## 3 Discussions and Results
### TW Profiles  
TW solutions are obtained as \(N(x,t)=\varphi_{1}(\xi)\,\ \xi=x\cdot d-\lambda t\in\mathrm{R}\), where \(d\) is a unitary vector in the TW direction, \(\lambda\) corresponds to the TW-speed and \(\varphi_{1}\in L^{\infty}(\mathrm{R}^{3})\cap L^{1}_{loc}(\mathrm{R}^{3})\) is the TW profile. Note that TW profiles are equivalent under translation \(\xi\to\xi+\xi_{1}\) and symmetry \(\xi\to-\xi\). We admit the vector in the TW propagation direction is expressed as \(d=(1,0,0)\in\mathbb{R}^{3}\) and the TW moves from \(-\infty\) to \(\infty\). Hence:  
\[N(x,t)=\varphi_{1}(\xi),\ \ \ \Theta(x,t)=\varphi_{2}(\xi),\ \ \xi=x-\lambda t\,. \tag{18}\]  
After replacement and standard operations, the problem (15) reads:  
\[ \begin{array}{c}N_{t}=\varphi_{1}^{\prime\prime}+a\varphi_{1}^{\prime}- \varphi_{2}^{n}(\varphi_{1}-r)=\varphi_{1}^{\prime\prime}+a\varphi_{1}^{ \prime}+\varphi_{2}^{n}(r-\varphi_{1}),\\ \Theta_{t}=\varphi_{2}^{\prime\prime}+a\varphi_{2}^{\prime\prime}-\varphi_{1} ^{n}\varphi_{2},\\ \varphi_{1}^{0}=\varphi_{2}^{0}=H(-x)\,,\\ N_{t}=-\lambda\varphi_{1}^{\prime};\ \ \Theta_{t}=-\lambda\varphi_{2}^{\prime}. \end{array}  \tag{19}\]  
Note that the TW profile \(\varphi_{2}\) asymptotically tends towards zero whilst keeping the positivity condition. In addition, \(\varphi_{1}\) increases to \(r>H(-x)\), so that \((r-\varphi_{1})>0\). These particular behaviours will be shown afterwards, but shall be taken into account during the coming assessments. A linearisation of (19) close to the equilibrium condition \((\varphi_{1}=r,\varphi_{2}=0)\) is difficult in general due to the non-Lipschitz character of \(\varphi_{2}^{n}\). Note that close to the critical point \(\varphi_{2}=0\), \(\varphi_{2}^{n}>\varphi_{2}\), hence, the following alternative problem is formulated: (19):  
\[ \begin{array}{c}\varphi_{1}^{\prime\prime}+(\lambda+a)\varphi_{1}^{\prime}+ \varphi_{2}(r-\varphi_{1})=0,\\ \varphi_{2}^{\prime\prime}+(\lambda+a)\varphi_{2}^{\prime}-r^{m}\varphi_{2}= 0,\\ \varphi_{1}^{0}=\varphi_{2}^{0}=H(-x).\end{array}  \tag{20}\]  
TW profiles resulting from (20) are lower solutions to (19). This statement will be shown afterwards, but it is relevant to ensure the positivity of any TW profile. Indeed, any positive solution to (20) will provide evidence to ensure the positivity of any upper solution in (19). Considering \(r-\varphi_{1}=\phi_{1}\), then:  
\[ \begin{array}{c}-\varphi_{1}^{\prime\prime}-(\lambda+a)\varphi_{1}^{\prime }+\varphi_{2}\phi_{1}=0,\\ \varphi_{2}^{\prime\prime}+(\lambda+a)\varphi_{2}^{\prime}-r^{m}\varphi_{2}= 0,\\ \hat{\varphi}_{1}^{0}=r-H(-x),\\ \varphi_{2}^{0}=H(-x).\end{array}  \tag{21}\]  
With the boundary conditions expressing the asymptotic behaviour of solutions:  
\[ \begin{array}{c}\phi_{1}\to 0,\ \ \varphi_{2}\to 0,\ \ \ \xi>>1,\\ \phi_{1}\to r-1,\ \ \ \varphi_{2}\to 1\ r>1,\ \ \xi>>1.\end{array}  \tag{22}\]  
The fundamental TW problem consist of finding a positive and monotone profile for a certain interval of TW speeds. Particularly, solutions are expressed in the proximity of the equilibrium points \(\varphi_{1}=0\) and \(\varphi_{2}=0\). When solutions approach the boundary conditions in (22), a linearisation approach can be followed, so that solutions can be exponentially expressed as  
\[\phi_{1}=De^{-\gamma_{1}\xi},\ \ \ \varphi_{2}=Ee^{-\gamma_{2}\xi}. \tag{23}\]  
Then, (21) reads:  
\[ \begin{array}{c}-D\gamma_{1}^{2}e^{-\gamma_{1}\xi}+(\lambda+a)\gamma_{1}De^ {-\gamma_{1}\xi}+DEe^{-(\gamma_{1}+\gamma_{2})\xi}=0,\\ E\gamma_{2}^{2}e^{-\gamma_{2}\xi}-(\lambda+a)\gamma_{2}Ee^{-\gamma_{2}\xi}-E ^{m}e^{-\gamma_{2}\xi}=0.\end{array}