Theory and Modern Applications
Table 5 Conservation laws for Eq. (1)
From: Lie symmetry and μ-symmetry methods for nonlinear generalized Camassa–Holm equation
Λ | |
|---|---|
\(\Lambda _{1}=1\) | \(\Upsilon _{u}^{(x)}=-\frac{1}{2}(p+1)(p+2)u^{p+1}+\frac{1}{2}(p)(p+1)u^{p-1}u_{x}^{2} +(p+1)u^{p}u_{xx}-\frac{2}{3}u_{xt}\) |
\(\Upsilon _{u}^{(t)}=u- \frac{1}{3}u_{xx} \) | |
\(\rho =-\frac{1}{2}(p+2)u^{p+1}+\frac{1}{2}(p)u^{p-1}u_{x}^{2} + u^{p}u_{xx}-\frac{2}{3}u_{xt} \) | |
\(\varrho =u- \frac{1}{3}u_{xx} \) | |
\(\Lambda _{2}=U\) | \(\Upsilon _{u}^{(x)}=-\frac{1}{2}(p+1)(p+2)u^{p+2}+\frac{1}{2}(p-1)(p+2)u^{p}u_{x}^{2} + (p+2)u^{p+1}u_{xx}-\frac{4}{3}uu_{xt}+\frac{2}{3}u_{x}u_{t}\) |
\(\Upsilon _{u}^{(t)}=u^{2}+\frac{1}{3}u_{x}^{2}-\frac{2}{3}u_{xx}u\) | |
\(\rho =-\frac{1}{2}(p+1)(p+2)u^{p+2}+\frac{1}{2}(p-1)(p)u^{p-1}u_{x}^{3} + u^{p+1}u_{xx}-\frac{2}{3}uu_{xt}+\frac{1}{3}u_{x}u_{t} \) | |
\(\varrho =\frac{1}{2}u^{2}+\frac{1}{6}u_{x}^{2}-\frac{1}{3}u_{xx}u\) |