## 3 Small Amplitude Wave Approximation
### Case 2 (\(\beta=2\))  
In this case, the least order of \(\varepsilon\) provides relations (7) and (8), while the next order of \(\varepsilon\) leads to the following relations:  
\[V_{12}=-\frac{1}{\lambda}\phi_{2}+\frac{1}{2\lambda^{3}}\phi_{1}^{2}, \tag{11a}\] \[N_{12}=-\frac{1}{\lambda^{2}}\phi_{2}+\frac{3}{2\lambda^{4}}\phi_{1}^{2},\] (11b) \[V_{22}=\frac{\lambda}{(\lambda^{2}\mu-3\sigma)}\phi_{2}+\frac{\lambda\big{(} \lambda^{2}\mu+9\sigma\big{)}}{2(\lambda^{2}\mu-3\sigma)^{3}}\phi_{1}^{2},\] (11c) \[N_{22}=\frac{1}{(\lambda^{2}\mu-3\sigma)}\phi_{2}+\frac{3\big{(}\lambda^{2}\mu+ \sigma\big{)}}{2(\lambda^{2}\mu-3\sigma)^{3}}\phi_{1}^{2},\] (11d) \[P_{2}=\frac{3}{(\lambda^{2}\mu-3\sigma)}\phi_{2}+\frac{9\big{(}\lambda^{2}\mu+ \sigma\big{)}}{2(\lambda^{2}\mu-3\sigma)^{3}}\phi_{1}^{2}, \tag{11e}\]  
By substituting Equation (11) into the Poisson Equation (3), the following relationship can be obtained  
\[\bigg{(}\frac{1}{\lambda^{2}}+\frac{\mu_{+}}{(\lambda^{2}\mu-3\sigma)}+\alpha _{1}\bigg{)}\phi_{2}+\bigg{(}-\frac{3}{2\lambda^{4}}+\frac{3\mu_{+}\big{(} \lambda^{2}\mu+\sigma\big{)}}{(\lambda^{2}\mu-3\sigma)}+\alpha_{2}\bigg{)} \phi_{1}^{2}=0 \tag{12}\]  
In Equation (12), the coefficient of \(\phi_{2}(\xi,\tau)\) is complementary (Equation (8a)) is equal to zero and \(\phi_{1}(\xi,\tau)\
eq 0\); therefore, the coefficient of \(\phi_{1}^{2}(\xi,\tau)\) can be considered equal to zero, and in this case, the following relation can be obtained:  
\[\bigg{(}-\frac{3}{2\lambda^{4}}+\frac{3\mu_{+}\big{(}\lambda^{2}\mu+\sigma \big{)}}{(\lambda^{2}\mu-3\sigma)}+\alpha_{2}\bigg{)}=0, \tag{13}\]  
The third order of \(\varepsilon\) represents the following evolution equation:  
\[\frac{\partial}{\partial\tau}\phi_{1}(\xi,\tau)+A^{\prime}\phi_{1}^{2}(\xi, \tau)\frac{\partial}{\partial\xi}\phi_{1}(\xi,\tau)+B\varepsilon^{2\alpha+1} \frac{\partial^{3}}{\partial\xi^{3}}\phi_{1}(\xi,\tau)+C\varepsilon^{\alpha+ \gamma+1}\frac{\partial^{2}}{\partial\xi^{2}}\phi_{1}(\xi,\tau)=0 \tag{14}\]  
where  
\[A^{\prime}=\frac{1}{A_{0}}\bigg{[}3\alpha_{3}+\frac{15}{2\lambda^{6}}+\frac{3 \mu_{+}\big{(}5\lambda^{4}\mu^{2}+30\lambda^{2}\mu\sigma+9\sigma^{2}\big{)}}{2 \big{(}\lambda^{2}\mu-3\sigma\big{)}^{5}}\bigg{]}, \tag{15a}\] \[\alpha_{3}=-\frac{1}{6}\Bigg{[}\mu_{i}\frac{\bigg{(}4k_{i}^{2}+8k_{i}+3\big{)} \big{(}2k_{i}-1\big{)}}{\big{(}2k_{i}-3\big{)}^{3}}+\mu_{e}\sigma_{e}^{3} \frac{\big{(}4k_{e}^{2}+8k_{e}+3\big{)}\big{(}2k_{e}-1\big{)}}{\big{(}2k_{e}- 3\big{)}^{3}}\Bigg{]} \tag{15b}\]  
Equation (14) represents the modified Kortweg-de Vreis -Burger equation, with dispersion and dissipation terms depending mainly on the arbitrary parameters \(\alpha\), and \(\gamma\). At \(\alpha=1\), \(\gamma>1\), Equation (14) tends to the modified KdV equation, but at \(\alpha=2\), \(\gamma=0\), Equation (14) converts into the modified Burgers' equation, while at \(\alpha=1\), \(\gamma=1\), the modified KdV-Burgers' equation appears. In Equation (12) the coefficient of \(\phi_{1}^{2}(\xi,\tau)\) can be considered of the order of \(\varepsilon\) and not equal to zero, so \(\left(-\frac{3}{2\lambda^{4}}+\frac{3\mu_{+}\left(\lambda^{2}\mu+\sigma\right)}{ \left(\lambda^{2}\mu-3\sigma\right)}+\alpha_{2}\right)\phi_{1}^{2}(\xi,\tau)\) becomes an order of \(\varepsilon^{3}\), and it must appear in the higher order of [PERSON]'s equation. Therefore, in this case, the higher order of \(\varepsilon\), it yields the Further-Burgers equation as:  
\[N_{12}=-\frac{1}{\lambda^{2}}\phi_{2}+\frac{3}{2\lambda^{4}}\phi_{1}^{2},\]0  
At \(\alpha=1,\;\gamma>1\), Equation (16) tends toward the so-called Further equation, but at \(\alpha=2,\;\gamma=0,\;\text{and}\;A=0\), Equation (16) converts into the modified Burgers' equation, while at \(\alpha=1,\;\gamma=1\), the Further-Burgers equation appears as:  
\[N_{12}=-\frac{1}{\lambda^{2}}\phi_{2}+\frac{3}{2\lambda^{4}}\phi_{1}^{2},\]1