## 2 Analysis of Rational Solitary Solutions Methods (RSSM)
### Rational Soliton Solution to NPSEs with Power Law \(H(\psi)=\psi^{k}\), \(0<k<0\)  
The nonlinearity power law PNSEs take the form:  
\[iQ_{t}+\sigma Q_{xt}+\varepsilon Q_{xx}+\zeta Q\left|Q\right|^{2k}=i\,\left[ \alpha Q_{x}+\eta[\left|Q\right|^{2k}Q]_{x}+v\,Q\,\left(\left|Q\right|^{2k} \right)_{x}\right]. \tag{22}\]  
Equations (3) and (4) lead to write PNSEs in Equation (22) as follows:  
\[\left(\varepsilon-\sigma v\right)\psi^{\prime\prime}-\left(w+\alpha s_{0}- \alpha ws_{0}+\varepsilon s_{0}^{2}\right)\psi+\left(\zeta-s_{0}\eta\right) \psi^{2k+1}\,=0, \tag{23}\]  
when the soliton velocity is given by \(v=\frac{2\varepsilon s_{0}+\alpha-\omega v}{\left(\alpha s_{0}-1\right)}\) and \(\eta=\frac{-2\omega v}{2k+1}\). From balance the highest power of \(\psi^{\prime\prime}\) and \(\psi^{2k+1}\), we get \(N=\frac{1}{k}\) as rational number, and we take the transformation  
\[\psi(\xi)=\left[\varphi(\xi)\right]^{1/2k}. \tag{24}\]  
The transformation in Equation (24) leads to obtain  
\[\left(\varepsilon-\sigma v\right)\left(1-2k\right)\varphi^{\prime 2}+2k( \varepsilon-\sigma v)\varphi\varphi^{\prime\prime}-4k^{2}(w+\alpha s_{0}- \alpha ws_{0}+\varepsilon s_{0}^{2})\,\varphi^{2}+4k^{2}(\zeta-s_{0}\eta)\, \varphi^{3}\,=0. \tag{25}\]  
From \(\varphi\varphi^{\prime\prime}\) and \(\varphi^{3}\), the solution of Equation (21) can be written in the form:  
\[\psi(\xi)=a_{0}+a_{1}f(\xi)+b_{1}g(\xi)+a_{2}f(\xi)g(\xi)+b_{2}g^{2}(\xi),\; \alpha_{2}\
eq 0,\;or\;b_{2}\
eq 0, \tag{26}\]  
where \(g(\xi)\), \(f(\xi)\) satisfies the conditions in Equation (8), and \(a_{0},a_{1},a_{2},b_{1},b_{2}\) are arbitrary constants. The solution formula in Equation (26) is a solution to Equation (25) under conditions (8) if and only if:  
**Family 3. **\(\delta=1\)  
\[ \begin{array}{c}a_{2}=\frac{A(k+1)\left(\upsilon\sigma-\varepsilon\right) \sqrt{A^{2}+B^{2}}}{2k^{2}\left(\eta s_{0}-\zeta\right)},\;a_{1}=\;a_{0}=0,\; \eta s_{0}-\zeta\
eq 0\\ b_{1}=\frac{B(k+1)\left(\upsilon\sigma-\varepsilon\right)}{2k^{2}\left(\eta s _{0}-\zeta\right)},\;w=\frac{\upsilon\sigma-\varepsilon+4k^{2}s_{0}\alpha+4k^ {2}s_{0}^{2}\varepsilon}{4k^{2}\left(\eta s_{0}-1\right)},\\ b_{2}=-\frac{\left(A^{2}+B^{2}\right)\left(k+1\right)\left(\upsilon\sigma- \varepsilon\right)}{2k^{2}\left(\eta s_{0}-\zeta\right)}.\end{array}  \tag{27}\]  
In this family the solution of PNSEs in Equation (22) takes the form:  
\[ \begin{array}{c}Q_{5}(x,t)=e^{i\left(-\varsigma Qx+\upsilon t+\rho\right)} \bigg{[}\frac{B(k+1)\left(\upsilon\sigma-\varepsilon\right)\text{sech}(x- \upsilon t)}{2k^{2}\left(\eta s_{0}-\zeta\right)\left[A\tanh(x-\upsilon t)+B \text{sech}(x-\upsilon t)\right]}\\ +\frac{A(k+1)\left(\upsilon\sigma-\varepsilon\right)\sqrt{A^{2}+B^{2}}\text{ sech}(x-\upsilon t)}{2k^{2}\left(\eta s_{0}-\zeta\right)\left[A\tanh(x- \upsilon t)+B\text{sech}(x-\upsilon t)\right]^{2}}-\frac{\left(A^{2}+B^{2} \right)\left(k+1\right)\left(\upsilon\sigma-\varepsilon\right)\text{sech}^{2 }(x-\upsilon t)}{2k^{2}\left(\eta s_{0}-\zeta\right)\left[A\tanh(x-\upsilon t )+B\text{sech}(x-\upsilon t)\right]^{2}}\bigg{]}^{1/2k}\end{array}  \tag{28}\]  
or  
\[ \begin{array}{c}Q_{6}(x,t)=e^{i\left(-\varsigma Qx+\upsilon t+\rho\right)} \bigg{[}\frac{B(k+1)\left(\upsilon\sigma-\varepsilon\right)\text{coth}(x- \upsilon t)}{2k^{2}\left(\eta s_{0}-\zeta\right)\left[A+B\text{sech}(x- \upsilon t)\right]}\\ +\frac{A(k+1)\left(\upsilon\sigma-\varepsilon\right)\sqrt{A^{2}+B^{2}}\text{ coth}(x-\upsilon t)\text{sech}(x-\upsilon t)}{2k^{2}\left(\eta s_{0}-\zeta\right) \left[A+B\text{ sech}(x-\upsilon t)\right]^{2}}-\frac{\left(A^{2}+B^{2} \right)\left(k+1\right)\left(\upsilon\sigma-\varepsilon\right)\text{sech}^{2 }(x-\upsilon t)}{2k^{2}\left(\eta s_{0}-\zeta\right)\left[A+B\text{ sech}(x- \upsilon t)\right]^{2}}\bigg{]}^{1/2k}\end{array}  \tag{29}\]  
**Family 4.