## 3 Results
#### 3.1.2 The Transition Layer Functions
From the equalities in Equation (20), algebraic equations are obtained, from which we define the functions \(Q_{k}^{(\mp)}u(\tau,\mathbf{y})\): \[\left(\varepsilon^{2}\frac{\partial^{2}}{\partial\tau^{2}}- \varepsilon^{3}k(\mathbf{y})\frac{\partial}{\partial\tau}+\sum_{i\geq 0} \varepsilon^{i+4}L_{i}(\tau,\mathbf{y})\right)\left[\sum_{i=0}^{n}\varepsilon^{i} Q_{i}^{(\mp)}u(\tau,\mathbf{y})\ \right] =Q^{(\mp)}f(\tau,\mathbf{y},\varepsilon), \tag{20}\]1 where \(Q_{k}^{(\mp)}f(\tau,\mathbf{y})\) are known functions decreasing exponentially to zero as \(\tau\to\mp\infty\); in the first order they are as follows: \[\left(\varepsilon^{2}\frac{\partial^{2}}{\partial\tau^{2}}- \varepsilon^{3}k(\mathbf{y})\frac{\partial}{\partial\tau}+\sum_{i\geq 0} \varepsilon^{i+4}L_{i}(\tau,\mathbf{y})\right)\left[\sum_{i=0}^{n}\varepsilon^{i} Q_{i}^{(\mp)}u(\tau,\mathbf{y})\ \right] =Q^{(\mp)}f(\tau,\mathbf{y},\varepsilon), \tag{20}\]2 \[\left(\varepsilon^{2}\frac{\partial^{2}}{\partial\tau^{2}}- \varepsilon^{3}k(\mathbf{y})\frac{\partial}{\partial\tau}+\sum_{i\geq 0} \varepsilon^{i+4}L_{i}(\tau,\mathbf{y})\right)\left[\sum_{i=0}^{n}\varepsilon^{i} Q_{i}^{(\mp)}u(\tau,\mathbf{y})\ \right] =Q^{(\mp)}f(\tau,\mathbf{y},\varepsilon), \tag{20}\]3 Here, we have taken the identities \(f^{(\mp)}(\varphi(v(\mathbf{x}),\mathbf{x}),v(\mathbf{x}),\mathbf{x},0)\equiv 0\) into account and the first part of Equation (19) for \(k=1\). 3. From the equalities in Equation (21) with known expressions for \(Q_{k}^{(\mp)}u\), we obtain equations for the functions \(Q_{k}^{(\mp)}v(\tau,\mathbf{y})\). We solve them on the half-lines \(\tau<0\) and \(\tau>0\) respectively, with the supplementary conditions at infinity and at \(\tau=0\): \[\left(\varepsilon^{2}\frac{\partial^{2}}{\partial\tau^{2}}- \varepsilon^{3}k(\mathbf{y})\frac{\partial}{\partial\tau}+\sum_{i\geq 0} \varepsilon^{i+4}L_{i}(\tau,\mathbf{y})\right)\left[\sum_{i=0}^{n}\varepsilon^{i} Q_{i}^{(\mp)}u(\tau,\mathbf{y})\ \right] =Q^{(\mp)}f(\tau,\mathbf{y},\varepsilon), \tag{20}\]4 \[\left(\varepsilon^{2}\frac{\partial^{2}}{\partial\tau^{2}}- \varepsilon^{3}k(\mathbf{y})\frac{\partial}{\partial\tau}+\sum_{i\geq 0} \varepsilon^{i+4}L_{i}(\tau,\mathbf{y})\right)\left[\sum_{i=0}^{n}\varepsilon^{i} Q_{i}^{(\mp)}u(\tau,\mathbf{y})\ \right] =Q^{(\mp)}f(\tau,\mathbf{y},\varepsilon), \tag{20}\]5 The functions \(Q_{k}^{(\mp)}g(\tau,\mathbf{y})\) are known at each step; in the first order they are expressed as \[\left(\varepsilon^{2}\frac{\partial^{2}}{\partial\tau^{2}}- \varepsilon^{3}k(\mathbf{y})\frac{\partial}{\partial\tau}+\sum_{i\geq 0} \varepsilon^{i+4}L_{i}(\tau,\mathbf{y})\right)\left[\sum_{i=0}^{n}\varepsilon^{i} Q_{i}^{(\mp)}u(\tau,\mathbf{y})\ \right] =Q^{(\mp)}f(\tau,\mathbf{y},\varepsilon), \tag{20}\]6 \[\left(\varepsilon^{2}\frac{\partial^{2}}{\partial\tau^{2}}- \varepsilon^{3}k(\mathbf{y})\frac{\partial}{\partial\tau}+\sum_{i\geq 0} \varepsilon^{i+4}L_{i}(\tau,\mathbf{y})\right)\left[\sum_{i=0}^{n}\varepsilon^{i} Q_{i}^{(\mp)}u(\tau,\mathbf{y})\ \right] =Q^{(\mp)}f(\tau,\mathbf{y},\varepsilon), \tag{20}\]7Here, the identities \(g^{(\mp)}(\varphi(\psi^{(\mp)}(\mathbf{x}),\mathbf{x}),\psi^{(\mp)}(\mathbf{x}), \mathbf{x},0)\equiv 0\) and the second part of Equation (19) for \(k=1\) are taken into account. The functions \(Q_{k}^{(\mp)}g(\tau,\mathbf{y})\), \(k=0,1,\ldots\) decrease exponentially to zero as \(\tau\to\mp\infty\). 4. We determine the \(M_{k}^{(\mp)}u(\sigma,\mathbf{y})\) functions. The equations for them are obtained from the equalities in Equation (22).