Nonlocal symmetries and conservation laws of nonlocal Camassa-Holm type equations

Abstract

The two-component µ-Camassa-Holm equation, the µ-Camassa-Holm equation and the modified µ-Camassa-Holm equation are three nonlocal integrable models. In this paper, it is shown that the two-component µ-Camassa-Holm equation and the µ-Camassa-Holm equation with a linear dispersion admit a kind of nonlocal symmetries, and the modified µ-Camassa-Holm equation does not have such kind of nonlocal symmetry. An number of conservation laws to the modified µ-Camassa-Holm equation and µ-Camassa-Holm equation are obtained.