Traveling pulse solutions to Fitzhugh-Nagumo equations
type: Item
instance of: scholarly article
title: Traveling pulse solutions to Fitzhugh-Nagumo equations
zbMATH Open document ID: 1328.34037
author: Chao-Nien Chen
author: Y. S. Choi
published in: Calculus of Variations and Partial Differential Equations
publication date: 2015-09-11T00:00:00Z
review text: The aim of this paper is to obtain a traveling pulse solution to the system of FitzHugh-Nagumo equations. Specifically they are interested in the traveling pulse of \[ \begin{aligned} u_t = u_{xx} + \frac{1}{d}(f(u)-v),\\ v_t=v_{xx}+u-\gamma v, \end{aligned}\eqno{(1)} \] where \(f(u) = u(u-\beta )(1-u), \beta \in (0, 1/2), d > 0\) and \(\gamma>0\). The authors look for a traveling wave of (1) which is decaying to \((0, 0)\) at \(|x| = \infty\) and to obtain it they study the homoclinic solution of \[ \begin{aligned} d c^2 u_{xx} + d c^2 u_{x} +f(u)-v=0 ,\\ c^2 v_{xx} + c^2 v_{x} + u - \gamma v=0.\end{aligned}\eqno{(2)} \] They state the following results. { Theorem 1.1} Given \(\beta \in (0, 1/2)\) and \(\gamma < 4/(1 - \beta )^2\), there is a \(d_3 = d_3(\gamma )\) such that if \(d \leq d_3\) then for some \(c > 0\), (2) has a solution which is denoted by \((u_0, v_0)\). In fact \(u_0, v_0 \in C^\infty(\mathbb{R})\) and are exponentially decaying to \(0\) as \(|x| \to \infty\); that is, (1) possesses a traveling pulse solution. The authors investigate the profile for the fastest traveling pulse and its asymptotical speed for \(d\) being sufficiently small. {Theorem 1.2} Let \((3 -\sqrt{6})/6 < \beta < 1/2\), \(\gamma \in (0, \beta /3)\) and \((c_0, u_0, v_0)\) be the fastest traveling pulse obtained by Theorem 1.1. Let \(\beta_1\) be the unique point in \((\beta , 1)\) to satisfy \(\int_0^{\beta_1} f (\eta )\, d\eta = 0.\) Then {\parindent=6mm \begin{itemize} \item[(i)] There exist \(z_1 < z_2 < z_4 < z_5\) such that \(u_0 < 0\) on \((-\infty, z_2)\), \(u_0 > 0\) on \((z_2, z_4)\) and \(u_0 < 0\) on \((z_4,\infty)\). In each of such intervals, \(u_0\) has only one critical point; that is, a minimum at \(z_1\) in \((-\infty, z_2)\) and a minimum at \(z_5\) in \((z_4,\infty)\), and \(u_0(y_1)\) is the unique maximum located in \((z_2, z_4)\). \item [(ii)] \(\beta_1 < u_0(y_1) < 1\) and \(\min\{u_0(z_1), u_0(z_5)\} > -1.\) \item [(iii)] \(v_0\) attains its maximum at …
reviewed by: Petr Tomiczek
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35C07
Mathematics Subject Classification ID: 58E50
zbMATH DE Number: 6481158
zbMATH Keywords: reaction-diffusion system
zbMATH Keywords: travelling pulse solution
zbMATH Keywords: speed of pulse
zbMATH Keywords: Fitzhugh-Nagumo equations
MaRDI profile type: Publication
full work available at URL: s00526-014-0776-z
OpenAlex ID: W2089396557
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DOI: 10.1007/S00526-014-0776-Z
description: scientific article; zbMATH DE number 6481158