## 2. Instructions for the Methods  
Assume a nonlinear partial differential equation (NLPDE) as follows:  
\[P\left(U,U_{x},U_{t},U_{xx},U_{tt},U_{tx},\ldots\right)=0, \tag{5}\]  
and define the traveling wave transformation as follows,  
\[U\left(x,y,t\right)=\phi\left(\xi\right),\quad\zeta=x-vt. \tag{6}\]  
Putting Equation (6) into Equation (5), the outcome is:  
\[N\left(\phi,\phi^{\prime},\phi^{\prime\prime},\,\ldots\right)=0. \tag{7}\]  
For the \(m+\frac{C^{\prime}\left(\xi\right)}{G\left(\xi\right)}\) expansion method, we take the trial solution for Equation (7) as follows:  
\[\phi\left(\xi\right)=\sum_{i=-n}^{n}a_{i}\left(m+F\right)^{i}=a_{-n}\left(m+F \right)^{-n}+\ldots+m\,a_{0}+a_{1}\left(m+F\right)+\ldots+a_{n}\left(m+F \right)^{n}, \tag{8}\]  
where \(a_{i},\ i=0,\,1,\,\ldots,\,n\) and \(m\) are nonzero constants. According to the principles of balance, we find the value of \(n\). In this manuscript, we define \(F\) to be a function as:  
\[F=\frac{G^{\prime}\left(\xi\right)}{G\left(\xi\right)}, \tag{9}\]  
where \(G\left(\xi\right)\) satisfy \(G^{\prime\prime}+\left(\lambda+2m\right)G^{\prime}+\mu G=0\). Putting Equation (8) into Equation (7) by using Equation (9), then collecting all terms with the same order of \(\left(m+F\right)^{n}\), we get the system of algebraic equations for \(\
u\), \(a_{n}\), \(n=0,\,1,\,\ldots,\,n\), \(\lambda\), and \(\mu\). As a result, solving the obtained system, we get the explicit and exact solutions of Equation (5). For the \(\left(\exp-\varphi\left(\xi\right)\right)\) expansion method, we use the trial solution as follows:  
\[\phi\left(\xi\right)=\sum_{i=0}^{n}b_{i}(\exp\left(-\varphi\left(\xi\right) \right))^{i},\quad i=1,2,\ldots,n \tag{10}\]where \(b_{i}\) are non-zero constants. The auxiliary ODE \(\varphi\left(\xi\right)\) is defined as follows:  
\[\varphi^{\prime}\left(\xi\right)=\exp\left(-\varphi\left(\xi\right)\right)+\mu\, \exp\left(\varphi\left(\xi\right)\right)+\lambda. \tag{11}\]  
Solving Equation (11), we have:  
Case 1. When \(\Delta>0\) and \(\mu\
eq 0\), we get the hyperbolic function solution:  
\[\varphi\left(\xi\right)=\ln\left(\frac{-\lambda-\sqrt{\Delta}\tanh\left(\frac {1}{2}\sqrt{\Delta}\left(\xi+c\right)\right)}{2\mu}\right). \tag{12}\]  
Case 2. When \(\Delta<0\) and \(\mu\
eq 0\), we get the trigonometric function solution:  
\[\varphi\left(\xi\right)=\ln\left(\frac{-\lambda+\sqrt{-\Delta}\tan\left(\frac {1}{2}\sqrt{-\Delta}\left(\xi+c\right)\right)}{2\mu}\right). \tag{13}\]  
Case 3. When \(\Delta>0\), \(\mu=0\), and \(\lambda\
eq 0\), we get hyperbolic function solution  
\[\varphi\left(\xi\right)=-\ln\left(\frac{\lambda}{-1+\cosh\left(\lambda\left( \xi+c\right)\right)+\sinh\left(\lambda\left(\xi+c\right)\right)}\right). \tag{14}\]  
Case 4. When \(\Delta=0\), \(\mu\
eq 0\) and \(\lambda\
eq 0\), we get the rational function solution:  
\[U\left(x,y,t\right)=\phi\left(\xi\right),\quad\zeta=x-vt. \tag{6}\]0  
Case 5. When \(\Delta=0\), \(\mu=0\), and \(\lambda=0\), we get:  
\[U\left(x,y,t\right)=\phi\left(\xi\right),\quad\zeta=x-vt. \tag{6}\]1  
where \(c\) is the non-zero constant of integration and \(\Delta=\lambda^{2}-4\mu\). **3. Application to the \(\left(m+\frac{G^{\prime}}{G}\right)\) Expansion Method**  
In this section, we use the \(\left(m+\frac{G^{\prime}}{G}\right)\) expansion method for the cubic-quartic nonlinear Schrodinger and cubic-quartic resonant nonlinear Schrodinger equations. _3.1. The Cubic-Quartic Nonlinear Schrodinger Equation_  
To solve Equation (3), by the \(\left(m+\frac{G^{\prime}}{G}\right)\) expansion method, we use the following transformation:  
\[U\left(x,y,t\right)=\phi\left(\xi\right),\quad\zeta=x-vt. \tag{6}\]2  
In the above equation, \(\theta\left(x,t\right)\) symbolize the phase component of the soliton, \(\kappa\) represent the soliton frequency, while \(\omega\) denote the wave number, and \(\upsilon\) symbolize the velocity of the soliton. Substitute wave transformation into Equation (3), and separate the outcome equation into real and imaginary parts.