Nonlocal Symmetries of the Camassa-Holm Type Equations

Abstract

A class of nonlocal symmetries of the Camassa-Holm type equations with bi-Hamiltonian structures, including the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation and Degasperis-Procesi equation, is studied. The nonlocal symmetries are derived by looking for the kernels of the recursion operators and their inverse operators of these equations. To find the kernels of the recursion operators, the authors adapt the known factorization results for the recursion operators of the KdV, modified KdV, Sawada-Kotera and Kaup-Kupershmidt hierarchies, and the explicit Liouville correspondences between the KdV and Camassa-Holm hierarchies, the modified KdV and modified Camassa-Holm hierarchies, the Novikov and Sawada-Kotera hierarchies, as well as the Degasperis-Procesi and Kaup-Kupershmidt hierarchies.

Keywords

Nonlocal symmetry / Recursion operator / Camassa-Holm equation / Modified Camassa-Holm equation / Novikov equation / Degasperis-Procesi equation / Liouville correspondence