Orbital stability of periodic peakons for a new higher-order $\mu$-Camassa-Holm equation

Consideration here is a higher-order $\mu$-Camassa-Holm equation, which is a higher-order extension of the $\mu$-Camassa-Holm equation and retains some properties of the $\mu$-Camassa-Holm equation and the modified $\mu$-Camassa-Holm equation. By utilizing the inequalities with the maximum and minimum of the solution related to the first three conservation laws, we establish that the periodic peakons of this equation are orbitally stable under small perturbations in the energy space.