Theory and Modern Applications
Table 4 Reduction of Eq. (1)
From: Lie symmetry and μ-symmetry methods for nonlinear generalized Camassa–Holm equation
Operator | Reduced equations |
|---|---|
\({\alpha \nu }_{1}\) | \(w_{y}=0\) |
\(\alpha {\nu }_{1}+{\nu }_{2}\) | \(\begin{array}[t]{l} -\alpha w_{y}+\alpha w_{yyy}-\frac{1}{2}(p+1)(p+2)w^{p}w_{y}+ \frac{1}{2}p(p-1)w^{p-2}w_{y}^{3} \\ \quad {}+2pw^{p-1}w_{y}w_{yy}+w^{p}w_{yy}=0 \end{array}\) |
\(\alpha {\nu }_{1}+{\nu }_{3}\) | \(\begin{array}[t]{l} -\frac{1}{\alpha ^{3}}(y^{3}+w^{p}y^{4})w_{yyy}+(\frac{2}{\alpha ^{3}}w^{p}y^{2} -\frac{3p+2}{\alpha p}y^{2}-\frac{3p+3}{\alpha ^{3}p}w^{p}y^{3})w_{yy} \\ \quad {}-\frac{2p}{\alpha ^{3}}w^{p-1}y^{3}w_{y}w_{yy} -\frac{p(p-1)}{2\alpha ^{3}}w^{p-2}y^{4}w_{y}^{3}-\frac{(p+1)}{2\alpha ^{3}}w^{p-1}y^{3}w_{y}^{2} \\ \quad {}+(\frac{\alpha ^{2}p^{2} -(p+1)^{2}}{\alpha ^{2}p^{2}}y+\frac{p+7}{2\alpha ^{3}p}-\frac{p^{2}+3p+3}{\alpha ^{3}p^{2}})w^{p} y^{2}w_{y} g \\ \quad {}+(\frac{p+5}{2\alpha ^{3}p^{2}}-\frac{1}{\alpha ^{3}p^{3}}-\frac{(p+1)(p+2)}{2\alpha p})w^{p+1}y=0\end{array} \) |