46,396 research outputs found
Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces
The aim of this manuscript is to present for the first time the application of the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces. Furthermore we present pattern formation generated by the reaction-diffusion systemwith cross-diffusion on evolving domains and surfaces. A two-component reaction-diffusion system with linear cross-diffusion in both u and v is presented. The finite element method is based on the approximation of the domain or surface by a triangulated domain or surface consisting of a union of triangles. For surfaces, the vertices of the triangulation lie on the continuous surface. A finite element space of functions is then defined by taking the continuous functions which are linear affine on each simplex of the triangulated domain or surface. To demonstrate the role of cross-diffusion to the theory of pattern formation, we compute patterns with model kinetic parameter values that belong only to the cross-diffusion parameter space; these do not belong to the standard parameter space for classical reaction-diffusion systems. Numerical results exhibited show the robustness, flexibility, versatility, and generality of our methodology; the methodology can deal with complicated evolution laws of the domain and surface, and these include uniform isotropic and anisotropic growth profiles as well as those profiles driven by chemical concentrations residing in the domain or on the surface
Drift Laws for Spiral Waves on Curved Anisotropic Surfaces
Rotating spiral waves organize spatial patterns in chemical, physical and
biological excitable systems. Factors affecting their dynamics such as
spatiotemporal drift are of great interest for par- ticular applications. Here,
we propose a quantitative description for spiral wave dynamics on curved
surfaces which shows that for a wide class of systems, including the BZ
reaction and anisotropic cardiac tissue, the Ricci curvature scalar of the
surface is the main determinant of spiral wave drift. The theory provides
explicit equations for spiral wave drift direction, drift velocity and the
period of rotation. Depending on the parameters, the drift can be directed to
the regions of either maximal or minimal Ricci scalar curvature, which was
verified by direct numerical simulations.Comment: preprint before submission to Physical Review
The Matzoh Ball Soup Problem: a complete characterization
We characterize all the solutions of the heat equation that have their
(spatial) equipotential surfaces which do not vary with the time. Such
solutions are either isoparametric or split in space-time. The result gives a
final answer to a problem raised by M. S. Klamkin, extended by G. Alessandrini,
and that was named the Matzoh Ball Soup Problem by L. Zalcman. Similar results
can also be drawn for a class of quasi-linear parabolic partial differential
equations with coefficients which are homogeneous functions of the gradient
variable. This class contains the (isotropic or anisotropic) evolution
p-Laplace and normalized p-Laplace equations
Spin Conduction in Anisotropic 3-D Topological Insulators
When topological insulators possess rotational symmetry their spin lifetime
is tied to the scattering time. We show that in anisotropic TIs this tie can be
broken and the spin lifetime can be very large. Two different mechanisms can
obtain spin conduction over long distances. The first is tuning the Hamiltonian
to conserve a spin operator ,
while the second is tuning the Fermi energy to be near a local extremum of the
energy dispersion. Both mechanisms can produce persistent spin helices. We
report spin lifetimes and spin diffusion equations.Comment: Added a page of additional text and refined the presentation. Main
content unchange
A Meshfree Generalized Finite Difference Method for Surface PDEs
In this paper, we propose a novel meshfree Generalized Finite Difference
Method (GFDM) approach to discretize PDEs defined on manifolds. Derivative
approximations for the same are done directly on the tangent space, in a manner
that mimics the procedure followed in volume-based meshfree GFDMs. As a result,
the proposed method not only does not require a mesh, it also does not require
an explicit reconstruction of the manifold. In contrast to existing methods, it
avoids the complexities of dealing with a manifold metric, while also avoiding
the need to solve a PDE in the embedding space. A major advantage of this
method is that all developments in usual volume-based numerical methods can be
directly ported over to surfaces using this framework. We propose
discretizations of the surface gradient operator, the surface Laplacian and
surface Diffusion operators. Possibilities to deal with anisotropic and
discontinous surface properties (with large jumps) are also introduced, and a
few practical applications are presented
Roughening of ion-eroded surfaces
Recent experimental studies focusing on the morphological properties of
surfaces eroded by ion-bombardment report the observation of self-affine
fractal surfaces, while others provide evidence about the development of a
periodic ripple structure. To explain these discrepancies we derive a
stochastic growth equation that describes the evolution of surfaces eroded by
ion bombardment. The coefficients appearing in the equation can be calculated
explicitly in terms of the physical parameters characterizing the sputtering
process. Exploring the connection between the ion-sputtering problem and the
Kardar-Parisi-Zhang and Kuramoto-Sivashinsky equations, we find that
morphological transitions may take place when experimental parameters, such as
the angle of incidence of the incoming ions or their average penetration depth,
are varied. Furthermore, the discussed methods allow us to calculate
analytically the ion-induced surface diffusion coefficient, that can be
compared with experiments. Finally, we use numerical simulations of a one
dimensional sputtering model to investigate certain aspects of the ripple
formation and roughening.Comment: 20 pages, LaTeX, 5 ps figures, contribution to the 4th CTP Workshop
on Statistical Physics "Dynamics of Fluctuating Interfaces and Related
Phenomena", Seoul National University, Seoul, Korea, January 27-31, 199
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