## 1 Introduction
### Evans Function  
The Evans function is an important tool for studying and investigating the stability of nonlinear waves (see, for example, [3,4]). The Evans function is an analytic function whose zeros correspond exactly, in location and multiplicity, to the eigenvalues of the linearized operator. The Evans function was first formulated by [PERSON] for a specific class of systems and is a generalization which is suited for systems of partial differential equations of the transmission coefficient from quantum mechanics (see, for details, [5, 6, 7]). [PERSON] paid remarkable attention toward studying the stability of nerve impulses, which he then classified as the category of nerve impulse equations. This class has an important property that leads naturally to the formulation of blue the Evans function in a clear and straightforward manner. The \(D(\alpha)\) notation was used by [PERSON] to refer to the determinant which, in fact, played the same role as the determinant of an eigenvalue matrix in problems of finite dimensions. [PERSON] in [8] applied the stability of the traveling pulse (nerve impulse) of the Fitzhugh-Nagumo system by following [PERSON] idea. In fact, [PERSON] gave the name _\"Evans function\"_ as well as the notation \(E(\alpha)\) which is now in common usage. The authors in [9] presented the first general definition of the Evans function \(E(\alpha)\), which is based upon the idea of [PERSON], with its placement in a new conceptual form to clearly give a general definition. We consider the following eigenvalue problem:  
\[\frac{d\vec{u}}{dz}=A(z;\alpha)\vec{u}\qquad\big{(}\vec{u}\in\mathbb{R}^{n}\ \ ( \text{or}\ \ \mathbb{C}^{n});\ -\ \in<z<\in\ \big{)} \tag{5}\]  
and assume that (5) is rewritten in the form given by  
\[\frac{d\vec{u}}{dz}=\big{(}A_{0}(\alpha)+A_{1}(z)\big{)}\vec{u}. \tag{6}\]  
Firstly, we consider the following equation:  
\[\frac{d\vec{u}}{dz}=A_{0}(\alpha)\vec{u}\qquad\big{(}\alpha\in\mathbb{R}\ \ ( \text{or}\ \ \mathbb{C})\big{)} \tag{7}\]  
and assume that the eigenvalues of \(A_{0}(\alpha)\) are \(\beta_{1}(\alpha),\beta_{2}(\alpha),\cdots,\beta_{n}(\alpha)\) and that the corresponding eigenvectors are \(\gamma_{1}(\alpha),\gamma_{2}(\alpha),\cdots,\gamma_{n}(\alpha).\) Whenever \(\Re(\beta_{j})\
eq 0\) and the \(\beta_{j}\) are distinct for \(j=1,\cdots,n\), the space \(\mathbb{C}^{n}\) is represented as follows:  
\[\mathbb{C}^{n}=E^{s}\oplus E^{u}.\]  
We assume that  
\[\Re(\beta_{j})<0\qquad(j=1,\cdots,k)\]  
and that  
\[\Re(\beta_{j})>0\qquad(j=k+1,\cdots,n),\]while  
\[E^{s}=\{\vec{u}_{j}(\alpha)\quad\text{and}\quad\Re\big{(}\beta_{j}(\alpha)\big{)} <0\quad(j=1,\cdots,k)\}\]  
and  
\[E^{u}=\{\vec{u}_{j}(\alpha)\quad\text{and}\quad\Re\big{(}\beta_{j}(\alpha)\big{)} >0\quad(j=k+1,\cdots,n)\}.\]  
Solutions belonging to \(E^{s}\) are bounded as \(z\to\in\), while the solutions belonging to \(E^{u}\) are bounded as \(z\to-\in\). We may then label these solutions as \(u_{j}^{+}(\alpha)\) or \(u_{j}^{-}(\alpha)\), according to whether they are bounded as \(z\to\in\) or \(z\to-\in\). Also, these solutions satisfy the following limit relationship:  
\[\lim_{z\to\pm\in}\vec{u}_{j}^{\pm}(z,\alpha)e^{\beta^{\mp}(\alpha)z}=\gamma^{ \mp}. \tag{8}\]  
We mention that there exist a set of values \(\alpha_{0}\) and a common subspace of \(E^{s}\) and \(E^{u}\) such that  
\[\lim_{z\to\pm\in}\vec{u}(z,\alpha_{0})=0.\]  
We now give the following definition. Definition 1: \(\alpha=\alpha_{0}\in\mathbb{C}\) is an eigenvalue of (5) if the following condition holds true:  
\[\frac{d\vec{u}}{dz}=\big{(}A_{0}(\alpha)+A_{1}(z)\big{)}\vec{u}. \tag{6}\]0  
We mention here that the values of \(\alpha\), which satisfy the equation \(\Re\big{(}\beta_{j}(\alpha)\big{)}=0\) determines the essential spectrum of the Equation (5). The spectrum of the Equation (5) generally consists of the pure point spectrum, isolated eigenvalues of finite multiplicity, and the essential spectrum. The essential spectrum is contained within the parabolic curves of the continuous spectrum (see [10]).