Index theory for traveling waves in reaction diffusion systems with skew gradient structure

The setting of this paper is the systems of reaction-diffusion equations of the form

where $x,t\in\mathbb{R}$ are space and time, respectively, $w\in\mathbb{R}^{n}$ and $\nabla F$ are the gradients of the function $F:\mathbb{R}^{n}\rightarrow\mathbb{R}$, and $Q\in\mathrm{Mat}\,(n,\mathbb{R})$ has the form $Q=\begin{bmatrix}I_{r}&0\\ 0&-I_{n-r}\end{bmatrix}$, where $\mathrm{Mat}\,(n,\mathbb{R})$ is the set of all $n\times n$ matrices. Such a system is of the activator-inhibitor type and will be referred to as a skew-gradient system [18]. A traveling wave solution $w^{*}$ of one variable $\xi=x-ct$ is a solution to (1.1) with asymptotic behavior

Here, $w_{\pm}$ are the constant equilibria of (1.1): that is, $\nabla F(w_{\pm})=0$. If $w_{+}=w_{-}$, then $w^{*}$ is called a pulse (and a front otherwise). In this paper, we are focused primarily on fronts. We will always consider the case of $c>0$: otherwise, the direction of motion can be reversed.

Traveling waves being solutions of (1.1) is an important subjection in dynamical systems, and many aspects of traveling waves have been studied in extensive works[11, 6]. One of the key issues of concern is whether a steady state is stable with respect to disturbances under initial conditions since this directly determines whether it can be observed in nature. Motivated by [9], in this paper, we establish a unified geometric approach for the stability analysis of traveling front solutions for (1.1). Written in a moving frame, a traveling front solution $w^{*}$ of (1.1) can be regarded as a heteroclinic solution $w^{*}$ of the following equation:

The stability analysis is directly related to the spectral information of the operator $L:=\frac{\operatorname{d}^{2}}{\operatorname{d}\xi^{2}}+c\frac{\operatorname{d}}{\operatorname{d}\xi}+QB(\xi)$ by linearizing (1.2) along $w^{*}$, where $B(\xi)=\nabla^{2}F(w^{*})$; note that the matrices $B_{\pm}\coloneqq\lim\limits_{\xi\to\pm\infty}B(\xi)$ are well-defined. Moreover, the previous description guarantees that there exists $C>$, such that

For $z\in\mathbb{C}$, denote by $\Re z$ and $\mathrm{\mathfrak{I}}z$ the real and imaginary parts of $z$, respectively. $\mathbb{R}^{+}\coloneqq(0,+\infty)$, $\mathbb{R}^{-}\coloneqq(-\infty,0)$, $\mathbb{C}^{+}\coloneqq\{z\in\mathbb{C}|\Re z>0\}$ and $\mathbb{C}^{-}\coloneqq\{z\in\mathbb{C}|\Re z<0\}$. The pair $(\mathbb{R}^{n},\langle,\rangle)$ denotes the $n$-dimensional Euclidean space. Moreover, $\overline{\#}$ denotes the closure of the set $\#$.

Since a wave solution of (1.1) possesses a translation invariance property, it is said to be nondegenerate if zero is a simple eigenvalue of $L$.

In this paper, we study the following special case.

Thus, $w_{+}$ and $w_{-}$ are both stable equilibria of (1.1).

From now on, we focus on studying the eigenvalue problem

Denoted by $\sigma_{p}(L)$ is the set of isolated eigenvalues with finite multiplicity, and $\sigma_{ess}(L)=\sigma(L)\backslash\sigma_{p}(L)$ is the essential spectrum of $L$. It is known (Cf. Lemma 2.3) that $\sigma_{ess}(L)\subset\mathbb{C}^{-}$. As was commonly performed in [9], we set $y=\begin{bmatrix}\dot{\phi}\\ \phi\end{bmatrix}$ to convert (1.4) to

where $A_{\lambda}(\xi)=\begin{bmatrix}-c&\lambda I-QB(\xi)\\ I&0\end{bmatrix}$, and there are well-defined matrices $A_{\lambda}(\pm\infty)=\lim\limits_{\xi\to\pm\infty}A_{\lambda}(\xi)$. Throughout this paper, the dots denote differentiation with respect to $\xi$.

For any $M\in\mathrm{Mat}\,(\mathbb{R},\mathbb{R}^{n})$, denote by $V^{+}(M)$ ($V^{-}(M)$) the positive (negative) spectral space corresponding to the eigenvalues of $M$ having positive (negative) real parts. We provide the next two conditions:

Letting $B\in\mathrm{Mat}\,(n,\mathbb{R})$, in the following remark, we will explain the relationship between (H(H1)) and (H(H2’)).

Following Definition 1.1, we focus our attention on $\lambda\in\overline{\mathbb{R}^{+}}$: suppose that $\Phi_{\tau,\lambda}(\xi)$ is the matrix solution of (1.5), such that $\Phi_{\tau,\lambda}(\tau)=I$. We recall that the stable and unstable subspaces of (1.5) are

Throughout this paper, we abbreviate $E_{0}^{s}(\tau)$ and $E^{u}_{0}(\tau)$ as $E^{s}(\tau)$ and $E^{u}(\tau)$, respectively.

To realize the symplectic structure, we introduce the matrix $J\coloneqq\begin{bmatrix}0&-Q\\ Q&0\end{bmatrix}$. Since $Q^{2}=I$ and $Q^{T}=Q$, it follows that $J^{2}=-I$ and $J^{T}=-J$. Therefore, $J$ is a complex structure on $\mathbb{R}^{2n}$, and $\omega(\cdot,\cdot)\coloneqq\left\langle J\cdot,\cdot\right\rangle$ defines a syplectic form on $\mathbb{R}^{2n}$. $\mathrm{Lag}(n)$ denotes the set of all Lagrangian subspaces of $(\mathbb{R}^{2n},\omega)$.

By invoking [13, Lemma 3.1], Remarks 2.4 and 2.6 imply that the subspaces $E^{s}_{\lambda}(\tau)$ and $E^{u}_{\lambda}(\tau)$ are both Lagrangian for $(\tau,\lambda)\in\mathbb{R}\times\overline{\mathbb{R}^{+}}$. Moreover, those results can be obtained from the same discussion in [9, Theorem 2.3].

We denote by $\mathscr{P}([0,1];\mathbb{R}^{2n})$ the space of all ordered pairs of continuous maps of Lagrangian subspaces $L:[0,1]\ni t\longmapsto L(t)\coloneqq\big{(}L_{1}(t),L_{2}(t)\big{)}\in\mathrm{Lag}(n)\times\mathrm{Lag}(n)$ equipped with the compact-open topology. Following the authors in [4], we are in the position to briefly recall the definition of the Maslov index for pairs of Lagrangian subspaces, which will be denoted throughout the paper by the symbol $\operatorname{\iota^{\scriptscriptstyle{\mathrm{CLM}}}}$. Loosely speaking, given the pair $L=(L_{1},L_{2})\in\mathscr{P}([0,1];\mathbb{R}^{2n})$, this index counts with signs and multiplicities the number of instants $t\in[0,1]$ that $L_{1}(t)\cap L_{2}(t)\neq\{0\}$.

For the sake of the reader, we list a couple of properties of the $\operatorname{\iota^{\scriptscriptstyle{\mathrm{CLM}}}}$-index that we shall use throughout the paper.

• •

  (Reversal) Let $L\coloneqq(L_{1},L_{2})\in\mathscr{P}([a,b];\mathbb{R}^{2n})$. Denoting by $\widehat{L}\in\mathscr{P}([-b,-a];\mathbb{R}^{2n})$ the path traveled in the opposite direction, and by setting $\widehat{L}\coloneqq(L_{1}(-s),L_{2}(-s))$, we obtain

  $$\operatorname{\iota^{\scriptscriptstyle{\mathrm{CLM}}}}(\widehat{L};[-b,-a])=-\operatorname{\iota^{\scriptscriptstyle{\mathrm{CLM}}}}(L;[a,b]).$$

• •

  (Stratum homotopy relative to the ends) Given a continuous map

  $$L:[a,b]\ni s\rightarrow L(s)\in\mathscr{P}([a,b];\mathbb{R}^{2n})\text{ where }L(s)(t)\coloneqq(L_{1}(s,t),L_{2}(s,t))$$

  such that $\dim L_{1}(s,a)\cap L_{2}(s,a)$ and $\dim L_{1}(s,b)\cap L_{2}(s,b)$ are both constant, and then,

  $$\operatorname{\iota^{\scriptscriptstyle{\mathrm{CLM}}}}(L(0);[a,b])=\operatorname{\iota^{\scriptscriptstyle{\mathrm{CLM}}}}(L(1);[a,b]).$$

Moreover, one efficient way to study the Maslov index is via the crossing form introduced by [17] as follows.

Let $L(t):[0,1]\to\mathrm{Lag}(n)$ be a smooth curve with $L(0)=L_{0}$. Let $W$ be a fixed Lagrangian complement of $L(t)$. For $v\in L_{0}$ and small $t$, define $w(t)\in W$ by $v+w(t)\in L(t)$. The form $Q(v)=\left.\frac{\operatorname{d}}{\operatorname{d}t}\right|_{t=0}\omega(v,w(t))$ is independent of the choice of $W$[17]. A crossing for $L(t)$ is some $t$ for which $L(t)$ intersects $V\in\mathrm{Lag}(n)$ nontrivially. At each crossing, the crossing form is defined as

A crossing is called regular if the crossing form is nondegenerate. For a quadratic form $Q$, we use the notation $\text{sign}(Q)$ for its signature. We also write $m^{+}(Q)$ and $m^{-}(Q)$ for the positive and negative indices of inertia of $Q$, so that

From [20], if the path $L(t)$ has only regular crossing with respect to $V$, then we have that

We recall the definition of Maslov index for a traveling pulse wave $w^{*}$ of (1.1) defined in [9]. The only requirement is that $\tau_{0}$ is large enough so that

Moreover, since the transformation used in the conversion of (1.4) to (1.5) is different from that in [9], then the crossing form defined in the previous paper [9] differs from ours by a negative sign, and so, we provide [9, Definition 3.6] in the following form:

By using the Maslov index, this paper [9] provides a unified geometric treatment for the stability analysis of the traveling pulses solution and has been successfully employed in [11, 10] to obtain some elegant stability results. Motivated by [9], it is natural to consider the case of the traveling front solutions. We remark that Definition 1.4 is dependent on the condition (1.12). It is easy to check that this condition may be inefficient for the traveling front solutions, and we sidestep this difficulty by following [13, Definition 1.2] and define the Maslov index for the traveling waves as following.

Lemmas 2.10 and 2.5 show that the set of nonnegative, real eigenvalues of $L$ is bounded above, and then, the spectrum of $L$ in $\overline{\mathbb{C}^{+}}$ consists of isolated eigenvalues of finite multiplicity (Cf. page 172 of [2]), and so, it follows that the quantities

• •

  $N_{+}(L)\coloneqq\text{the number of real, positive eigenvalues of $L$ counting algebraic multiplicity}$

• •

  $\overline{N}_{+}(L)\coloneqq\text{the number of real, nonnegative eigenvalues of $L$ counting algebraic multiplicity}$

are both well-defined.

Now, we introduce a symplectic invariant called triple index (Cf. Definition 4.3) which can be used to compute the Maslov index, and we denote by $\operatorname{\iota}(L_{1},L_{2};L_{3})$ the triple index for any $L_{1},L_{2},L_{3}\in\mathrm{Lag}(n)$.

Now, we present the central result of this paper.

For the following FitzHugh-Nagumo equations

where $d,\ \gamma>0$ and $f(u)=u(1-u)(u-a)$ with $0<a<\frac{1}{2}$. If $\gamma$ is large enough, then the equation $u=\gamma f(u)$ has three solutions $0=u_{1}<u_{2}<u_{3}$, and we remark that it is easy to check that $f^{\prime}(0)<0$ and $f^{\prime}(u_{3})<0$. Now, we consider a traveling front solution $w^{*}$ of the FitzHugh-Nagumo equation (1.16): its existence has been established in [6], and we remark that such waves are obtained as local minimizers of an energy functional, and based on the variational characterization, authors [5] utilize the spectral flow to define and calculate a stability index. As a supplement, we provide a geometric insight into the stability of the traveling front solution for the FitzHugh-Nagumo equation (1.16).

Making a comparison with (1.1), then, (1.16) can be expressed as

where $D=\begin{bmatrix}\frac{1}{d}&0\\ 0&1\end{bmatrix}$, $Q=\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}$ and $\nabla F(w)=\begin{bmatrix}f(u)-v\\ \gamma v-u\end{bmatrix}.$ The eigenvalue problem

or its equivalent eigenvalue problem

is studied to determine the stability of $w^{*}$, where $L=\frac{\operatorname{d}^{2}}{\operatorname{d}\xi^{2}}+c\frac{\operatorname{d}}{\operatorname{d}\xi}+QB(\xi)$ and $B(\xi)=D^{\frac{1}{2}}\nabla^{2}F(w^{*})D^{\frac{1}{2}}$.

As is commonly done for the Sturm-Liouville operators [7, 14], in this section, by applying [13, Theorem 1], we derive an index formula (Cf. Proposition 2.12) for the Hamiltonian system (2.4), which plays a crucial role in obtaining the spectral information of $L$.

The notation of spectral flow was introduced by Atiyah, Patodi and Singer in their study of index theory on manifolds with a boundary [3]. For the reader’s convenience, we provide a brief description of the basic properties of spectral flow. Suppose that $E$ is a real separable Hilbert space, and denote by $\mathscr{CF}^{sa}(E)$ the space of all closed self-adjoint and Fredholm operators equipped with the gap topology. Let $A:[0,1]\rightarrow\mathscr{CF}^{sa}\left(E\right)$ be a continuous curve. The $\mathrm{Sf\,}(A_{t};t\in[0,1)]$ counts the algebraic multiplicities of the spectral flow of $A_{t}$ across the line $t=-\epsilon$ with some small positive number $\epsilon$.

For each $A_{t}$, there is an orthogonal splitting

where $E_{0}$ is the null space of $A_{t}$, $\langle A_{t}v,v\rangle\geqslant 0$ if $v\in E_{+}$ and $\langle A_{t}v,v\rangle\leqslant 0$ if $v\in E_{-}$. There is an efficient way to compute the spectral flow through what are called crossing forms. Let $P_{t}$ be the orthogonal projector from $E$ to $E_{0}\big{(}A_{t}\big{)}$. When $E_{0}\big{(}A_{t_{0}}\big{)}\neq\{0\}$, we call the instant $t_{0}$ a crossing instant. In this case, we defined the crossing from $\mathrm{Cr}[A_{t_{0}}]$ as

We call the crossing instant $t_{0}$ regular if the crossing form $\mathrm{Cr}(A_{t_{0}})$ is nondegenerate. In this case we define the signature simply as

We assume that all of the crossings are regular. Then, the crossing instants are isolated (and, hence, those on a compact interval are finite in number), and the spectral flow is given by the following formula

where $\mathcal{S}_{*}\coloneqq\mathcal{S}\cap(a,b)$, and $\mathcal{S}$ denotes the set of all crossings.

We list some basic properties of spectral flow:

Now, we return to the problem of stability analysis. If there is not a Hamiltonian structure for $L$ by the presence of the $\frac{\operatorname{d}}{\operatorname{d}\xi}$ term, we can circumvent this by considering instead the operator

as is done in [12, 9]. We now proceed to the study of the following eigenvalue problem

Setting $\psi=\dot{\varphi}-\frac{1}{2}c\varphi$ and $z=\begin{bmatrix}\psi\\ \varphi\end{bmatrix}$ converts Equation (2.3) to the following Hamiltonian system

where $H_{\lambda}=-JA_{\lambda}-\frac{1}{2}cJ=\begin{bmatrix}Q&\frac{1}{2}cQ\\ \frac{1}{2}cQ&B-\lambda Q\end{bmatrix}.$ We define the self-adjoint operators

where $\operatorname{dom}{F}_{\lambda}=W^{1,2}(\mathbb{R},\mathbb{R}^{2n})$ and $\lambda\in[0,C]$. Here, $C$ is given in (1.3). The following Proposition 2.2 shows that the Hamiltonian system (2.4) has close contact with the system (1.5).

It is well-known [16, Lemma 3.1.10] that the essential spectrum is given by

Following the same method as that in paper [11, Section 2], we here give the proof of the following Lemma for convenience of the reader.