## 2 Theoretical Model
### Method of Multiple Scales, MOMS  
We employed the Method of Multiple Scales (MOMS) to analyze nonlinear equations. The MOMS divides the time into two scales: fast time change and slow time change. In this context, the term with fast time change is denoted as \(T_{0}=\tau\), and the term with slow time change is denoted as \(T_{1}=\epsilon^{2}\tau\), with \(\epsilon\) representing the time scale of small disturbance. Assuming \(W(x,\tau,\epsilon)=\epsilon^{1}W_{0}(T_{0}T_{1}T_{2},\ldots\ldots)+\epsilon^{3 }W_{1}(T_{0}T_{1}T_{2},\ldots\ldots)\), Equation (24) can be derived:  
\[\begin{split}(\epsilon^{1}\frac{\partial^{2}W_{0}}{\partial T_{0} }^{2}+\epsilon^{3}\frac{\partial^{2}W_{1}}{\partial T_{0}}^{2}+2\epsilon^{3} \frac{\partial^{2}W_{0}}{\partial T_{0}\partial T_{1}}+2\epsilon^{5}\frac{ \partial^{2}W_{1}}{\partial T_{0}\partial T_{1}}+\ldots\ldots)+(\epsilon W_{0} ^{iv}+\epsilon^{3}W_{1}{}^{iv})\\ +\epsilon^{2}\mu(\epsilon\frac{\partial W_{0}}{\partial T_{0}}+ \epsilon^{3}\frac{\partial W_{1}}{\partial T_{0}}+\epsilon^{3}\frac{\partial W_ {0}}{\partial T_{1}}+\ldots\ldots)+(\epsilon^{3}W_{0}{}^{*3}+2\epsilon^{5}W_{0 }{}^{*2}W_{1}{}^{\prime\prime})\\ +(\epsilon^{3}W_{0}{}^{iv}W_{0}{}^{\prime\prime}+2\epsilon^{5}W_{0 }{}^{\prime\prime}W_{0}{}^{\prime\prime}W_{1}{}^{\prime}+\epsilon^{5}W_{0}{}^{ \prime\prime}W_{0}{}^{\prime\prime}+\dots\ldots)\\ +4(\epsilon^{3}W_{0}{}^{\prime}W_{0}{}^{\prime\prime}W_{0}{}^{ \prime\prime\prime}+\epsilon^{5}W_{0}{}^{\prime}W_{0}{}^{\prime\prime\prime}W_{1} {}^{\prime\prime}+\epsilon^{5}W_{0}{}^{\prime\prime}W_{0}{}^{\prime\prime}W_{1} {}^{\prime\prime}+\epsilon^{5}W_{0}{}^{\prime\prime}W_{0}{}^{\prime\prime}W_{1} {}^{\prime})\\ -\frac{k\eta^{2}}{e^{\pi\xi^{2}}}\int_{b}^{a}\left(\epsilon W_{0} {}^{\prime\prime}+\epsilon^{3}W_{1}{}^{\prime\prime}\right)dx\int_{0}^{\tau} \left[\int_{b}^{a}\left(\frac{\partial^{3}(\epsilon W_{0}{}^{\prime\prime}+ \epsilon^{3}W_{1}{}^{\prime\prime})}{\partial T_{0}\partial x^{2}}\right)dxe ^{R_{B}\tau}\right]d\tau+F_{M}\\ +(\epsilon^{1}\frac{\partial^{2}W_{0}}{\partial T_{0}}^{2}+\epsilon^{3} \frac{\partial^{2}W_{1}}{\partial T_{0}}^{2}+2\epsilon^{3}\frac{\partial^{2}W_{ 1}}{\partial T_{0}\partial T_{1}}+2\epsilon^{5}\frac{\partial^{2}W_{1}}{ \partial T_{0}\partial T_{1}}+\ldots\ldots\ldots)=-\frac{1}{2}\epsilon^{3}[W_{0} ^{\prime}\int_{1}^{x}\left(\int_{0}^{x}W_{0}{}^{\prime}dx\right)^{**}dx]^{ \prime}+\epsilon^{3}q\end{split} \tag{25}\]  
In Equation (25), \(\epsilon\) is considered an extreme, small value, and we thus neglect the impact of higher-order terms, such as \(\epsilon^{5},\epsilon^{7}\ldots\ldots\) etc., on the overall system. The terms composed of order \(\epsilon^{1}\) are then expressed as:  
\[2\frac{\partial^{2}W_{0}}{\partial T_{0}}^{2}+W_{0}{}^{iv}=0 \tag{26}\]The terms composed of order \(\varepsilon^{3}\) are then expressed as:  
\[\begin{array}{l}2\frac{\partial^{2}W_{1}}{\partial T_{0}}+W_{1}{}^{iv}=\!\!4 \frac{\partial^{2}W_{0}}{\partial T_{0}\partial T_{1}}-\mu\frac{\partial W_{0}} {\partial T_{0}}-W_{0}{}^{v}{}^{3}-W_{0}{}^{iv}W_{0}{}^{\prime}{}^{2}-4W_{0}{ }^{\prime}W_{0}{}^{\prime\prime}W_{0}{}^{\prime\prime}\\ -\frac{1}{2}[W_{0}{}^{\prime}\!\int_{1}^{x}(\int_{0}^{x}W^{\prime}{}^{2}dx\ )^{**}dx]^{\prime}-F_{M}\!+\!\frac{ \hbar q^{2}}{e^{kpt}}\!\int_{b}^{a}W^{\prime\prime}dx\!\int_{0}^{x}[\int_{b}^{ a}(\frac{\partial^{3}W_{0}}{\partial T_{0}\partial x^{2}})dxe^{k_{P}\tau}]d\tau+q \end{array} \tag{27}\]