# American Institute of Mathematical Sciences

September  2019, 39(9): 5017-5083. doi: 10.3934/dcds.2019205

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

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

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

Received  May 2018 Revised  February 2019 Published  May 2019

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

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.

Citation: Willem M. Schouten-Straatman, Hermen Jan Hupkes. Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5017-5083. doi: 10.3934/dcds.2019205
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##### References:
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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