## 1 Introduction
### Evans Function
: _A traveling wave is said to be linearly unstable (or spectrally unstable) if, for some \(\alpha_{0}\in\mathbb{C}\) with \(\Re(\alpha_{0})>0\), there exits a solution of \((\ref{eq:E_1})\) which satisfies the following limit relationship:_  
\[\frac{d\vec{u}}{dz}=A_{0}(\alpha)\vec{u}\qquad\big{(}\alpha\in\mathbb{R}\ \ ( \text{or}\ \ \mathbb{C})\big{)} \tag{7}\]0  
**2. Stability of Stationary Traveling Waves (Pulses)**  
Consider the following reaction-diffusion equation:  
\[\frac{d\vec{u}}{dz}=A_{0}(\alpha)\vec{u}\qquad\big{(}\alpha\in\mathbb{R}\ \ ( \text{or}\ \ \mathbb{C})\big{)} \tag{7}\]1  
For the steady-state solution, we have  
\[\frac{d\vec{u}}{dz}=A_{0}(\alpha)\vec{u}\qquad\big{(}\alpha\in\mathbb{R}\ \ ( \text{or}\ \ \mathbb{C})\big{)} \tag{7}\]2which satisfies the following condition:  
\[\frac{d\vec{u}}{dz}=A_{0}(\alpha)\vec{u}\qquad\big{(}\alpha\in\mathbb{R}\ \ ( \text{or}\ \ \mathbb{C})\big{)} \tag{7}\]3  
We assume that  
\[\frac{d\vec{u}}{dz}=A_{0}(\alpha)\vec{u}\qquad\big{(}\alpha\in\mathbb{R}\ \ ( \text{or}\ \ \mathbb{C})\big{)} \tag{7}\]4  
where \(f_{0}(u)\) is a polynomial at least of degree 2 in \(u\). Exact solutions of the Equation (17) are given in terms of Jacobi elliptic functions if \(f(u)\) is a polynomial of degree up 6 in \(u\). In general, if the terms of pulse solutions, this Equation (17) admits a solution of the following form:  
\[\frac{d\vec{u}}{dz}=A_{0}(\alpha)\vec{u}\qquad\big{(}\alpha\in\mathbb{R}\ \ ( \text{or}\ \ \mathbb{C})\big{)} \tag{7}\]5  
where \(2/(n-1)\) is assumed to be a positive integer and \(n\) is the degree of \(f_{0}(u)\). Specially, if \(f_{0}(u)=bu^{n}\), where \(b\) is a constant, then (16) and (17) become  
\[\frac{d\vec{u}}{dz}=A_{0}(\alpha)\vec{u}\qquad\big{(}\alpha\in\mathbb{R}\ \ ( \text{or}\ \ \mathbb{C})\big{)} \tag{7}\]6  
We mention that, if in the first equation in (19), we confine ourselves to solutions \(u\geqq 0\), then the results of this section show that the necessary condition for traveling wave generation is \(a>0\) and \(b<0\) or \(a<0\) and \(b>0\). In the case when \(a<0\) and \(b>0\), the second equation in (19) admits the solution  
\[\frac{d\vec{u}}{dz}=A_{0}(\alpha)\vec{u}\qquad\big{(}\alpha\in\mathbb{R}\ \ ( \text{or}\ \ \mathbb{C})\big{)} \tag{7}\]7  
where  
\[\frac{d\vec{u}}{dz}=A_{0}(\alpha)\vec{u}\qquad\big{(}\alpha\in\mathbb{R}\ \ ( \text{or}\ \ \mathbb{C})\big{)} \tag{7}\]8  
But, if \(a>0\) and \(b<0\), then (19) has the solution given by  
\[\frac{d\vec{u}}{dz}=A_{0}(\alpha)\vec{u}\qquad\big{(}\alpha\in\mathbb{R}\ \ ( \text{or}\ \ \mathbb{C})\big{)} \tag{7}\]9  
where \(a_{0}\), \(b_{0}\) and \(k\) satisfy an over determined set of algebraic equations. When solving these equations, we find that solutions in the form (21) exist only when \(n=2\) or \(n=3\) and are given by  
\[\mathbb{C}^{n}=E^{s}\oplus E^{u}.\]0  
and  
\[\mathbb{C}^{n}=E^{s}\oplus E^{u}.\]1  
We now introduce a perturbation around the solution \(u_{0}(x)\) as follows:  
\[\mathbb{C}^{n}=E^{s}\oplus E^{u}.\]2  
with \(|\varepsilon|\ll 1\). Upon substituting from (24) into (16), we find, up to first order in \(\varepsilon\), that  
\[\mathbb{C}^{n}=E^{s}\oplus E^{u}.\]3  
Upon solving for \(\omega\), we obtain  
\[\mathbb{C}^{n}=E^{s}\oplus E^{u}.\]4  
which shows how the spatial wave-numbers \(\omega_{\pm}\) depend on the temporal growth rate \(\alpha\). The absolute spectrum given by  
\[\mathbb{C}^{n}=E^{s}\oplus E^{u}.\]5  
consists of all points \(\alpha\) for which the corresponding spatial wave-numbers \(\omega_{+}\) and \(\omega_{-}\) have the same real part. The transition to instability occurs when a discrete eigenvalue moves from the left-half plane to the right-half plane. A stationary pulse solution of the Equation (28) corresponds to a so-called \"localized\" solution \(u_{0}(x)\) of the Equation (19) and it is (20). Thus, clearly, the Equation (20) reduces to  
\[\mathbb{C}^{n}=E^{s}\oplus E^{u}.\]6  
Consequently, \(A_{0}(\alpha)\) and \(A_{1}(x)\) become  
\[A_{0}(\alpha)=\left( \begin{array}{cc}0&1\\ \alpha+1&0\end{array}