronts in nitrogen under normal conditions are
investigated both in a particle model and in a fluid model in local field
approximation. The parameter functions for the fluid model are derived from
swarm experiments in the particle model. The front structure on the inner scale
is investigated in a 1D setting, allowing reasonable run-time and memory
consumption and high numerical accuracy without introducing super-particles. If
the reduced electric field immediately before the front is >= 50kV/(cm bar),
solutions of fluid and particle model agree very well. If the field increases
up to 200kV/(cm bar), the solutions of particle and fluid model deviate, in
particular, the ionization level behind the front becomes up to 60% higher in
the particle model while the velocity is rather insensitive. Particle and fluid
model deviate because electrons with high energies do not yet fully run away
from the front, but are somewhat ahead. This leads to increasing ionization
rates in the particle model at the very tip of the front. The energy overshoot
of electrons in the leading edge of the front actually agrees quantitatively
with the energy overshoot in the leading edge of an electron swarm or avalanche
in the same electric field.Comment: The paper has 17 pages, including 15 figures and 3 table
On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow
In this article, we study the axisymmetric surface diffusion flow (ASD), a
fourth-order geometric evolution law. In particular, we prove that ASD
generates a real analytic semiflow in the space of (2 + \alpha)-little-H\"older
regular surfaces of revolution embedded in R^3 and satisfying periodic boundary
conditions. We also give conditions for global existence of solutions and prove
that solutions are real analytic in time and space. Further, we investigate the
geometric properties of solutions to ASD. Utilizing a connection to
axisymmetric surfaces with constant mean curvature, we characterize the
equilibria of ASD. Then, focusing on the family of cylinders, we establish
results regarding stability, instability and bifurcation behavior, with the
radius acting as a bifurcation parameter for the problem.Comment: 37 pages, 6 figures, To Appear in SIAM J. Math. Ana
Upscaled phase-field models for interfacial dynamics in strongly heterogeneous domains
We derive a new effective macroscopic Cahn-Hilliard equation whose
homogeneous free energy is represented by 4-th order polynomials, which form
the frequently applied double-well potential. This upscaling is done for
perforated/strongly het- erogeneous domains. To the best knowledge of the
authors, this seems to be the first attempt of upscaling the Cahn-Hilliard
equation in such domains. The new homog- enized equation should have a broad
range of applicability due to the well-known versatility of phase-field models.
The additionally introduced feature of systemati- cally and reliably accounting
for confined geometries by homogenization allows for new modeling and numerical
perspectives in both, science and engineering. Our results are applied to
wetting dynamics in porous media and to a single channel with strongly
heterogeneous walls
Partial differential equations for self-organization in cellular and developmental biology
Understanding the mechanisms governing and regulating the emergence of structure and heterogeneity within cellular systems, such as the developing embryo, represents a multiscale challenge typifying current integrative biology research, namely, explaining the macroscale behaviour of a system from microscale dynamics. This review will focus upon modelling how cell-based dynamics orchestrate the emergence of higher level structure. After surveying representative biological examples and the models used to describe them, we will assess how developments at the scale of molecular biology have impacted on current theoretical frameworks, and the new modelling opportunities that are emerging as a result. We shall restrict our survey of mathematical approaches to partial differential equations and the tools required for their analysis. We will discuss the gap between the modelling abstraction and biological reality, the challenges this presents and highlight some open problems in the field
The Speed of Fronts of the Reaction Diffusion Equation
We study the speed of propagation of fronts for the scalar reaction-diffusion
equation \, with . We give a new integral
variational principle for the speed of the fronts joining the state to
. No assumptions are made on the reaction term other than those
needed to guarantee the existence of the front. Therefore our results apply to
the classical case in , to the bistable case and to cases in
which has more than one internal zero in .Comment: 7 pages Revtex, 1 figure not include
Traveling wave solutions in the Burridge-Knopoff model
The slider-block Burridge-Knopoff model with the Coulomb friction law is
studied as an excitable medium. It is shown that in the continuum limit the
system admits solutions in the form of the self-sustained shock waves traveling
with constant speed which depends only on the amount of the accumulated stress
in front of the wave. For a wide class of initial conditions the behavior of
the system is determined by these shock waves and the dynamics of the system
can be expressed in terms of their motion. The solutions in the form of the
periodic wave trains and sources of counter-propagating waves are analyzed. It
is argued that depending on the initial conditions the system will either tend
to synchronize or exhibit chaotic spatiotemporal behavior.Comment: 12 pages (ReVTeX), 7 figures (Postscript) to be published in Phys.
Rev.
Application of elastostatic Green function tensor technique to electrostriction in cubic, hexagonal and orthorhombic crystals
The elastostatic Green function tensor approach, which was recently used to
treat electrostriction in numerical simulation of domain structure formation in
cubic ferroelectrics, is reviewed and extended to the crystals of hexagonal and
orthorhombic symmetry. The tensorial kernels appearing in the expressions for
effective nonlocal interaction of electrostrictive origin are derived
explicitly and their physical meaning is illustrated on simple examples. It is
argued that the bilinear coupling between the polarization gradients and
elastic strain should be systematically included in the Ginzburg-Landau free
energy expansion of electrostrictive materials.Comment: 4 page
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