Fredholm properties of nonlocal differential operators via spectral flow

Faye, Grégory; Scheel, Arnd

doi:10.1512/iumj.2014.63.5383

Cited by 22 publications

(56 citation statements)

References 23 publications

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“…We remark that both linearized operators are automatically Fredholm of index zero [13], so that the algebraic multiplicity of the eigenvalue λ = 0 is finite. Since the derivatives of front and back profile contribute to the kernel, multiplicity is at least one.…”

Section: Hypothesis (H2)mentioning

confidence: 98%

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Existence of pulses in excitable media with nonlocal coupling

Faye

Scheel

2015

Advances in Mathematics

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We prove the existence of fast traveling pulse solutions in excitable media with non-local coupling. Existence results had been known, until now, in the case of local, diffusive coupling and in the case of a discrete medium, with finite-range, non-local coupling. Our approach replaces methods from geometric singular perturbation theory, that had been crucial in previous existence proofs, by a PDE oriented approach, relying on exponential weights, Fredholm theory, and commutator estimates.

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Section: Hypothesis (H2)mentioning

confidence: 98%

“…(i) We first remark that the operator L(u f ) is a Fredholm operator from H 1 η to L 2 η whenever η < η 0 , and its Fredholm index is 0 [13]. Its kernel is spanned by u f and its cokernel is spanned by e * f ∈ H 1 η , a solution of the adjoint equation…”

Section: Invertibility Of L I F and L I Bmentioning

confidence: 99%

Existence of pulses in excitable media with nonlocal coupling

Faye

Scheel

2015

Advances in Mathematics

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“…This analysis relies on some recent work [16] on Fredholm properties of nonlocal differential operators with infinite range interactions. Theorem 3 can be seen as preliminary result towards the nonlinear stability of traveling fronts of equation (1.1) (see the Discussion 5).…”

Section: Resultsmentioning

confidence: 99%

“…We are going to apply [16,Theorems 2 & 3]. To do so, we shall verify that all hypotheses of [16,Theorems 2 & 3] are satisfied for the operator T :…”

Section: Spectral Analysis -Proof Of Theoremmentioning

confidence: 95%

Traveling fronts for lattice neural field equations

Faye

2018

Physica D: Nonlinear Phenomena

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We show existence and uniqueness of traveling front solutions to a class of neural field equations set on a lattice with infinite range interactions in the regime where the kinetics of each individual neuron is of bistable type. The existence proof relies on a regularization of the traveling wave problem allowing us to use well-known existence results for traveling front solutions of continuous neural field equations. We then show that the traveling front solutions which have nonzero wave speed are unique (up to translation) by constructing appropriate sub and super solutions. The spectral properties of the traveling fronts are also investigated via a careful study of the linear operator around a traveling front in co-moving frame where we crucially use Fredholm properties of nonlocal differential operators previously obtained by the author in an earlier work. For the spectral analysis, we need to impose an extra exponential localization condition on the interactions.

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“…We show that for λ in a suitable right half-plane the operator L h,∞;λ is hyperbolic in the sense of [21,48], i.e. we write…”

Section: )mentioning

confidence: 99%

Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions

Schouten-Straatman¹,

Hupkes²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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We establish the existence and nonlinear stability of travelling pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions close to the continuum limit. For the verification of the spectral properties, we need to study a functional differential equation of mixed type (MFDE) with unbounded shifts. We avoid the use of exponential dichotomies and phase spaces, by building on a technique developed by Bates, Chen and Chmaj for the discrete Nagumo equation. This allows us to transfer several crucial Fredholm properties from the PDE setting to our discrete setting.

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Fredholm properties of nonlocal differential operators via spectral flow

Cited by 22 publications

References 23 publications

Existence of pulses in excitable media with nonlocal coupling

Existence of pulses in excitable media with nonlocal coupling

Traveling fronts for lattice neural field equations

Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions

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