Fast-slow asymptotic for semi-analytical ignition criteria in FitzHugh-Nagumo system

We study the problem of initiation of excitation waves in the FitzHugh-Nagumo model. Our approach follows earlier works and is based on the idea of approximating the boundary between basins of attraction of propagating waves and of the resting state as the stable manifold of a critical solution. Here, we obtain analytical expressions for the essential ingredients of the theory by singular perturbation using two small parameters, the separation of time scales of the activator and inhibitor, and the threshold in the activator's kinetics. This results in a closed analytical expression for the strength-duration curve.Comment: 10 pages, 5 figures, as accepted to Chaos on 2017/06/2

Macroscopic description of particle systems with non-local density-dependent diffusivity

In this paper we study macroscopic density equations in which the diffusion coefficient depends on a weighted spatial average of the density itself. We show that large differences (not present in the local density-dependence case) appear between the density equations that are derived from different representations of the Langevin equation describing a system of interacting Brownian particles. Linear stability analysis demonstrates that under some circumstances the density equation interpreted like Ito has pattern solutions, which never appear for the Hanggi-Klimontovich interpretation, which is the other one typically appearing in the context of nonlinear diffusion processes. We also introduce a discrete-time microscopic model of particles that confirms the results obtained at the macroscopic density level.Comment: 4 pages, 3 figure

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

On a Conjecture of Goriely for the Speed of Fronts of the Reaction--Diffusion Equation

In a recent paper Goriely considers the one--dimensional scalar reaction--diffusion equation $u_t = u_{xx} + f(u)$ with a polynomial reaction term $f(u)$ and conjectures the existence of a relation between a global resonance of the hamiltonian system $u_{xx} + f(u) = 0$ and the asymptotic speed of propagation of fronts of the reaction diffusion equation. Based on this conjecture an explicit expression for the speed of the front is given. We give a counterexample to this conjecture and conclude that additional restrictions should be placed on the reaction terms for which it may hold.Comment: 9 pages Revtex plus 4 postcript figure

Magnetic properties of NaV2O5, a one-dimensional spin 1/2 antiferromagnet with finite chains

We have performed measurements of the magnetic susceptibility of NaV$_2$O$_5$ between 2 and 400 K. The high temperature part is typical of spin 1/2 chains with a nearest--neighbour antiferromagnetic exchange integral $J$ of 529 K. We develop a model for the susceptibility of a system with finite chains to account for the low temperature part of the data, which cannot be fitted by a standard Curie-Weiss term. These results suggest that the next nearest--neighbour exchange integral $J_2$ in CaV$_4$O$_9$ should be of the order of 500 K because, like $J$ in NaV$_2$O$_5$, it corresponds to corner sharing VO$_5$ square pyramids.Comment: An early version of the manuscript was mistakenly submitted. Although relatively minor, the changes concern the list of authors, the main text, the references and the figure captions. 10 pages of latex, 2 figure

Spin glass behavior in URh_2Ge_2

URh_2Ge_2 occupies an extraordinary position among the heavy-electron 122-compounds, by exhibiting a previously unidentified form of magnetic correlations at low temperatures, instead of the usual antiferromagnetism. Here we present new results of ac and dc susceptibilities, specific heat and neutron diffraction on single-crystalline as-grown URh_2Ge_2. These data clearly indicate that crystallographic disorder on a local scale produces spin glass behavior in the sample. We therefore conclude that URh_2Ge_2 is a 3D Ising-like, random-bond, heavy-fermion spin glass.Comment: 10 pages, RevTeX, with 4 postscript figures, accepted by Physical Review Letters Nov 15, 199