Stability of Traveling Pulses with Oscillatory Tails in the FitzHugh–Nagumo System

Abstract

The FitzHugh–Nagumo equations are known to admit fast traveling pulses that have monotone tails and arise as the concatenation of Nagumo fronts and backs in an appropriate singular limit, where a parameter $\varepsilon$ goes to zero. These pulses are known to be nonlinearly stable with respect to the underlying PDE. Recently, the existence of fast pulses with oscillatory tails was proved for the FitzHugh–Nagumo equations. In this paper, we prove that the fast pulses with oscillatory tails are also nonlinearly stable. Similar to the case of monotone tails, stability is decided by the location of a nontrivial eigenvalue near the origin of the PDE linearization about the traveling pulse. We prove that this real eigenvalue is always negative. However, the expression that governs the sign of this eigenvalue for oscillatory pulses differs from that for monotone pulses, and we show indeed that the nontrivial eigenvalue in the monotone case scales with $\varepsilon$, while the relevant scaling in the oscillatory case is $\varepsilon ^{2/3}$.

Appendices

Appendix 1: Corner Estimates

In this section, we provide a proof of Theorem 4.5, based on a theorem in Eszter (1999), regarding the nature of solutions upon entry to a neighborhood of a slow manifold.

Proof of Theorem 4.5

This proof is based on an argument in Eszter (1999). In the box

$$\begin{aligned} \mathcal {U}_E' := \left\{ (U,V,W): U,V\in [-\Delta ,\Delta ], W\in [-\Delta , W^*+\Delta ] \right\} , \end{aligned}$$

for sufficiently small $\varepsilon >0$, there exist constants $\alpha ^{u/s}_\pm >0$ such that

$$\begin{aligned} 0<\alpha ^s_-<\Lambda (U,V,W;c,a,\varepsilon )<\alpha ^s_+,\\ 0<\alpha ^u_-<\Gamma (U,V,W;c,a,\varepsilon )<\alpha ^u_+, \end{aligned}$$

We first consider the V-coordinate. For any $\xi >\xi _1$, we have

$$\begin{aligned} \left| V(\xi )\right| \ge \left| V(\xi _1)\right| e^{\alpha ^u_-(\xi -\xi _1)}. \end{aligned}$$

Since $V(\xi _2)\in N_2$, we also have

$$\begin{aligned} \left| V(\xi _1)\right| \le \Delta e^{-\alpha ^u_-(\xi _2-\xi _1)}. \end{aligned}$$

We note that since the solution enters $\mathcal {U}_E'$ via $N_1$ and reaches $N_2$ at $\xi _2(\varepsilon )$, using the equation for W in (3.9), we have that $\xi _2(\varepsilon )$ satisfies $\xi _2(\varepsilon )\ge (C\varepsilon )^{-1}$. Therefore, using the upper bound on $\Gamma$ we have that

$$\begin{aligned} \left| V(\xi )\right| \le \Delta e^{-\alpha ^u_-\xi _2+\alpha ^u_+\xi -(\alpha ^u_+-\alpha ^u_-)\xi _1}\le Ce^{-\tfrac{1}{C\varepsilon }}, \end{aligned}$$

for $\xi \in \left[ \xi _1,\Xi (\varepsilon )\right]$.

The solution in the slow W-component may be written as

$$\begin{aligned} W(\xi ) = W(\xi _1,\varepsilon )+\int _{\xi _1}^\xi \varepsilon (1+H(U(s),V(s),W(s),c,a,\varepsilon )U(s)V(s))\hbox {d}s, \end{aligned}$$

from which we infer that

$$\begin{aligned} |W(\xi ) -W(\xi _1,\varepsilon )|\le C\varepsilon (\xi -\xi _1)\le C\varepsilon \Xi (\varepsilon ),\quad \text {for } \xi \in \left[ \xi _1,\Xi (\varepsilon )\right] , \end{aligned}$$

and hence

$$\begin{aligned} |W(\xi )|\le C\varepsilon \Xi (\varepsilon )+|W(\xi _1,\varepsilon )|,\quad \text {for } \xi \in \left[ \xi _1,\Xi (\varepsilon )\right] . \end{aligned}$$

Finally, we consider the U-component. We have that the difference $(U(\xi )-U_0(\xi ))$ satisfies

$$\begin{aligned} U'-U_0'&=-(\Lambda (U,V,W,c,a,\varepsilon )U-\Lambda (U_0,0,0,c,a,0)U_0)\\&= -\Lambda (U_0,0,0,c,a,0)(U-U_0)+\mathcal {O}\left( \varepsilon + |U-U_0|+|V|+|W|\right) U. \end{aligned}$$

with $U(\xi _1)-U_0(\xi _1)=\tilde{U}_0$ where $|\tilde{U}_0|\ll \Delta$. By possibly taking $\Delta$ smaller if necessary and using the fact that the rate of contraction in the U-component is stronger than $\alpha ^s_-$, we deduce that $(U(\xi )-U_0(\xi ))$ satisfies a differential equation

$$\begin{aligned} X'&=b_1(\xi )X +b_2(\xi ), \quad X(\xi _1)=\tilde{U}_0, \end{aligned}$$

where $b_1(\xi )<-\alpha ^s_-/2<0$ and

$$\begin{aligned} |b_2(\xi )|\le C\left( \varepsilon \Xi (\varepsilon )+|W(\xi _1,\varepsilon )|\right) e^{-\alpha ^s_-\xi } , \end{aligned}$$

for $\xi \in \left[ \xi _1,\Xi (\varepsilon )\right]$. Hence, it holds

$$\begin{aligned} |U(\xi )-U_0(\xi )| \le C\left( \varepsilon \Xi (\varepsilon )+|\tilde{U}_0|+|W(\xi _1,\varepsilon )|\right) , \end{aligned}$$

for $\xi \in \left[ \xi _1,\Xi (\varepsilon )\right]$, which completes the proof. $\square$

Appendix 2: Exponential Dichotomies and Trichotomies

It is well known that exponential separation is an important tool in studying spectral properties of traveling waves Sandstede (2002). Below we provide the definitions of exponential dichotomies and trichotomies to familiarize the reader with our notation. For an extensive introduction, we refer to Coppel (1978), Sandstede (1993).

Definition 9.1

Let $n \in \mathbb {Z}_{> 0}$, $J \subset \mathbb {R}$ an interval and $A \in C(J,\mathrm {Mat}_{n \times n}(\mathbb {C}))$. Denote by T(x, y) the evolution operator of

$$\begin{aligned} \varphi _x = A(x)\varphi . \end{aligned}$$

(9.2)

Equation (9.2) has an exponential dichotomy on J with constants $K,\mu > 0$ and projections $P^s(x), P^u(x) :\mathbb {C}^n \rightarrow \mathbb {C}^n, x \in J$ if for all $x,y \in J$ it holds

• $P^u(x) + P^s(x) = 1$;

• $P^{u,s}(x)T(x,y) = T(x,y) P^{u,s}(y)$;

• $\Vert T(x,y)P^s(y)\Vert , \Vert T(y,x)P^u(x)\Vert \le Ke^{-\mu (x-y)}$ for $x \ge y$.

Equation (9.2) has an exponential trichotomy on J with constants $K,\mu ,\nu > 0$ and projections $P^u(x),P^s(x),P^c(x):\mathbb {C}^n \rightarrow \mathbb {C}^n, x \in J$ if for all $x,y \in J$ it holds

• $P^u(x) + P^s(x) + P^c(x) = 1$;

• $P^{u,s,c}(x)T(x,y) = T(x,y) P^{u,s,c}(y)$;

• $\Vert T(x,y)P^s(y)\Vert , \Vert T(y,x)P^u(x)\Vert \le Ke^{-\mu (x-y)}$ for $x \ge y$;

• $\Vert T(x,y)P^c(y)\Vert \le Ke^{\nu |x-y|}$.

Often we use the abbreviations $T^{u,s,c}(x,y) = T(x,y)P^{u,s,c}(y)$ leaving the associated projections of the dichotomy or trichotomy implicit.