## 3 Discussions and Results
### TW Profiles
 \tag{24}\]  
From the second identity, it is possible to find an expression to relate the given speeds (TW speed and advection terms) with the exponential factor:  
\[\gamma_{2}=\frac{\lambda+a+\sqrt{(\lambda+a)^{2}+4r^{m}}}{2}. \tag{25}\]  
Similarly, for \(\gamma_{1}\):  
\[\gamma_{1}=\frac{\lambda+a+\sqrt{(\lambda+a)^{2}+4 Ee^{-\gamma_{2}\xi}}}{2}. \tag{26}\]Note that for \(\xi>>1\), the term \(4 Ee^{-\gamma_{2}\xi}\to 0^{+}\), then:  
\[\gamma_{1}=\frac{\lambda+a+\sqrt{(\lambda+a)^{2}+4E\omega}}{2},\ \ \omega\to 0^{+}. \tag{27}\]  
Observe that the results for \(\gamma_{1}\) and \(\gamma_{2}\) establish that the exponential decaying terms are bigger than the advection and TW-speed terms \((\lambda+a)\). Hence, any TW profiles evolves towards the asymptotic equilibrium and this evolution is convergent, independently of the advection and TW speed terms. Note that the existence, uniqueness and comparison of solutions to problems (19) and (20) follow a standard procedure, as proposed in [21] (Chapter 7). Along the following analysis, a numerical exercise is considered to provide particular forms of TW solutions to problem \(P_{T}\) (19). Furthermore, solutions to (20), close to the critical points \((\varphi_{1}=r,\varphi_{2}=0)\), are effectively shown to be lower solutions to (19). As solutions to problem (20) are lower compared with solutions to (19), a monotone (increasing or decreasing) TW profile and tip exist. The numerical exercise has been pursued with the following features:  
* The methodology is based on the Matlab function bvp4c which provides an implicit Runge-Kutta approach with an interpolant extension [22]. The collocation method requires the specification of boundary conditions, in this case, given by the stationary solutions at \(-\infty\) and \(+\infty\);
* The integration interval is \((0,100)\), large enough to study the TW evolution in their domain, and not impacted by the boundary conditions that act as tractors;
* The considered error for computation is \(10^{-6}\);
* The integration domain has been divided into 100,000 nodes;
* To make the numerical assessment tractable and without loss of generality, particular values were taken for the involved parameters: \(r=2\), \((a+\lambda)=4\)--and different values of \(n\) and \(m\). Results are represented in Figures 1-5 with the following remarks:  
* Case with \(n=0.9\) and \(m=0.1\)--according to the results in Figures 1-3, TW profiles to (20) are lower solutions compared to those for (19). Note that \(\varphi_{2m}\) (as solution to (20)) is positive along the domain. Then, any TW moving with the exponential decaying term \(\gamma_{2}\) as per (25) is positive in the whole domain. In addition, we note that the solutions to (19) are very close to solutions to the linearised (20), which permits the validation of the goodness of the linearised exercise;
* Similarly, the same exercise is repeated for \(n=0.2\) and \(m=0.9\). TW profiles, as solutions to (20), are lower compared with solutions to (19) (refer to Figures 4 and 5). In the same way, note that solutions to (19) evolve close to solutions for the linearised (20). Figure 1: TW profiles for \(a+\lambda=4\) with \(n=0.9\), \(m=0.1\). For the sake of simplicity, \(r=2\). The numerical exercise was performed for an integration interval \((0,100)\); nonetheless, the interval was cut for representation purposes. Note that \(f_{1}\) and \(f_{2}\) represent \(\varphi_{1}\) and \(\varphi_{2}\) solutions to (19) while \(f_{1m}\) and \(f_{2m}\) are solutions to (20). Figure 2: TW profiles for \(a+\lambda=4\) with \(n=0.9\), \(m=0.1\). For the sake of simplicity, \(r=2\). Note the lower solution \(\varphi_{1m}\) as the solution for the problem (20). The numerical exercise was performed for an integration interval \((0,100)\); nonetheless, the interval was cut for representation purposes. Note that \(f_{1}\) and \(f_{2}\) represent \(\varphi_{1}\) and \(\varphi_{2}\) solutions to (19) while \(f_{1m}\) and \(f_{2m}\) are solutions to (20). Figure 4: TW profiles for \(a+\lambda=4\) with \(n=0.2\), \(m=0.9\).