Computation of $\mu$-symmetry and $\mu$-conservation law for the Camassa-Holm and Hunter-Saxton equations

Abstract

This work is intended to compute the $\mu$-symmetry and $\mu$-conservation laws for the Cammasa-Holm (CH) equation and the Hunter-Saxton (HS) equation. In other words, $\mu$-symmetry, $\mu$-symmetry reduction, variational problem, and $\mu$-conservation laws for the CH equation and the HS equation are provided. Since the CH equation and the HS equation are of odd order, they do not admit a variational problem. First we obtain $\mu$-conservation laws for both of them in potential form because they admit a variational problem and then using them, we obtain $\mu$-conservation laws for the CH equation and the HS equation.

Keywords

• symmetry
• $\mu$-symmetry
• conservation law
• $\mu$-conservation law
• variational problem
• order reduction