Theory and Modern Applications
Table 7 Order reduction of Eq. (1)
From: Lie symmetry and μ-symmetry methods for nonlinear generalized Camassa–Holm equation
Symmetry of exponential type | Order reduction |
|---|---|
\(V=\exp ({\int -\frac{G_{x}(x,t)}{G(x,t)}\,dx-\frac{G_{t}(x,t)}{G(x,t)}\,dt} )X\) | \(-G(x,t)u_{x}=0\) |
\(V=\exp ({\int -\frac{G_{x}(x,t)}{G(x,t)}\,dx-(\frac{G_{t}(x,t)}{G(x,t)}-\frac{p}{p t-c_{1}})\,dt} )X\) | \(\frac{-G(x,t)}{p t-c_{1}} ((p t-c_{1})u_{t}+u )=0\) |
\(V=\exp { (\int -\frac{G_{x}(x,t)}{G(x,t)}\,dx-(\frac{G_{t}(x,t)}{G(x,t)} -\frac{c_{1}}{c_{1}t+c_{2}})\,dt} )X\) | \(\begin{array}[t]{l} \frac{-G(x,t)}{(c_{1}t+c_{2})p} ((c_{1}t+c_{2})pu_{t} \\ \quad {}+c_{1}u-pu_{x} )=0\end{array}\) |