Existence of traveling wavefront solutions for the discrete Nagumo equation

Zinner, Bertram

doi:10.1016/0022-0396(92)90142-a

Cited by 267 publications

(133 citation statements)

References 9 publications

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“…The classical results on the traveling wavefront problem to spatially continuous [1], [11], [12] or spatially discrete reaction-diffusion equations [21], [27], [28] are under the assumption that the space in which the waves propagate is homogeneous. The simplest nonhomogeneous medium is a periodic medium.…”

Section: Existence Uniqueness and Stability Of Traveling Fronts In mentioning

confidence: 99%

Traveling Wavefronts in an Antidiffusion Lattice Nagumo Model

Brucal-Hallare¹,

Vleck²

2011

SIAM J. Appl. Dyn. Syst.

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Abstract. We consider a system of lattice Nagumo equations with cubic nonlinearity, but with a negative discrete diffusion coefficient. We are interested in the existence, uniqueness, stability, and nonexistence of the traveling wavefront solutions of this system, and we shall call this problem the antidiffusion lattice Nagumo problem. By rewriting this system as a spatially periodic system with inhomogeneous but positive periodic diffusion coefficients and periodic nonlinearities, we uncover a rich solution behavior that includes many possible connecting orbits in the antidiffusion case. Second, we observe the presence of bistable and monostable dynamics. In the bistable region, we study the phenomenon of propagation of failure while in the monostable region, we compute the minimum wave speed.

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Section: Existence Uniqueness and Stability Of Traveling Fronts In mentioning

confidence: 99%

Traveling Wavefronts in an Antidiffusion Lattice Nagumo Model

Brucal-Hallare¹,

Vleck²

2011

SIAM J. Appl. Dyn. Syst.

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“…For traveling waves, we refer the readers to [11,34,12,42,1,4] and the references therein. The chaotic properties of solutions for such systems have been investigated by [11] and [14,40,13,22].…”

Section: Introductionmentioning

confidence: 99%

Attractors for stochastic lattice dynamical systems with a multiplicative noise

Caraballo

2008

Front. Math. China

110

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“…They studied the long time behavior of solutions to (1.1) for some nonlinear function f. When the nonlinear term f is a monostable/bistable type, there are extensive results about the traveling wave solutions for equation (1.1), some of which have revealed some essential differences between a discrete model and its corresponding continuous one. For details, see for example, [4,5,23,24]. Taking into account time delay in population dynamics, Wu and Zou [21] considered the delayed lattice differential equations and studied the existence of traveling wave solutions.…”

Section: Introductionmentioning

confidence: 99%

Existence of traveling wave solutions in m-dimensional delayed lattice dynamical systems with competitive quasimonotone and global interaction

Zhou¹

2017

J. Nonlinear Sci. Appl.

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This paper deals with the existence of traveling wave solutions for m-dimensional delayed lattice dynamical systems with competitive quasimonotone and global interaction. By using Schauder's fixed point theorem and a cross-iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The general results obtained will be applied to m-dimensional delayed lattice dynamical systems with Lotka-Volterra type competitive reaction terms and global interaction.

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Existence of traveling wavefront solutions for the discrete Nagumo equation

Cited by 267 publications

References 9 publications

Traveling Wavefronts in an Antidiffusion Lattice Nagumo Model

Traveling Wavefronts in an Antidiffusion Lattice Nagumo Model

Attractors for stochastic lattice dynamical systems with a multiplicative noise

Existence of traveling wave solutions in m-dimensional delayed lattice dynamical systems with competitive quasimonotone and global interaction

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