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

Willem M. Schouten-Straatman; Hermen Jan Hupkes

doi:10.3934/dcds.2019205

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2019, Volume 39, Issue 9: 5017-5083. Doi: 10.3934/dcds.2019205

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

Willem M. Schouten-Straatman^, and
Hermen Jan Hupkes

Mathematisch Instituut - Universiteit Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands

^* Corresponding author: w.m.schouten@math.leidenuniv.nl
^* Corresponding author: w.m.schouten@math.leidenuniv.nl

Received: April 30, 2018

Revised: January 31, 2019

Published: May 2019

Both authors acknowledge support from the Netherlands Organization for Scientific Research (NWO) (grant 639.032.612)

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Abstract

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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Mathematics Subject Classification: 34A33, 34D35, 34K08, 34K26, 34K31.

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Figure 1. Illustration of the regions $ R_1,R_2,R_3 $ and $ R_4 $. Note that the regions $ R_2 $ and $ R_3 $ grow when $ h $ decreases, while the regions $ R_1 $ and $ R_4 $ are independent of $ h $

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