Orbital Stability of Periodic Peakons to a Generalized μ-Camassa-Holm Equation

In this paper, we study the orbital stability of the periodic peaked solitons of the generalized mu-Camassa-Holm equation with nonlocal cubic and quadratic nonlinearities. The equation is a mu-version of a linear combination of the Camassa-Holm equation and the modified Camassa-Holm equation. It is also integrable with the Lax-pair and bi-Hamiltonian structure and admits the single peakons and multi-peakons. By constructing an inequality related to the maximum and minimum of solutions with the conservation laws, we prove that, even in the case that the Camassa-Holm energy counteracts in part the modified Camassa-Holm energy, the shapes of periodic peakons are still orbitally stable under small perturbations in the energy space.