## 4 Derivation of Multi-Wave Exact Solutions of Equation (1) Using a Fractional Wave Transformation
Although numerous variants of solutions can be generated using symbolic computation, our aim here is to extract those particular solutions in which the coefficients of the constructed solution (as in Equation (40)) are expressed by the coefficients of the simple equations and  
* _Family 2: A solution combined distinct generalized trigonometric functions with different independent variables._ \[\xi_{1}=x+\frac{\omega_{1}t^{\phi}}{\Gamma(\phi+1)},\ \ \xi_{2}=x+\frac{\omega_{2}t^{\phi}}{ \Gamma(\phi+1)} \tag{37}\]5 (47) \[\xi_{1}=x+\frac{\omega_{1}t^{\phi}}{\Gamma(\phi+1)},\ \ \xi_{2}=x+\frac{\omega_{2}t^{\phi}}{ \Gamma(\phi+1)} \tag{37}\]6 \[\xi_{1}=x+\frac{\omega_{1}t^{\phi}}{\Gamma(\phi+1)},\ \ \xi_{2}=x+\frac{\omega_{2}t^{\phi}}{ \Gamma(\phi+1)} \tag{37}\]7 \[\xi_{1}=x+\frac{\omega_{1}t^{\phi}}{\Gamma(\phi+1)},\ \ \xi_{2}=x+\frac{\omega_{2}t^{\phi}}{ \Gamma(\phi+1)} \tag{37}\]8 \[\xi_{1}=x+\frac{\omega_{1}t^{\phi}}{\Gamma(\phi+1)},\ \ \xi_{2}=x+\frac{\omega_{2}t^{\phi}}{ \Gamma(\phi+1)} \tag{37}\]9 where \[\left(\frac{df_{1}}{d\xi_{1}}\right)^{2}=\sum_{j_{1}=0}^{4}\mu_{j_{1}}[f_{1}( \xi_{1})]^{j_{1}},\ \ \ \left(\frac{df_{2}}{d\xi_{2}}\right)^{2}=\sum_{j_{2}=0}^{4}v_{j_{2}}[f_{2}( \xi_{2})]^{j_{2}} \tag{38}\]0 (48) for \(\mu_{0}=\frac{\mu_{2}^{2}}{4\mu_{4}},\mu_{2}>0,\ \mu_{4}<0\) and \[\left(\frac{df_{1}}{d\xi_{1}}\right)^{2}=\sum_{j_{1}=0}^{4}\mu_{j_{1}}[f_{1}( \xi_{1})]^{j_{1}},\ \ \ \left(\frac{df_{2}}{d\xi_{2}}\right)^{2}=\sum_{j_{2}=0}^{4}v_{j_{2}}[f_{2}( \xi_{2})]^{j_{2}} \tag{38}\]1 \[\left(\frac{df_{1}}{d\xi_{1}}\right)^{2}=\sum_{j_{1}=0}^{4}\mu_{j_{1}}[f_{1}( \xi_{1})]^{j_{1}},\ \ \ \left(\frac{df_{2}}{d\xi_{2}}\right)^{2}=\sum_{j_{2}=0}^{4}v_{j_{2}}[f_{2}( \xi_{2})]^{j_{2}} \tag{38}\]2 \[\left(\frac{df_{1}}{d\xi_{1}}\right)^{2}=\sum_{j_{1}=0}^{4}\mu_{j_{1}}[f_{1}( \xi_{1})]^{j_{1}},\ \ \ \left(\frac{df_{2}}{d\xi_{2}}\right)^{2}=\sum_{j_{2}=0}^{4}v_{j_{2}}[f_{2}( \xi_{2})]^{j_{2}} \tag{38}\]3 \[\left(\frac{df_{1}}{d\xi_{1}}\right)^{2}=\sum_{j_{1}=0}^{4}\mu_{j_{1}}[f_{1}( \xi_{1})]^{j_{1}},\ \ \ \left(\frac{df_{2}}{d\xi_{2}}\right)^{2}=\sum_{j_{2}=0}^{4}v_{j_{2}}[f_{2}( \xi_{2})]^{j_{2}} \tag{38}\]4 \[\left(\frac{df_{1}}{d\xi_{1}}\right)^{2}=\sum_{j_{1}=0}^{4}\mu_{j_{1}}[f_{1}( \xi_{1})]^{j_{1}},\ \ \ \left(\frac{df_{2}}{d\xi_{2}}\right)^{2}=\sum_{j_{2}=0}^{4}v_{j_{2}}[f_{2}( \xi_{2})]^{j_{2}} \tag{38}\]5 \[\left(\frac{df_{1}}{d\xi_{1}}\right)^{2}=\sum_{j_{1}=0}^{4}\mu_{j_{1}}[f_{1}( \xi_{1})]^{j_{1}},\ \ \ \left(\frac{df_{2}}{d\xi_{2}}\right)^{2}=\sum_{j_{2}=0}^{4}v_{j_{2}}[f_{2}( \xi_{2})]^{j_{2}} \tag{38}\]6 \[\left(\frac{df_{1}}{d\xi_{1}}\right)^{2}=\sum_{j_{1}=0}^{4}\mu_{j_{1}}[f_{1}( \xi_{1})]^{j_{1}},\ \ \ \left(\frac{df_{2}}{d\xi_{2}}\right)^{2}=\sum_{j_{2}=0}^{4}v_{j_{2}}[f_{2}( \xi_{2})]^{j_{2}} \tag{38}\]7 \[\left(\frac{df_{1}}{d\xi_{1}}\right)^{2}=\sum_{j_{1}=0}^{4}\mu_{j_{1}}[f_{1}( \xi_{1})]^{j_{1}},\ \ \ \left(\frac{df_{2}}{d\xi_{2}}\right)^{2}=\sum_{j_{2}=0}^{4}v_{j_{2}}[f_{2}( \xi_{2})]^{j_{2}} \tag{38}\]8  
where \(E_{\phi}(\xi_{1})=\sum_{k=0}^{\infty}\frac{\xi_{1}^{k}}{\Gamma(1+k\phi)}\) and \(E_{\phi}(\xi_{2})=\sum_{k=0}^{\infty}\frac{\xi_{2}^{k}}{\Gamma(1+k\phi)}\) are Mittag-Leffler functions. Illustrative examples of two exact solutions of Equation (1) are given in Figures 4 and 5. **Remark 8**. : _The hyperbolic, trigonometric, and rational functions that appear in the solutions of this section are defined as \"fractional generalized\" functions with respect to the fractional parameter \(\phi\in(0,1]\). They differ from the generalized hyperbolic and trigonometric functions of the form that were introduced in the previous section but reduce back to their corresponding classical counterparts at \(\phi=1\). It is obvious that the fractional generalized functions so defined can easily be reduced to the standard functions when the fractional order \(\phi=1\), thus restoring the classical (integer) case. Therefore, for the particular case \(\phi=1\), all the exact solutions presented above also are valid the classical (integer) version of the system (2) at \(\phi=1\)._  
**Remark 9**.