Traveling Waves in Discrete Periodic Media for Bistable Dynamics

Chen, Xinfu; Guo, Jong‐Shenq; Wu, Chin-Chin

doi:10.1007/s00205-007-0103-3

Cited by 90 publications

(104 citation statements)

References 37 publications

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“…In this section, we restate the theorems of Chen, Guo, and Wu [7] on the existence, uniqueness, asymptotics, and stability of a traveling wave solution in spatially discrete periodic media. (See Theorems 2, 3, 4, and 6 in [7].)…”

Section: Resultsmentioning

confidence: 99%

“…We next summarize the existence, uniqueness, and stability results for traveling wavefronts in spatially discrete periodic media as presented by Chen, Guo, and Wu [7] and subsequently illustrate the consequences for the antidiffusion lattice Nagumo traveling wavefront problem.…”

Section: In This New System a Traveling Wavefront If It Exists Conmentioning

confidence: 92%

“…The simplest nonhomogeneous medium is a periodic medium. The theory presented by Chen, Guo, and Wu [7] deals with the periodic media case. Unlike the homogeneous medium, the periodic case requires that one solve for N wave profiles, where N is the period of the wave.…”

Section: Existence Uniqueness and Stability Of Traveling Fronts In mentioning

confidence: 99%

“…Section 3.2 defines the problem while section 3.3 contains the main results [7]. Consider a general system of spatially discrete reaction-diffusion equations for U (t) = {u n (t)} n∈Z :…”

Section: Assumptionsmentioning

confidence: 99%

“…The next section contains the results of the study of Chen, Guo, and Wu [7]. In particular, they proved the existence of traveling wave fronts in spatially discrete periodic media; however, unlike the spatially continuous homogeneous case, finding a closed form for such solutions is not immediate (see [9] for one technique to construct solutions in the spatially discrete homogeneous media case).…”

Section: Statement Of the Problemmentioning

confidence: 99%

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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: Resultsmentioning

confidence: 99%

Section: In This New System a Traveling Wavefront If It Exists Conmentioning

confidence: 92%

Section: Existence Uniqueness and Stability Of Traveling Fronts In mentioning

confidence: 99%

“…Section 3.2 defines the problem while section 3.3 contains the main results [7]. Consider a general system of spatially discrete reaction-diffusion equations for U (t) = {u n (t)} n∈Z :…”

Section: Assumptionsmentioning

confidence: 99%

Section: Statement Of the Problemmentioning

confidence: 99%

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Traveling Wavefronts in an Antidiffusion Lattice Nagumo Model

Brucal-Hallare¹,

Vleck²

2011

SIAM J. Appl. Dyn. Syst.

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Periodic traveling wave solutions of discrete nonlinear Klein‐Gordon lattices

Hennig

Karachalios

2023

Math Methods in App Sciences

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We prove the existence of periodic traveling wave solutions for general discrete nonlinear Klein‐Gordon systems, considering both cases of hard and soft on‐site potentials. In the case of hard on‐site potentials, we implement a fixed‐point theory approach, combining Schauder's fixed‐point theorem and the contraction mapping principle. This approach enables us to identify a ring in the energy space for nontrivial solutions to exist, energy (norm) thresholds for their existence, and upper bounds on their velocity. In the case of soft on‐site potentials, the proof of existence of periodic traveling wave solutions is facilitated by a variational approach based on the mountain pass theorem. Thresholds on the averaged kinetic energy for these solutions to exist are also derived.

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Traveling Waves and Pattern Formation for Spatially Discrete Bistable Reaction-Diffusion Equations

Hupkes

Morelli

Schouten-Straatman

et al. 2020

Springer Proceedings in Mathematics &Amp; Statistics

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We survey some recent results on traveling waves and pattern formation in spatially discrete bistable reaction-diffusion equations. We start by recalling several classic results concerning the existence, uniqueness and stability of travelling wave solutions to the discrete Nagumo equation with nearest-neighbour interactions, together with the Fredholm theory behind some of the proofs. We subsequently discuss extensions involving wave connections between periodic equilibria, long-range interactions and planar lattices. We show how some of the results can be extended to the two-component discrete FitzHugh-Nagumo equation, which can be analyzed using singular perturbation theory. We conclude by studying the behaviour of the Nagumo equation when discretization schemes are used that involve both space and time, or that are non-uniform but adaptive in space.

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Traveling Waves in Discrete Periodic Media for Bistable Dynamics

Cited by 90 publications

References 37 publications

Traveling Wavefronts in an Antidiffusion Lattice Nagumo Model

Traveling Wavefronts in an Antidiffusion Lattice Nagumo Model

Periodic traveling wave solutions of discrete nonlinear Klein‐Gordon lattices

Traveling Waves and Pattern Formation for Spatially Discrete Bistable Reaction-Diffusion Equations

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