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

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

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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.

    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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