Universality of Crystallographic Pinning

Hoffman, Aaron; Mallet‐Paret, John

doi:10.1007/s10884-010-9157-2

Cited by 37 publications

(39 citation statements)

References 26 publications

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“…A significant step towards confirming this was made by Hoffman and Mallet-Paret [19]. These authors provided a generic condition on the nonlinearity f in (1.1) that guarantees propagation failure, but unfortunately this condition is hard to verify in practice.…”

Section: Introductionmentioning

confidence: 99%

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Propagation failure in the discrete Nagumo equation

Hupkes

Pelinovsky

Sandstede

2011

Proc. Amer. Math. Soc.

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Abstract. We address the classical problem of propagation failure for monotonic fronts of the discrete Nagumo equation. For a special class of nonlinearities that support unpinned "translationally invariant" stationary monotonic fronts, we prove that propagation failure cannot occur. Properties of travelling fronts in the discrete Nagumo equation with such special nonlinear functions appear to be similar to those in the continuous Nagumo equation.

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Section: Introductionmentioning

confidence: 99%

“…In [19], Hoffman and Mallet-Paret study (1.1) with a general class of bistable nonlinearities f . Their results roughly state that the nondegenerate saddle-node bifurcation depicted in Figure 2(ii) occurs for almost every choice of f and always implies a − < a + .…”

Section: Introductionmentioning

confidence: 99%

Propagation failure in the discrete Nagumo equation

Hupkes

Pelinovsky

Sandstede

2011

Proc. Amer. Math. Soc.

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“…For anisotropic kernels, the resulting pinning region may well depend on the direction of propagation, an effect which is well known to lead to faceting of curved interfaces. For discrete, lattice systems, much progress has been achieved more recently by [10]. It would be interesting to study such problems in the nonlocal context presented here.…”

Section: Summary Of Resultsmentioning

confidence: 98%

Pinning and Unpinning in Nonlocal Systems

et al. 2016

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We investigate pinning regions and unpinning asymptotics in nonlocal equations. We show that phenomena are related to but different from pinning in discrete and inhomogeneous media. We establish unpinning asymptotics using geometric singular perturbation theory in several examples. We also present numerical evidence for the dependence of unpinning asymptotics on regularity of the nonlocal convolution kernel.

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“…It is proven that this critical value ρ * (θ) typically satisfies ρ * > 1 2 , depends continuously on θ when tan θ is irrational, and is discontinuous when tan θ is rational or infinite. By now there is plenty of numerical [14,25] and theoretical [33,21] evidence to suggest that this behavior is not just an artifact of the idealized nonlinearity g but also occurs in the case of the cubic nonlinearity (1.5).…”

Section: Lattice Differential Equationsmentioning

confidence: 99%

Negative Diffusion and Traveling Waves in High Dimensional Lattice Systems

Hupkes¹,

Vleck²

2013

SIAM J. Math. Anal.

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Abstract. We consider bistable reaction diffusion systems posed on rectangular lattices in two or more spatial dimensions. The discrete diffusion term is allowed to have positive spatially periodic coefficients, and the two spatially periodic equilibria are required to be well ordered. We establish the existence of traveling wave solutions to such pure lattice systems that connect the two stable equilibria. In addition, we show that these waves can be approximated by traveling wave solutions to systems that incorporate both local and nonlocal diffusion. In certain special situations our results can also be applied to reaction diffusion systems that include (potentially large) negative coefficients. Indeed, upon splitting the lattice suitably and applying separate coordinate transformations to each sublattice, such systems can sometimes be transformed into a periodic diffusion problem that fits within our framework. In such cases, the resulting traveling structure for the original system has a separate wave profile for each sublattice and connects spatially periodic patterns that need not be well ordered. There is no direct analogue of this procedure that can be applied to reaction diffusion systems with continuous spatial variables.

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Universality of Crystallographic Pinning

Cited by 37 publications

References 26 publications

Propagation failure in the discrete Nagumo equation

Propagation failure in the discrete Nagumo equation

Pinning and Unpinning in Nonlocal Systems

Negative Diffusion and Traveling Waves in High Dimensional Lattice Systems

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