Depinning Asymptotics in Ergodic Media

Scheel, Arnd; Tikhomirov, Sergey

doi:10.1007/978-3-319-64173-7_6

Cited by 3 publications

(3 citation statements)

References 30 publications

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“…We point out that, apart from the aforementioned works, the existence of pulsating fronts was investigated in [43,46,54,56,58]. References [43,46,58] were also concerned with generalized transition fronts in the sense of [9,10], for which the media can be non-periodic (see [3,14,20,24,25,41,45,47,49,52] for further propagation or blocking results in non-periodic bistable media).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Bistable pulsating fronts in slowly oscillating environments *

Ding,

Hamel,

Liang

2022

Preprint

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We consider reaction-diffusion fronts in spatially periodic bistable media with large periods. Whereas the homogenization regime associated with small periods had been well studied for bistable or Fisher-KPP reactions and, in the latter case, a formula for the limit minimal speeds of fronts in media with large periods had also been obtained thanks to the linear formulation of these minimal speeds and their monotonicity with respect to the period, the main remaining open question is concerned with fronts in bistable environments with large periods. In bistable media the unique front speeds are not linearly determined and are not monotone with respect to the spatial period in general, making the analysis of the limit of large periods more intricate. We show in this paper the existence of and an explicit formula for the limit of bistable front speeds as the spatial period goes to infinity. We also prove that the front profiles converge to a family of front profiles associated with spatially homogeneous equations. The main results are based on uniform estimates on the spatial width of the fronts, which themselves use zero number properties and intersection arguments.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Bistable pulsating fronts in slowly oscillating environments *

Ding,

Hamel,

Liang

2022

Preprint

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“…Middle: the saddle-node bifurcation at a = a ± . Right: the degenerate situation at a = 1 2 corresponding to the multi-site discretization (13) Indeed, pinning phenomena have also been investigated in nonlocal problems featuring smooth convolution kernels [4,140]. In certain cases the behaviour of c near the pinning boundary can be described by power laws with various exponents.…”

Section: Pinned Wavesmentioning

confidence: 99%

“…In particular, the dependence on z is now non-trivial. This means that one can no longer hope to find an exact solution similar to (140).…”

Section: Linear Theorymentioning

confidence: 99%

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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Depinning of Traveling Waves in Ergodic Media

Tikhomirov

2019

Trends in Mathematics

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Depinning Asymptotics in Ergodic Media

Cited by 3 publications

References 30 publications

Bistable pulsating fronts in slowly oscillating environments *

Bistable pulsating fronts in slowly oscillating environments *

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

Depinning of Traveling Waves in Ergodic Media

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