## 2 Analysis of Rational Solitary Solutions Methods (RSSM)  
They are many methods to solve nonlinear partial differential equations (NPDEs) such as transformed rational function (TRF) method [26] and the multiple exp-function (MEF) method [24; 25]. The RSSM is one of the important applications to the TRF method and MEF methods for solving the NPDEs. The transformation in Equation (2) is applied to transfer the NPDEs to NODEs. We suppose the solutions of the NODEs according to the RSSM have the shape  
\[W(\xi)=a_{0}+\sum_{i=0}^{N}a_{i}g^{i}(\xi)+\sum_{j=1}^{N}b_{j}g^{j-1}(\xi)f( \xi),\,\alpha_{N}\
eq 0,\,\,or\,\,b_{N}\
eq 0, \tag{7}\]  
where the conditions on the two functions \(g(\xi)\) and \(f(\xi)\) take shapes:  
\[ \begin{array}{c}f^{\prime}(\xi)=-Ag^{2}(\xi)+\frac{\delta A}{B}g(\xi)[1-Bg( \xi)],\\ g^{\prime}(\xi)=-Af(\xi)\,g(\xi),\\ f^{2}(\xi)=g^{2}(\xi)+\frac{\delta}{A^{2}}[1-Bg(\xi)]^{2},\end{array}  \tag{8}\]  
\(A,B,a_{i},i=0,1,2,\ldots\) and \(b_{i},i=0,1,2,\ldots\) are constants and can be evaluated later, while \(\delta=\pm 1\). The nonlinear auxiliary conditions in Equation (8) can be solved as follows:  
If \(\delta=1\), the hyperbolic solutions take the following form:  
\[f(\xi)=\frac{1}{A\text{tanh}(\xi)+B\text{sech}(\xi)},\,\,\,\,\,\,\,\,\,\,\,g( \xi)=\frac{\text{sech}(\xi)}{A\text{tanh}(\xi)+B\text{sech}(\xi)}. \tag{9}\]  
or  
\[f(\xi)=\frac{\text{coth}(\xi)}{A+B\text{csech}(\xi)},\,g(\xi)=\frac{\text{csch }(\xi)}{A+B\text{csch}(\xi)}. \tag{10}\]  
If \(\delta=-1\), the rational periodic solutions take the following form:  
\[f(\xi)=\frac{1}{A\,\text{tan}(\xi)+B\,\text{sec}(\xi)},\,\,\,\,\,\,\,\,\,\,g( \xi)=\frac{\text{sec}(\xi)}{A\,\text{tan}(\xi)+B\,\text{sec}(\xi)}, \tag{11}\]  
or  
\[f(\xi)=\frac{\text{cot}(\xi)}{A+B\,\text{csc}(\xi)},\,g(\xi)=\frac{\text{csc}( \xi)}{A+B\,\text{csc}(\xi)}. \tag{12}\]  
The homogenous idea of the balance power is used to determine the end sum of the series in Equation (7). Insert Equation (7) with the auxiliary Equation (8) into the NODEs and deduce the system of algebraic equations from comparison between the coefficients of \(f(\xi)\) and \(g(\xi)\). Resolve the aforesaid system of equations to \(A,B,a_{i},\,i=0,1,2,\,\cdot\,b_{i},\,i=0,1,2,\ldots\) and insert the results in the Equation (3) solution to get the solutions of the NPDEs. **Remark 1**. : _The system in Equation (8) can be presented by a single of NODE \(Bg^{\prime\prime}(\xi)=g(\xi)\)\([Bg^{2}\Big{[}(\delta+2)A^{2}+\delta B^{2}\Big{]}-\delta(A^{2}+2B^{2})g+\delta B]\) and it has the same solutions (9)-(12) when \(A^{2}=B^{2}\)._  
Rational Soliton Solution to Nonlinear Perturbed Schrodinger Equations NPSEs with Kerr Law \(H(\psi)=\psi\)  
The nonlinearity Kerr law PNSEs takes the form:  
\[iQ_{t}+\sigma Q_{xt}+\varepsilon Q_{xx}+\zeta Q|Q|^{2}=i\left[\alpha Q_{x}+ \eta\big{[}|Q|^{2k}Q\big{]}_{x}+\upsilon\ Q\ (|Q|^{2k})_{x}\right]. \tag{13}\]  
Equations (3) and (4) lead to write PNSEs in Equation (3) as follows  
\[(\varepsilon-\sigma v)\ \psi^{\prime\prime}-(w+\alpha s_{0}-\sigma ws_{0}+ \varepsilon s_{0}^{2})\ \psi+(\zeta-s_{0}\eta)\psi^{3}\ =0. \tag{14}\]  
when the soliton velocity is given by \(v=\frac{2\varepsilon s_{0}+\alpha-\sigma w}{(\varepsilon s_{0}-1)}\), the condition \(\eta=\frac{-2\upsilon}{3}\), and put \(k=1\) in the nonlinear media to be integrable. Balancing \(\psi^{\prime\prime}\) and \(\psi^{3}\), we get the Equation (14), which can be written in the form:  
\[\psi(\xi)=a_{0}+a_{1}f(\xi)+b_{1}g(\xi),\ \alpha_{1}\
eq 0,\ or\ b_{1}\
eq 0 \tag{15}\]  
where \(g(\xi)\), \(f(\xi)\) satisfies the conditions in Equation (8) and \(a_{0},a_{1},b_{1}\) are arbitrary constants. The solution formula in Equation (15) is a solution to Equation (14) under the conditions of Equation (8) if and only if:  
**Family 1**. : \(\delta=1\)  
\[