Differential and Integral Equations

Fronts on a lattice

Lambertus A. Peletier and José Antonio Rodríguez

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Motivated by a model of a system of many particles at low densities, we consider a lattice differential equation with two uniform steady states and we investigate the existence of travelling fronts connecting them. This leads to a two-point boundary-value problem for a nonlinear delay-differential equation. We replace the original parabolic nonlinearity by a piecewise-linear function, where explicit computations are possible. We find monotone and nonmonotone fronts. Finally we also describe all the fronts such that the $\alpha$-limit is the unstable uniform state. For different values of the wave speed $c$ of the front we find bounded and unbounded as well as eventually periodic orbits, i.e., orbits $u_c (x)$ that are periodic for $x\geqslant x_{\text{per}}(c)$ for some $x_{\text{per}}(c)\in{{\mathbb {R}}}$.

Article information

Differential Integral Equations, Volume 17, Number 9-10 (2004), 1013-1042.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37L60: Lattice dynamics [See also 37K60]
Secondary: 34C37: Homoclinic and heteroclinic solutions 82C32: Neural nets [See also 68T05, 91E40, 92B20]


Peletier, Lambertus A.; Rodríguez, José Antonio. Fronts on a lattice. Differential Integral Equations 17 (2004), no. 9-10, 1013--1042. https://projecteuclid.org/euclid.die/1356060312

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