Traveling pulse solutions to Fitzhugh-Nagumo equations
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 \(\hat{z}\) which is the only critical point of \(v_0.\) \item [(iv)] \(0 < v_0 < 1\) on \((-\infty,\infty)\) and \(z_2 < \hat{z} < y_1.\) \end{itemize}} The authors also consider a sequence \(d^{(n)}\to 0\) as \(n\to \infty\). For each \(d^{(n)}\), denoted by \((c^{(n)}_0, u^{(n)}_0 , v^{(n)}_0)\) a fastest traveling pulse. The next theorem demonstrates the asymptotical behavior of such traveling pulses. { Theorem 1.3} Suppose that \((c_0, u_0, v_0)\) is the fastest traveling pulse solution obtained by Theorem 1.1. Then {\parindent=6mm \begin{itemize} \item[(i)] \(dc^2_0 \leq \frac{(1-2\beta )^2}{2}\), and along any sequence \(d^{(n)} \to 0^+\) as \(n\to \infty\), \(dc^2_0 \to \frac{(1-2\beta )^2}{2}\) always holds. \item [(ii)] If \(y_1 \equiv \sup_x\{ x : u_0 \text{ attains a positive maximum at } x\}\), then \(u^{(n)}_0 (y_1) \to 1, u^{(n)}_0 (z_5) \to 0\) and \(v^{(n)}_0 (y_1) \to 0\) as \(d^{(n)} \to 0^+\), where the locations of \(y_1\) and \(z_5\) depend on the choice of \(d^{(n)}\). \end{itemize}} To show the existence of traveling pulse solutions, the authors employ a variational formulation with a nonlocal term.
scientific article; zbMATH DE number 6481158
scientific article