## 2 Analysis of Rational Solitary Solutions Methods (RSSM)
### Quadratic-Cubic Law \(H(q)=E_{1}\sqrt{Q}+E_{2}Q\)  
From Type 1, we suppose the solution to NPSEs in Equation (34) in the form in Equation (53). The formula in Equation (53) is a solution to Equation (34) when \(n=1\), and Equation (34) is integrable when \(k=1\). Suppose a solution of Equation (34) in the following form:  
\[\begin{array}{c}\psi(\xi)=A_{0}+A_{1}{\rm sech}_{q}\xi+B_{1}{\rm tanh}_{q} \xi.\end{array} \tag{72}\]  
Equation (72) is a solution of Equation (34) when  
**Set 1.**  
\[\begin{array}{c}A_{0}=\frac{-6(\sigma v-\varepsilon)}{E_{1}},\;A_{1}=0,\;B _{1}=\frac{6(\sigma v-\varepsilon)}{E_{1}},\;w=\frac{\varepsilon s_{0}^{2}+ \alpha s_{0}+4\sigma v-4\varepsilon}{\alpha s_{0}-1}\,,\;E_{2}=\frac{18s_{0} \eta(\sigma v-\varepsilon)+E_{1}^{2}}{18(\sigma v-\varepsilon)}.\end{array} \tag{73}\]  
In this set, the solution to NPSEs in Equation (33) is  
\[\begin{array}{c}Q_{24}(x,t)=\left[\frac{-6(\sigma v-\varepsilon)}{E_{1}}+ \frac{6(\sigma v-\varepsilon)}{E_{1}}{\rm tanh}_{q}(x-vt)\right]e^{i(-s_{0}x+wt+ \rho)}\,.\end{array} \tag{74}\]  
**Set 2.**  
\[\begin{array}{c}A_{0}=\frac{3(\sigma v-\varepsilon)}{E_{1}},\;A_{1}=\frac{3 \sqrt{2q}(\sigma v-\varepsilon)}{E_{1}},\;B_{1}=0,\;w=\frac{\varepsilon s_{0}^{ 2}+\alpha s_{0}-2\sigma v+2\varepsilon}{\alpha s_{0}-1}\,,\;E_{2}=-\frac{-9s_{0 }\eta(\sigma v-\varepsilon)+E_{1}^{2}}{9(\sigma v-\varepsilon)},q>0.\end{array} \tag{75}\]In this set, the solution to NPSEs in Equation (33) is  
\[Q_{25}(x,t)=\left[\frac{3(\sigma v-\varepsilon)}{E_{1}}+\frac{3\sqrt{2q}(\sigma v -\varepsilon)}{E_{1}}\text{sech}_{q}(x-vt)\right]e^{i(-s_{0}x+wt+\rho)}. \tag{76}\]  
**Set 3.**: \[\begin{array}{c}A_{0}=\frac{-3(\sigma v-\varepsilon)}{2E_{1}},\,A_{1}=\frac{3 (\sigma v-\varepsilon)\sqrt{q}}{2E_{1}},\,B_{1}=\frac{3(\sigma v-\varepsilon) }{2E_{1}},\\ w=\frac{\epsilon s_{0}^{2}+4s_{0}+\sigma v-\varepsilon}{s_{0}-1}\,,\,E_{2}= \frac{9s_{0}\eta(\sigma v-\varepsilon)+2E_{1}^{2}}{9(\sigma v-\varepsilon)},\, q<0\end{array}\] (77)  
In this set, the solution to NPSEs in Equation (33) is  
\[Q_{26}(x,t)=\left[\frac{-3(\sigma v-\varepsilon)}{2E_{1}}+\frac{3(\sigma v- \varepsilon)\sqrt{-q}}{2E_{1}}\text{sech}_{q}(x-vt)+\frac{3(\sigma v- \varepsilon)}{2E_{1}}\text{tanh}_{q}(x-vt)\right]e^{i(-s_{0}x+wt+\rho)}. \tag{78}\]  
**For the type 2.** The solutions of the NPSEs in Equation (33) are given by  
\[Q_{27}(x,t)=\left[-\frac{6(\sigma v-\varepsilon)}{E_{1}}-\frac{6(\sigma v- \varepsilon)}{E_{1}}\text{coth}_{q}(x-vt)\right]e^{i(s_{0}x+wt+\rho)}, \tag{79}\]  
where \(w=\frac{\epsilon s_{0}^{2}+4s_{0}+4(\sigma v-\varepsilon)}{(s_{0}-1)}\) and \(E_{2}=\frac{18s_{0}\eta(\sigma v-\varepsilon)+E_{1}^{2}}{18(\sigma v- \varepsilon)}\).  
\[Q_{28}(x,t)=\left[\frac{3(\sigma v-\varepsilon)}{E_{1}}+\frac{3\sqrt{-2q}( \sigma v-\varepsilon)}{E_{1}}\text{csch}_{q}(x-vt)\right]e^{i(s_{0}x+wt+\rho)}. \tag{80}\]  
where \(w=\frac{\epsilon s_{0}^{2}+4s_{0}-2\sigma v+2\varepsilon}{s_{0}-1}\) and \(E_{2}=-\frac{-9s_{0}\eta(\sigma v-\varepsilon)+E_{1}^{2}}{9(\sigma v- \varepsilon)},q<0\).  
\[Q_{29}(x,t)=\left[\frac{-3(\sigma v-\varepsilon)}{2E_{1}}+\frac{3(\sigma v- \varepsilon)\sqrt{q}}{2E_{1}}\text{csch}_{q}(x-vt)-\frac{3(\sigma v- \varepsilon)}{2E_{1}}\text{coth}_{q}(x-vt)\right]e^{i(-s_{0}x+wt+\rho)}. \tag{81}\]  
where \(w=\frac{\epsilon s_{0}^{2}+4s_{0}+(\sigma v-\varepsilon)}{(s_{0}-1)}\) and \(E_{2}=\frac{9s_{0}\eta(\sigma v-\varepsilon)+2E_{1}^{2}}{9(\sigma v- \varepsilon)},q>0\).