## 4 A Direct Approach for Analyzing the Stability of Traveling Wave Solutions  
Here, in this section, we present an approach for solving the Sturm-Liouville problem arising from a stability analysis for traveling waves. The idea behind this approach is inspired by some results for the Legendre functions and the Legendre polynomials (see also the recent investigations [23; 24] involving applications of the substantially more general Jacobi polynomials). The Legendre operator is given by  
\[\hat{\mathcal{L}}=\frac{d}{dx}(1-x^{2})\frac{d}{dx}=(1-x^{2})\frac{d^{2}}{dx^{ 2}}-2x\frac{d}{dx}.\]  
We consider the following eigenvalue problem:  
\[\hat{\mathcal{L}}u=\alpha u \tag{37}\]  
or, equivalently,  
\[(1-x^{2})u^{{}^{\prime\prime}}-2 xu^{{}^{\prime}}=\alpha u. \tag{38}\]  
This equation has regular singular points at \(x=\pm 1\). However, if  
\[\alpha=-n(n+1)\ \ (n=0,1,\cdots),\]  
then the solution of the Equation (37) or (38) is finite as \(x\to\pm 1\). In this case, the solution of (32) is known as a Legendre polynomial \(P_{n}(x)\). We also note that \(\alpha=0\) is an eigenvalue of the problem (38). We now consider the associated Legendre operator  
\[\hat{\mathcal{L}}_{a}=(1-x^{2})\frac{d}{dx}(1-x^{2})\frac{d}{dx}\]  
and consider the following eigenvalue problem:  
\[\hat{\mathcal{L}}_{a}u=\alpha u. \tag{39}\]  
We note that \(x=\pm 1\) are regular singular points of (39). The solution of (39) is finite as \(x\to\pm 1\) if \(\alpha=0\), so that \(\alpha=0\) is an eigenvalue of (39). In order to find the relationship between the stability analysis and the results for the Legendre functions, we consider the general example given by the  
Figure 2: Numerical evaluation of the Evans function \(E(\alpha)\) for the Equation (35). Equation (19). Thus, for instance, we assume that \(a<0\) and \(b>0\), so that the solution \(u_{0}(x)\) of the second equation in (19) is given by  
\[u_{0}(x)=\alpha[\text{sech}(kx)]^{\frac{2}{\alpha-1}}\qquad(n>1), \tag{40}\]  
where  
\[k^{2}=-\frac{a(n-1)^{2}}{4}\qquad\text{and}\qquad\alpha=\left(\frac{2k^{2}(n+1 )}{b(n-1)^{2}}\right)^{\frac{1}{n-1}}.\]  
Now, the perturbed equation for (19) is given by  
\[U^{\prime\prime}-(\alpha+|a|)U+\frac{|a|n(n+1)}{2}\text{ sech}^{2}(kx)U=0. \tag{41}\]  
By using the transformation \(\zeta=\tanh(kx)\), (41) becomes  
\[\hat{\mathcal{L}}u=\alpha u \tag{37}\]0  
\[\hat{\mathcal{L}}u=\alpha u \tag{37}\]1  
Equation (42) has two regular singular points at \(\zeta=\pm 1\). In the original variable \(x\), they correspond to \(x\to\pm\in\) and are regular singular points of (19). Indeed, (42) can be written in the form:  
\[\hat{\mathcal{L}}u=\alpha u \tag{37}\]2  
Equation (43) has the solution given by  
\[\hat{\mathcal{L}}u=\alpha u \tag{37}\]3  
In the Equation (44), the finite solutions require that \(\beta=\frac{n+1}{n-1}\) and \(\delta\) are positive integers. We search now for solutions of (44) which satisfy the following limit relationship:  
\[\hat{\mathcal{L}}u=\alpha u \tag{37}\]4  
This condition determines the solution of the eigenvalue problem (41). We remark that this solution does not depend on the coefficient of the nonlinear term in (19), namely, \(b\). If \(\beta=\frac{n+1}{n-1}\) is a positive integer, then we find from the properties of associated Legendre polynomials that  
\[\hat{\mathcal{L}}u=\alpha u \tag{37}\]5  
or  
\[\hat{\mathcal{L}}u=\alpha u \tag{37}\]6  
The finite solutions of (42) as \(\zeta\to\pm 1\) (or \(x\to\pm\in\)) are given by  
\[\hat{\mathcal{L}}u=\alpha u \tag{37}\]7  
where \(\delta\) and \(\beta\) are integers and \(a_{0}\) is a constant. The required values of \(\alpha\) are determined by the following equation:  
\[\hat{\mathcal{L}}u=\alpha u \tag{37}\]8Among the values of \(\alpha\) which satisfy (45), we select the values which satisfy the eigenvalue condition. To make these results clear, we consider the following special cases:  
1. If \(n=3\), \(a=-1\) and \(b=2\), the first equation in (19) becomes the Newell-Whitehead equation. Then, from (45), we obtain \(\alpha=-1,0,3\).