We consider traveling wave solutions to spatially discrete reaction-diiusion equations with nonlocal variable diiusion and bistable nonlinearities. For the case of spatially periodic diiusion we obtain analytic solutions for the traveling wave problem using a piecewise linear nonlinearity. The formula for the wave forms is implicitly deened in the general periodic case and we provide an explici...

This paper builds on the past study of single-spike waves in one-dimensional integrate-and-fire networks to provide a framework for the study of waves with arbitrary (finite or countably infinite) collections of spike times. Based on this framework, we prove an existence theorem for single-spike traveling waves, and we combine analysis and numerics to study two-spike traveling waves, periodic t...

We study traveling wave solutions to nonlinear Schrödinger (NLS) and nonlinear Klein-Gordon (NLKG) equations on a compact Riemannian manifold M , with a Killing field X, generating a group of isometries. The emphasis is on NLKG; then if X has length < 1 everywhere, one gets a semilinear elliptic PDE on M , to which standard variational techniques apply (for a natural class of nonlinearities), a...

The Quota Traveling Salesman Problem is a generalization of the well known Traveling Salesman Problem. The goal of the traveling salesman is, in this case, to reach a given quota of the sales, minimizing the amount of time. In this paper we address the on-line version of the problem, where requests are given over time. We present algorithms for various metric spaces, and analyze their performan...

This paper describes a catalytic theory that grounds cognition in biology. An enzyme is a biological catalyst that increases the speed of a molecular reaction; the reaction is thought to occur via a traveling wave, a soliton, whose formation and persistence depend on the structure or invariance of the catalytic environment. Generalizing to cognition, the invariance of perceptual events plays th...

Piezoelectric traveling wave rotary ultrasonic motors are motors that generate torque by using the friction force between a piezoelectric composite ring (or disk-shaped stator) and a metallic ring (or disk-shaped rotor) when a traveling wave is excited in the stator. The motor speed is proportional to the amplitude of the traveling wave and, in order to obtain large amplitudes, the stator is ex...

We use averaging and homogenization theory to study the propagation of traveling pulses in an inhomogeneous excitable neural network. The network is modeled in terms of a nonlocal integrodifferential equation, in which the integral kernel represents the spatial distribution of synaptic weights. We show how a spatially periodic modulation of homogeneous synaptic connections leads to an effective...

This paper treats a free boundary problem in two-dimensional excitable media arising from a singular limiting problem of a FitzHugh–Nagumo-type reaction-diffusion system. The existence and uniqueness up to translations of two-dimensional traveling curved waves solutions is shown. To study the stability of the waves, the local and global existence and uniqueness of solutions to the free boundary...