Theory and Modern Applications
Table 6 μ-symmetry of Eq. (1)
From: Lie symmetry and μ-symmetry methods for nonlinear generalized Camassa–Holm equation
N(x), P(t) | \(\lambda _{1}\), \(\lambda _{2}\) | μ-symmetry |
|---|---|---|
N(x)=0 | \(\lambda _{1}=-\frac{G_{x}(x,t)}{G(x,t)}\) | \(X=G(x,t)\partial _{x}\) |
P(t)=0 | \(\lambda _{2}=-\frac{G_{t}(x,t)}{G(x,t)}\) | |
N(x)=0 | \(\lambda _{1}=-\frac{G_{x}(x,t)}{G(x,t)}\) | \(X=G(x,t) (\partial _{t}+\frac{u}{c_{1}-pt} \partial _{u} )\) |
\(h(t)=\frac{p}{p t-c_{1}}\) | \(\lambda _{2}=-\frac{G_{t}(x,t)}{G(x,t)}+\frac{p}{p t-c_{1}}\) | |
N(x)=0 | \(\lambda _{1}=-\frac{G_{x}(x,t)}{G(x,t)}\) | \(\begin{array}[t]{l} X=G(x,t) (\frac{-1}{c_{1}t+c_{2}}\partial _{x} \\ \hphantom{X={}}{}+\partial _{t}-\frac{c_{1}u}{(c_{1}t+c_{2})p}\partial _{u} )\end{array}\) |
\(P(t)=\frac{c_{1}}{c_{1}t+c_{2}}\) | \(\lambda _{2}=-\frac{G_{t}(x,t)}{G(x,t)}+\frac{c_{1}}{c_{1}t+c_{2}}\) |