## 2 Analysis of Rational Solitary Solutions Methods (RSSM)
### Rational Soliton Solution to NPSEs with Quadratic-Cubic Law \(H(Q)=E_{1}\sqrt{Q}+E_{2}Q\)  
The nonlinearity quadratic law PNSEs take the form:  
\[iQ_{t}+\sigma Q_{xt}+\varepsilon Q_{xx}+\zeta[E_{1}|Q|+E_{2}|Q|^{2}]Q=i\,[ \alpha Q_{x}+\eta[|Q|^{2k}Q]_{x}+v\,Q\,\,(|Q|^{2k})_{x}], \tag{33}\]  
where \(E_{1}\) and \(E_{2}\) are constants. Equations (3) and (4) lead to write PNSEs in Equation (33) as follows:  
\[(\varepsilon-\sigma v)\,\psi^{\prime\prime}-(w+\alpha s_{0}-\sigma ws_{0}+ \varepsilon s_{0}^{2})\,\psi+E_{1}\,\psi^{2}+\,E_{2}\,\psi^{3}-s_{0}\eta\, \psi^{2k+1}=0. \tag{34}\]  
When the soliton velocity is given by \(v=\frac{2\varepsilon s_{0}+\alpha-\alpha w}{(\alpha s_{0}-1)}\), the condition \(\eta=\frac{-2 kv}{2k+1}.\) Balancing the power of \(\psi^{\prime\prime}\) and \(\psi^{3}\) in the series solution in Equation (7) with the restriction in Equation (8), we have \(N=1\) and Equation (28) is integrable when \(k=1\). Consequently, we set the solution in the following form:  
\[\psi(\xi)=a_{0}+a_{1}f(\xi)+b_{1}g(\xi)\,\,\alpha_{1}\
eq 0,\,\,or\,\,b_{1} \
eq 0, \tag{35}\]  
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 (35) is a solution to Equation (34) under conditions in Equation (8) if and only if:  
**Family 5. **\(\delta=1\)  
\[ \begin{array}{c}a_{0}=\frac{-E_{1}}{3(E_{2}-s_{0}\eta)},\,a_{1}=\frac{AE_{1 }}{3(E_{2}-s_{0}\eta)},\\ b_{1}=\frac{E_{1}\sqrt{A^{2}+B^{2}}}{3(E_{2}-s_{0}\eta)},\,\,\sigma=\frac{2E_{1 }^{2}+9E_{2}\varepsilon-\mathfrak{s}_{0}\,\varepsilon\,\eta}{9E_{2}(E_{2}-s_{ 0}\eta)},\,\,E_{2}-s_{0}\eta\
eq 0\\ w=\frac{v\,(-s\,\varepsilon s_{0}^{3}\,\eta+\,\varepsilon s_{0}^{2}E_{2}-s_{ 0}^{2}\eta^{3}+9\alpha s_{0}E_{2}+2E_{1}^{2})}{9\,s_{0}\varepsilon E_{2}+9 \varepsilon s_{0}\eta-9\varepsilon s_{0}^{2}\varepsilon t-9\varepsilon s_{0}^{ 2}\varepsilon t-9\varepsilon s_{0}E_{2}+2s_{0}\,E_{1}^{2}}.\end{array}  \tag{36}\]  
In this family the solution of PNSEs in Equation (33) take the form:  
\[ \begin{array}{c}Q_{9}(x,t)=e^{i(-s_{0}x+\imath vt+\rho)}\bigg{[}\frac{-E_{1}} {3(E_{2}-s_{0}\eta)}+\frac{AE_{1}}{3(E_{2}-s_{0}\eta)[A\tanh(x-\imath vt)+B \text{sech}(x-\imath vt)]}\\ \hskip 142.26378 pt+\frac{E_{1}\sqrt{A^{2}+B^{2}}\text{sech}(x-\imath vt)}{3(E_ {2}-s_{0}\eta)[A\tanh(x-\imath vt)+B\text{sech}(x-\imath vt)]}\bigg{]}.\end{array}  \tag{37}\]  
or  
\[ \begin{array}{c}Q_{10}(x,t)=e^{i(-s_{0}x+\imath vt+\rho)}\bigg{[}\frac{-E_{1}} {3(E_{2}-s_{0}\eta)}+\frac{AE_{1}\coth(x-\imath vt)}{(E_{2}-s_{0}\eta)[A+B \text{sech}(x-\imath vt)]}\\ \hskip 142.26378 pt+\frac{E_{1}\sqrt{A^{2}+B^{2}}\text{sech}(x-\imath vt)}{3(E_ {2}-s_{0}\eta)[A+B\text{sech}(x-\imath vt)]}\bigg{]}.\end{array}  \tag{38}\]  
**Family 6. **\(\delta=-1\)  
\[ \begin{array}{c}b_{1}=-\frac{3 Ba_{0}(2E_{2}a_{0}-2s_{0}\eta a_{0}+E_{1})}{3E_{2 }a_{0}+E_{1}-3s_{0}\eta a_{0}},\;A=\;\frac{B\sqrt{2E_{1}^{2}-3s_{0}E_{1}\eta a_{ 0}+3E_{1}E_{2}a_{0}}}{\sqrt{2}[3E_{2}a_{0}+E_{1}-3s_{0}\eta a_{0}]}\;,\\ w=\frac{v\left(-s_{0}\eta a_{0}^{2}+E_{1}a_{0}+E_{2}a_{0}^{2}-s_{0}^{2}-\alpha s _{0}\right)}{2\,s_{0}\,E_{2}a_{0}^{2}-2s_{0}^{2}\eta a_{0}^{2}-\alpha s_{0}+v+ s_{0}\,E_{1}a_{0}},\;\sigma=-\frac{1}{v}\left(2E_{2}a_{0}^{2}-2s_{0}\eta a_{0}^{2}+E_{1 }a_{0}-\varepsilon\right),\\ a_{1}=0.2E_{1}^{2}-3s_{0}E_{1}\,\eta a_{0}+3E_{1}E_{2}a_{0}>0\end{array}  \tag{39}\]  
In this family, the solution of PNSEs in Equation (33) takes the form:  
\[Q_{11}=\left[a_{0}-\frac{3a_{0}(2E_{2}a_{0}-2s_{0}\eta a_{0}+E_{1})sec(x-vt)}{ \left[\sqrt{2E_{1}^{2}-3s_{0}E_{1}\,\eta a_{0}+3E_{1}E_{2}a_{0}}\tan(x-vt)+(3E_ {2}a_{0}+E_{1}-3s_{0}\eta a_{0})sec(x-vt)\right]}\right]e^{i(-s_{0}x+wt+\rho)}. \tag{40}\]  
or  
\[Q_{12}(x,t)=\left[a_{0}-\frac{3a_{0}(2E_{2}a_{0}-2s_{0}\eta a_{0}+E_{1})\csc(x -vt)}{\left[\sqrt{2E_{1}^{2}-3s_{0}E_{1}\,\eta a_{0}+3E_{1}E_{2}a_{0}}+(3E_{2} a_{0}+E_{1}-3s_{0}\eta a_{0})\csc(x-vt)\right]}\right]e^{i(-s_{0}x+wt+\rho)}. \tag{41}\]  
There are many different other families, but we leave it as a kind of convenience to the readers.