## 3 The Conformable RPS Approach  
This section aims to construct the multiple CRPS solutions for Equations (1) and (2) based on fractional Taylor series in the sense of conformable derivative through substituting the series expansion within its truncated residual error functions. To achieve our goal, let \(u(x,t)=v(x,t)+ive(x,t)\) in Equation (1) so that the following equivalent coupled system can be obtained  
\[ \begin{array}{l}T_{t}^{a}v(x,t)+\xi w_{xx}(x,t)+\psi(x)w(x,t)+\eta\big{(}v^ {2}(x,t)w(x,t)+w^{3}(x,t)\big{)}=0,\\ T_{t}^{a}w(x,t)-\xi w_{xx}(x,t)-\psi(x)v(x,t)-\eta\big{(}v^{3}(x,t)+w^{2}(x,t) v(x,t)\big{)}=0,\end{array}  \tag{11}\]  
subject to the initial conditions  
\[v(x,0)=v_{0}(x),\text{ and }w(x,0)=w_{0}(x), \tag{12}\]  
in which \(u(x,0)=v_{0}(x)+iw_{0}(x)\). Now, let us assume that the solutions of Equations (11) and (13) for \(0\leq t,\ x\in\mathbb{R}\), have the following multiple CRPS about \(t_{0}=0\):  
\[ \begin{array}{l}v(x,t)=\sum\limits_{k=0}^{\infty}a_{k}(x)\frac{t^{ka}}{a^{k }k!},\\ w(x,t)=\sum\limits_{k=0}^{\infty}b_{k}(x)\frac{t^{ka}}{a^{k}k!},\end{array}  \tag{13}\]  
Evidently, the \(n\)th-truncated series solutions can be defined as follows  
\[ \begin{array}{l}v_{n}(x,t)=v_{0}(x)+\sum\limits_{k=1}^{n}a_{k}(x)\frac{t^{ka }}{a^{k}k!},\\ w_{n}(x,t)=w_{0}(x)+\sum\limits_{k=1}^{n}b_{k}(x)\frac{t^{ka}}{a^{k}k!},\end{array}  \tag{14}\]  
and the \(n\)th-residual functions can be defined as follows  
\[ \begin{array}{l}Res_{v}^{n}(x,t)=T_{t}^{a}v_{n}(x,t)+\xi\big{(}w_{n}\big{)} _{xx}(x,t)+\psi(x)w_{n}(x,t)+\eta\big{(}v_{n}^{2}(x,t)w_{n}(x,t)+\\ w_{n}^{3}(x,t)\big{)},\\ Res_{w}^{sa}(x,t)=T_{t}^{a}w_{n}(x,t)-\xi(v_{n})_{xx}(x,t)-\psi(x)v_{n}(x,t)\\ -\eta\big{(}v_{n}^{3}(x,t)+w_{n}^{2}(x,t)v_{n}(x,t)\big{)},\end{array}  \tag{15}\]  
whereas the residual functions  
\[Res_{v}(x,t)=\lim_{n\to\infty}Res_{v}^{n}(x,t)\text{ and }Res_{w}(x,t)=\lim_{n\to\infty}Res_{w}^{n}(x,t).\]  
Consequently, it can be noted that \(Res(x,t)\) is vanished for each \(x\in\mathbb{R}\) and \(0<t<\epsilon\), where \(\epsilon\) is the convergence radius for the multiple PS. Also, \(T_{t}^{na}Res(x,t)=0\) and \(T_{t}^{(n-1)a}Res(x,t)_{|t=0}=0\) for each\(n=1,2,...\). Therefore, by solving the corresponding system \(T_{t}^{(n-1)a}Res^{n}(x,t)_{|t=0}=0\), \(n=1,2,3,...\), the unknown coefficients of Equation (13) can be determined with the help of the fact that \(T_{t}^{a}t^{p}=0\) for \(p>\alpha\) at \(t=0\) after a little simplification. However, to find the unknown coefficients \(a_{k}(x)\) and \(b_{k}(x)\), \(k=1,2,...,n\), of \(n\)th-truncated series (Equation (14)), do the following: Firstly, substitute the first truncated series \(v_{1}(x,t)=v_{0}(x)+a_{1}(x)\frac{\rho^{a}}{\alpha}\) and \(w_{1}(x,t)=w_{0}(x)+b_{1}(x)\frac{\rho^{a}}{\alpha}\) into the first truncated residual functions \(Res_{  
Now, by operating \(T_{t}^{a}\) on both sides of Equation (18), using the fact that \(T_{t}^{a}t^{p}=0\) for \(p>a\) at \(t=0\) and equating the resulting equations to zero at \(t=0\) yields  
\[ \begin{array}{ll}T_{t}^{a}\,Res_{v}^{2}(x,t)_{|t=0}&=\ a_{2}(x)+\xi b_{1}^{ \prime\prime}(x)+\psi(x)b_{1}(x)\\ &+\eta\big{(}\big{(}2w_{0}(x)v_{0}(x)a_{1}(x)+v_{0}^{2}(x)b_{1}(x)\big{)}+ \big{(}3w_{0}^{2}(x)b_{1}(x)\big{)}\big{)}=0,\\ T_{t}^{a}\,Res_{uv}^{2}(x,t)_{|t=0}&=\ a_{2}(x)+\xi b_{1}^{\prime\prime}(x)+ \psi(x)b_{1}(x)\\ &=\ b_{2}(x)-\xi a_{1}^{\prime}(x)-\psi(x)a_{1}(x)\\ &-\eta\big{(}\big{(}3v_{0}^{2}(x)a_{1}(x)\big{)}+\big{(}2w_{0}(x)v_{0}(x)b_{1 }(x)+w_{0}^{2}(x)a_{1}(x)\big{)}\big{)}=0.\end{array} \]  
Thus, the send unknown coefficients \(a_{2}(x)\) and \(b_{2}(x)\) are given by  
\[ \begin{array}{ll}a_{2}(x)=-\big{(}\xi b_{1}^{\prime\prime}(x)+\psi(x)b_{1}(x )+\eta\big{(}2w_{0}(x)v_{0}(x)a_{1}(x)+\big{(}v_{0}^{2}(x)+3w_{0}^{2}(x)\big{)} b_{1}(x)\big{)}\big{)},\\ b_{2}(x)=\xi a_{1}^{\prime\prime}(x)+\psi(x)a_{1}(x)+\eta\big{(}\big{(}3v_{0}^{2 }(x)+w_{0}^{2}(x)\big{)}a_{1}(x)+2w_{0}(x)v_{0}(x)b_{1}(x)\big{)}.\end{array}