279 research outputs found
Interaction of Ising-Bloch fronts with Dirichlet Boundaries
We study the Ising-Bloch bifurcation in two systems, the Complex Ginzburg
Landau equation (CGLE) and a FitzHugh Nagumo (FN) model in the presence of
spatial inhomogeneity introduced by Dirichlet boundary conditions. It is seen
that the interaction of fronts with boundaries is similar in both systems,
establishing the generality of the Ising-Bloch bifurcation. We derive reduced
dynamical equations for the FN model that explain front dynamics close to the
boundary. We find that front dynamics in a highly non-adiabatic (slow front)
limit is controlled by fixed points of the reduced dynamical equations, that
occur close to the boundary.Comment: 10 pages, 8 figures, submitted to Phys. Rev.
Front propagation into unstable and metastable states in Smectic C* liquid crystals: linear and nonlinear marginal stability analysis
We discuss the front propagation in ferroelectric chiral smectics (SmC*)
subjected to electric and magnetic fields applied parallel to smectic layers.
The reversal of the electric field induces the motion of domain walls or fronts
that propagate into either an unstable or a metastable state. In both regimes,
the front velocity is calculated exactly. Depending on the field, the speed of
a front propagating into the unstable state is given either by the so-called
linear marginal stability velocity or by the nonlinear marginal stability
expression. The cross-over between these two regimes can be tuned by a magnetic
field. The influence of initial conditions on the velocity selection problem
can also be studied in such experiments. SmC therefore offers a unique
opportunity to study different aspects of front propagation in an experimental
system
Fixed-Node Monte Carlo Calculations for the 1d Kondo Lattice Model
The effectiveness of the recently developed Fixed-Node Quantum Monte Carlo
method for lattice fermions, developed by van Leeuwen and co-workers, is tested
by applying it to the 1D Kondo lattice, an example of a one-dimensional model
with a sign problem. The principles of this method and its implementation for
the Kondo Lattice Model are discussed in detail. We compare the fixed-node
upper bound for the ground state energy at half filling with
exact-diagonalization results from the literature, and determine several spin
correlation functions. Our `best estimates' for the ground state correlation
functions do not depend sensitively on the input trial wave function of the
fixed-node projection, and are reasonably close to the exact values. We also
calculate the spin gap of the model with the Fixed-Node Monte Carlo method. For
this it is necessary to use a many-Slater-determinant trial state. The
lowest-energy spin excitation is a running spin soliton with wave number pi, in
agreement with earlier calculations.Comment: 19 pages, revtex, contribution to Festschrift for Hans van Leeuwe
Fronts with a Growth Cutoff but Speed Higher than
Fronts, propagating into an unstable state , whose asymptotic speed
is equal to the linear spreading speed of infinitesimal
perturbations about that state (so-called pulled fronts) are very sensitive to
changes in the growth rate for . It was recently found
that with a small cutoff, for ,
converges to very slowly from below, as . Here we show
that with such a cutoff {\em and} a small enhancement of the growth rate for
small behind it, one can have , {\em even} in the
limit . The effect is confirmed in a stochastic lattice model
simulation where the growth rules for a few particles per site are accordingly
modified.Comment: 4 pages, 4 figures, to appear in Rapid Comm., Phys. Rev.
Shift in the velocity of a front due to a cut-off
We consider the effect of a small cut-off epsilon on the velocity of a
traveling wave in one dimension. Simulations done over more than ten orders of
magnitude as well as a simple theoretical argument indicate that the effect of
the cut-off epsilon is to select a single velocity which converges when epsilon
tends to 0 to the one predicted by the marginal stability argument. For small
epsilon, the shift in velocity has the form K(log epsilon)^(-2) and our
prediction for the constant K agrees very well with the results of our
simulations. A very similar logarithmic shift appears in more complicated
situations, in particular in finite size effects of some microscopic stochastic
systems. Our theoretical approach can also be extended to give a simple way of
deriving the shift in position due to initial conditions in the
Fisher-Kolmogorov or similar equations.Comment: 12 pages, 3 figure
New exact fronts for the nonlinear diffusion equation with quintic nonlinearities
We consider travelling wave solutions of the reaction diffusion equation with
quintic nonlinearities . If the parameters
and obey a special relation, then the criterion for the existence of a
strong heteroclinic connection can be expressed in terms of two of these
parameters. If an additional restriction is imposed, explicit front solutions
can be obtained. The approach used can be extended to polynomials whose highest
degree is odd.Comment: Revtex, 5 page
On the validity of the linear speed selection mechanism for fronts of the nonlinear diffusion equation
We consider the problem of the speed selection mechanism for the one
dimensional nonlinear diffusion equation . It has been
rigorously shown by Aronson and Weinberger that for a wide class of functions
, sufficiently localized initial conditions evolve in time into a monotonic
front which propagates with speed such that . The lower value is that predicted
by the linear marginal stability speed selection mechanism. We derive a new
lower bound on the the speed of the selected front, this bound depends on
and thus enables us to assess the extent to which the linear marginal selection
mechanism is valid.Comment: 9 pages, REVTE
Multiple Front Propagation Into Unstable States
The dynamics of transient patterns formed by front propagation in extended
nonequilibrium systems is considered. Under certain circumstances, the state
left behind a front propagating into an unstable homogeneous state can be an
unstable periodic pattern. It is found by a numerical solution of a model of
the Fr\'eedericksz transition in nematic liquid crystals that the mechanism of
decay of such periodic unstable states is the propagation of a second front
which replaces the unstable pattern by a another unstable periodic state with
larger wavelength. The speed of this second front and the periodicity of the
new state are analytically calculated with a generalization of the marginal
stability formalism suited to the study of front propagation into periodic
unstable states. PACS: 47.20.Ky, 03.40.Kf, 47.54.+rComment: 12 page
Perturbative Linearization of Reaction-Diffusion Equations
We develop perturbative expansions to obtain solutions for the initial-value
problems of two important reaction-diffusion systems, viz., the Fisher equation
and the time-dependent Ginzburg-Landau (TDGL) equation. The starting point of
our expansion is the corresponding singular-perturbation solution. This
approach transforms the solution of nonlinear reaction-diffusion equations into
the solution of a hierarchy of linear equations. Our numerical results
demonstrate that this hierarchy rapidly converges to the exact solution.Comment: 13 pages, 4 figures, latex2
Streamer Propagation as a Pattern Formation Problem: Planar Fronts
Streamers often constitute the first stage of dielectric breakdown in strong
electric fields: a nonlinear ionization wave transforms a non-ionized medium
into a weakly ionized nonequilibrium plasma. New understanding of this old
phenomenon can be gained through modern concepts of (interfacial) pattern
formation. As a first step towards an effective interface description, we
determine the front width, solve the selection problem for planar fronts and
calculate their properties. Our results are in good agreement with many
features of recent three-dimensional numerical simulations.Comment: 4 pages, revtex, 3 ps file
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