2,579 research outputs found
Structures and waves in a nonlinear heat-conducting medium
The paper is an overview of the main contributions of a Bulgarian team of
researchers to the problem of finding the possible structures and waves in the
open nonlinear heat conducting medium, described by a reaction-diffusion
equation. Being posed and actively worked out by the Russian school of A. A.
Samarskii and S.P. Kurdyumov since the seventies of the last century, this
problem still contains open and challenging questions.Comment: 23 pages, 13 figures, the final publication will appear in Springer
Proceedings in Mathematics and Statistics, Numerical Methods for PDEs:
Theory, Algorithms and their Application
A spin dynamics approach to solitonics
It is spatial dispersion which is exclusively responsible for the emergence
of exchange interaction and magnetic ordering. In contrast, magneto-crystalline
anisotropy present in any realistic material brings in a certain non-linearity
to the equation of motion. Unlike homogeneous ferromagnetic ordering a variety
of non-collinear ground state configurations emerge as a result of competition
among exchange, anisotropy, and dipole-dipole interaction. These particle-like
states, e.g. magnetic soliton, skyrmion, domain wall, form a spatially
localised clot of magnetic energy. In this paper we explore topologically
protected magnetic solitons that might potentially be applied for logical
operations and/or information storage in the rapidly advancing filed of
solitonics (and skyrmionics). An ability to easily create, address, and
manipulate such structures is among the prerequisite forming a basis of -onics
technology, and is investigated in detail here using numerical and analytical
tools
Two-dimensional structures in the quintic Ginzburg-Landau equation
By using ZEUS cluster at Embry-Riddle Aeronautical University we perform
extensive numerical simulations based on a two-dimensional Fourier spectral
method Fourier spatial discretization and an explicit scheme for time
differencing) to find the range of existence of the spatiotemporal solitons of
the two-dimensional complex Ginzburg-Landau equation with cubic and quintic
nonlinearities. We start from the parameters used by Akhmediev {\it et. al.}
and slowly vary them one by one to determine the regimes where solitons exist
as stable/unstable structures. We present eight classes of dissipative solitons
from which six are known (stationary, pulsating, vortex spinning, filament,
exploding, creeping) and two are novel (creeping-vortex propellers and spinning
"bean-shaped" solitons). By running lengthy simulations for the different
parameters of the equation, we find ranges of existence of stable structures
(stationary, pulsating, circular vortex spinning, organized exploding), and
unstable structures (elliptic vortex spinning that leads to filament,
disorganized exploding, creeping). Moreover, by varying even the two initial
conditions together with vorticity, we find a richer behavior in the form of
creeping-vortex propellers, and spinning "bean-shaped" solitons. Each class
differentiates from the other by distinctive features of their energy
evolution, shape of initial conditions, as well as domain of existence of
parameters.Comment: 19 pages, 19 figures, 8 tables, updated text and reference
Bifurcations in the wake of a thick circular disk
Using DNS, we investigate the dynamics in the wake of a circular disk of aspect ratio χ = d/w = 3(where d is the diameter and w the thickness) embedded in a uniform flow of magnitude U0 perpendicular to its symmetry axis. As the Reynolds number Re = U0d/ν is increased, the flow is shown to experience an original series of bifurcations leading to chaos. The range Re ∈ [150, 218] is analysed in detail. In this range, five different non-axisymmetric regimes are successively encountered, including states similar to those previously identified in the flow past a sphere or an infinitely thin disk, as well as a new regime characterised by the presence of two distinct frequencies. A theoretical model based on the theory of mode interaction with symmetries, previously introduced to explain the bifurcations in the flow past a sphere or an infinitely thin disk (Fabre et al. in Phys Fluids 20:051702, 2008), is shown to explain correctly all these results. Higher values of the Reynolds number, up to 270, are also considered. Results indicate that the flow encounters at least four additional bifurcations before reaching a chaotic state
"Mariage des Maillages": A new numerical approach for 3D relativistic core collapse simulations
We present a new 3D general relativistic hydrodynamics code for simulations
of stellar core collapse to a neutron star, as well as pulsations and
instabilities of rotating relativistic stars. It uses spectral methods for
solving the metric equations, assuming the conformal flatness approximation for
the three-metric. The matter equations are solved by high-resolution
shock-capturing schemes. We demonstrate that the combination of a finite
difference grid and a spectral grid can be successfully accomplished. This
"Mariage des Maillages" (French for grid wedding) approach results in high
accuracy of the metric solver and allows for fully 3D applications using
computationally affordable resources, and ensures long term numerical stability
of the evolution. We compare our new approach to two other, finite difference
based, methods to solve the metric equations. A variety of tests in 2D and 3D
is presented, involving highly perturbed neutron star spacetimes and
(axisymmetric) stellar core collapse, demonstrating the ability to handle
spacetimes with and without symmetries in strong gravity. These tests are also
employed to assess gravitational waveform extraction, which is based on the
quadrupole formula.Comment: 29 pages, 16 figures; added more information about convergence tests
and grid setu
Incorporating Inductances in Tissue-Scale Models of Cardiac Electrophysiology
In standard models of cardiac electrophysiology, including the bidomain and
monodomain models, local perturbations can propagate at infinite speed. We
address this unrealistic property by developing a hyperbolic bidomain model
that is based on a generalization of Ohm's law with a Cattaneo-type model for
the fluxes. Further, we obtain a hyperbolic monodomain model in the case that
the intracellular and extracellular conductivity tensors have the same
anisotropy ratio. In one spatial dimension, the hyperbolic monodomain model is
equivalent to a cable model that includes axial inductances, and the relaxation
times of the Cattaneo fluxes are strictly related to these inductances. A
purely linear analysis shows that the inductances are negligible, but models of
cardiac electrophysiology are highly nonlinear, and linear predictions may not
capture the fully nonlinear dynamics. In fact, contrary to the linear analysis,
we show that for simple nonlinear ionic models, an increase in conduction
velocity is obtained for small and moderate values of the relaxation time. A
similar behavior is also demonstrated with biophysically detailed ionic models.
Using the Fenton-Karma model along with a low-order finite element spatial
discretization, we numerically analyze differences between the standard
monodomain model and the hyperbolic monodomain model. In a simple benchmark
test, we show that the propagation of the action potential is strongly
influenced by the alignment of the fibers with respect to the mesh in both the
parabolic and hyperbolic models when using relatively coarse spatial
discretizations. Accurate predictions of the conduction velocity require
computational mesh spacings on the order of a single cardiac cell. We also
compare the two formulations in the case of spiral break up and atrial
fibrillation in an anatomically detailed model of the left atrium, and [...].Comment: 20 pages, 12 figure
Exact soliton solutions of the one-dimensional complex Swift-Hohenberg equation
Using Painlev\'e analysis, the Hirota multi-linear method and a direct ansatz
technique, we study analytic solutions of the (1+1)-dimensional complex cubic
and quintic Swift-Hohenberg equations. We consider both standard and
generalized versions of these equations. We have found that a number of exact
solutions exist to each of these equations, provided that the coefficients are
constrained by certain relations. The set of solutions include particular types
of solitary wave solutions, hole (dark soliton) solutions and periodic
solutions in terms of elliptic Jacobi functions and the Weierstrass
function. Although these solutions represent only a small subset of the large
variety of possible solutions admitted by the complex cubic and quintic
Swift-Hohenberg equations, those presented here are the first examples of exact
analytic solutions found thus far.Comment: 32 pages, no figures, elsart.cl
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