## 4 A Direct Approach for Analyzing the Stability of Traveling Wave Solutions
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**The Case**} & \multicolumn{2}{c}{**Values of \(\boldsymbol{\alpha}\) Where**} & \multirow{2}{*}{**Eigensolutions**} & \multicolumn{2}{c}{**Eigenvalues Where**} \\  & \(\boldsymbol{\lim_{\gamma\to\pm\in}U(\zeta,\boldsymbol{\alpha})\) **Exists**} & & \(\boldsymbol{\lim_{\zeta\to\pm\in}U(\zeta,\boldsymbol{\alpha})=0}\) \\ \hline \multirow{3}{*}{n = 3} & \(-1\) & \(U_{1}=a_{0}P_{1}^{0}\) & - \\  & \(0\) & \(U_{2}=a_{0}P_{2}^{2}\) & \(0\) \\  & \(0\) & \(U_{3}=a_{0}P_{2}^{2}\) & \(3\) \\ \hline \multirow{3}{*}{n = 2} & \(-1\) & \(U_{1}=a_{0}P_{1}^{0}\) & - \\  & \(-3/4\) & \(U_{2}=a_{0}P_{1}^{1}\) & \(-3/4\) \\  & \(0\) & \(U_{3}=a_{0}P_{2}^{2}\) & \(0\) \\  & \(5/4\) & \(U_{4}=a_{0}P_{3}^{3}\) & \(5/4\) \\ \hline \multirow{3}{*}{n = 3/2} & \(-1\) & \(U_{1}=a_{0}P_{1}^{0}\) & - \\  & \(-15/16\) & \(U_{2}=a_{0}P_{1}^{1}\) & \(-15/16\) \\ \cline{1-1}  & \(-3/4\) & \(U_{3}=a_{0}P_{2}^{2}\) & \(-3/4\) \\ \cline{1-1}  & \(-7/10\) & \(U_{4}=a_{0}P_{3}^{3}\) & - \\ \cline{1-1}  & \(0\) & \(U_{5}=a_{0}P_{2}^{4}\) & \(0\) \\ \cline{1-1}  & \(9/16\) & \(U_{6}=a_{0}P_{5}^{3}\) & \(9/16\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Special cases for the non-eigenvalue and eigenvalue which satisfied the eigenvalue condition and then it is the zeros of the Evans function \(E(\alpha)\). Certainly, when analyzing the Equation (47) by taking  
\[m=2,4,\cdots\qquad\text{and}\qquad 2k<\frac{3n-1}{n-1},\]  
we find the same results as before. So, instead, we can find results from (47) and (48). In view of (47), we have a lower bound for \(\alpha\) and from (47) we obtain the least upper bound for \(\alpha\) (or the dominant value of \(\alpha\)), namely, \(\alpha_{0}\) when \(m=2\). This is true because, for \(m>2\), the values of \(d\) decrease, and then \(\alpha\) decreases. In fact, when \(m=2\), we obtain  
\[\alpha_{0}=|a|\frac{(n+3)(n-1)}{4}\]  
and the following result holds true:  
\[-|a|<\alpha\leqq|a|\frac{(n+3)(n-1)}{4}. \tag{49}\]  
The last result suggests the introduction of the following general localization concepts of eigenvalues of the Sturm-Liouville problem. We now define what is meant by a dominant eigenvalue and a dominant solution. A dominant eigenvalue \(\alpha_{0}\) is the least upper bound of the eigenvalues \(a\leqq\alpha_{0}\). A dominant solution is a solution which corresponds to a dominant eigenvalue \(\alpha_{0}\). In the solution expansion near a regular singular point, we conjecture that a dominant solution is obtained at the lowest-order truncation of the series, namely, at \(m=2\). Consequently, the dominant eigenvalue is determined by the solution of the recursion formulas for \(m\geqq 2\). We return to (47) in order to check that we get the same results found previously. To this end, we reconsider the same examples. For \(n=3\), \(a=-1\) and \(b=2\), we find that (48) gives  
\[m=4-\sqrt{1+\alpha}\geqq 2.\]  
For \(m=2\) and \(m=4\), we have \(\alpha=3\) and \(\alpha=-1\), respectively. For \(n=2\), \(a=-1\) and \(b=1\), (48) gives rise to  
\[m=5-2\sqrt{1+\alpha}\geqq 2.\]  
Thus, we obtain \(\alpha=-\frac{3}{4}\), \(\frac{5}{4}\) for \(m=2,\ 4\), respectively. The corresponding eigensolution can be obtained as well. We remark that we have obtained the same results as above. On the other hand, polynomial solutions found through an expansion near a regular singular point gives rise to values of \(\alpha\
eq 0\). We mention that, when analyzing the stability of pulse solutions of the first equation in (19) when \(a>0\) and \(b<0\), we find that the eigenvalues are non-positive. These solutions are then stable.