The Hamiltonian Structures of μ-Equations Related to Periodic Peakons

The hierarchies of the μ-Camassa–Holm, two-component μ-Camassa–Holm and μ-modified Camassa–Holm equations are constructed from bi-Hamiltonian structures of the Korteweg-de Vries equation, the Ito equation and the modified Korteweg-de Vries equation. The key Hamiltonian operator that includes μ is ?_x μ??³_x, and the inner product used to define the Jacobi identity of the Hamiltonian operators is defined on the unit circle S¹.