## 2 Analysis of Rational Solitary Solutions Methods (RSSM)
### Rational Soliton Solution to NPSEs with Quadratic-Cubic Law \(H(Q)=E_{1}\sqrt{Q}+E_{2}Q\)
Rational Soliton Solution to NPS's with Anti-Cubic Law \(H(Q)=\frac{M_{1}}{Q^{2}}+M_{2}Q+M_{3}Q^{2}\)  
The nonlinearity anti cubic law PNSEs takes the form:  
\[iQ_{t}+\sigma Q_{xt}+\varepsilon Q_{xx}+\zeta[M_{1}|Q]^{-4}+M_{2}|Q|^{2}+M_{3 }|Q|^{4}]Q=i\left[\alpha Q_{x}+\eta[|Q|^{2k}Q]_{x}+v\,Q\,\left(|Q|^{2k}\right) _{x}\right], \tag{42}\]  
where \(M_{1}\), \(M_{2}\) and \(M_{3}\) are nonzero constants. Equations (3) and (4) lead to write PNSEs in Equation (34) as follows:  
\[(\varepsilon-\sigma v)\psi^{3}\,\psi^{\prime\prime}-(w+\alpha s_{0}-\sigma ws_{0 }+\varepsilon s_{0}^{2})\,\psi^{4}-4s_{0}\eta\psi^{2k+4}+M_{1}\,+\,M_{2}\,\psi^ {6}\,+\,M_{3}\,\,\psi^{8}=0. \tag{43}\]  
when the soliton velocity is given by \(v=\frac{2\varepsilon s_{0}+\alpha-\sigma w}{(\alpha s_{0}-1)}\), the condition \(\eta=\frac{-2 kv}{2k+1}\). From equating the power of terms \(\psi^{3}\,\psi^{\prime\prime}\) and We suppose \(N=1/2\), we have \(\psi^{8}\)  
\[\psi=\left[\Omega(\xi)\right]^{1/2}. \tag{44}\]  
From Equations (44) and (43) we get:  
\[(\varepsilon-\sigma v)[2\Omega\Omega^{\prime\prime}-{\Omega^{\prime}}^{2}]-4(w +\alpha s_{0}-\sigma ws_{0}+\varepsilon s_{0}^{2})\,\Omega^{2}-4s_{0}\eta\, \Omega^{4}+4(M_{1}\,+\,M_{2}\,\Omega^{3}\,+\,M_{3}\,\Omega^{4})=0. \tag{45}\]  
Consequently, we set the solution of Equation (45) in the following form:  
\[\Omega(\xi)=a_{0}+a_{1}f(\xi)+b_{1}g(\xi)\,\alpha_{1}\
eq 0,\;or\;b_{1}\
eq 0, \tag{46}\]  
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 (46) is a solution to Equation (45) under conditions of Equation (8) if and only if:  
**Family 7. **\(\delta=1\)  
\[ \begin{array}{c}M_{1}=-\frac{a_{0}^{2}(\alpha\upsilon-\varepsilon)}{4b_{1}^{2}} [2a_{0}b_{1}B+a_{0}^{2}(A^{2}+B^{2})+b_{1}^{2}],\ a_{1}=0,\\ M_{2}=-\frac{(\alpha\upsilon-\varepsilon)}{b_{1}^{2}}[2a_{0}(A^{2}+B^{2})+Bb_{1 }],\ b_{1}\
eq 0\\ M_{3}=\frac{1}{4b_{1}^{2}}[3(\sigma\upsilon-\varepsilon)(A^{2}+B^{2})+4s_{0} \eta b_{1}^{2}],\\ w=\frac{1}{4b_{1}^{2}(\alpha s_{0}-1)}[6a_{0}^{2}(\sigma\upsilon-\varepsilon)(A ^{2}+B^{2})+(\sigma\upsilon-\varepsilon)(b_{1}^{2}+6a_{0}b_{1}B)+4\varepsilon s _{0}^{2}b_{1}^{2}+4\alpha s_{0}b_{1}^{2}].\end{array}  \tag{47}\]  
In this family, the solution of PNSEs in Equation (42) takes the form:  
\[Q_{13}(x,t)=e^{i(-s_{0}x+\omega t+\rho)}\bigg{[}a_{0}+\frac{b_{1}\text{sech}(x -\upsilon t)}{[A\text{tanh}(x-\upsilon t)+B\sec h(x-\upsilon t)]}\bigg{]}^{1 /2}. \tag{48}\]  
or  
\[Q_{14}(x,t)=e^{i(-s_{0}x+\omega t+\rho)}\bigg{[}a_{0}+\frac{b_{1}\text{csch}(x -\upsilon t)}{[A+B\text{csch}(x-\upsilon t)]}\bigg{]}^{1/2}. \tag{49}\]  
**Family 8. **\(\delta=-1\)  
\[ \begin{array}{c}M_{1}=\frac{a_{0}^{2}(\alpha\upsilon-\varepsilon)}{4b_{1}^{ 2}}[2a_{0}b_{1}B-a_{0}^{2}(A^{2}-B^{2})+b_{1}^{2}],\ a_{1}=0,\\ M_{2}=\frac{(\alpha\upsilon-\varepsilon)}{b_{1}^{2}}[-2a_{0}(A^{2}-B^{2})+Bb_{ 1}],\\ M_{3}=\frac{1}{4b_{1}^{2}}[3(\sigma\upsilon-\varepsilon)(A^{2}-B^{2})+4s_{0} \eta b_{1}^{2}],\\ w=\frac{1}{4b_{1}^{2}(\alpha s_{0}-1)}[6a_{0}^{2}(\sigma\upsilon-\varepsilon)(A ^{2}-B^{2})-(\sigma\upsilon-\varepsilon)(b_{1}^{2}+6a_{0}b_{1}B)+4\varepsilon s _{0}^{2}b_{1}^{2}+4\alpha s_{0}b_{1}^{2}].\end{array}  \tag{50}\]  
In this family, the solution of PNSEs in Equation (42) takes the form:  
\[Q_{15}(x,t)=e^{i(-s_{0}x+\omega t+\rho)}\bigg{[}a_{0}+\frac{b_{1}\sec(x- \upsilon t)}{[A\tan(x-\upsilon t)+B\sec(x-\upsilon t)]}\bigg{]}^{1/2}, \tag{51}\]  
or  
\[Q_{16}(x,t)=e^{i(-s_{0}x+\omega t+\rho)}\bigg{[}a_{0}+\frac{b_{1}\sec(x- \upsilon t)}{[A+B\csc(x-\upsilon t)]}\bigg{]}^{1/2} \tag{52}\]  
There are many different other families, but we leave it as a kind convenience to the readers. **3. Q-Deformed Hyperbolic Functions Method to NPSEs**  
Many authors have introduced the definition of q-deformed hyperbolic functions (see, for example, [7; 12; 13; 14]). The q-deformed hyperbolic functions are defined by \(\sinh_{q}(\xi)=\frac{1}{2}(e^{\xi}-qe^{-\xi})\), and \(\cosh_{q}(\xi)=\frac{1}{2}(e^{\xi}+qe^{-\xi})\).