Stability of periodic peakons for the modified μμ-Camassa–Holm equation

Considered herein is the dynamical stability of periodic peaked solitons for the modified μμ-Camassa–Holm equation with the cubic nonlinearity. The equation is a μμ-version of the modified Camassa–Holm equation and is integrable with the Lax-pair and bi-Hamiltonian structure. The equation admits the periodic peakons. It is shown that the periodic peakons are dynamically stable under small perturbations in the energy space by finding inequalities related to the first three conservation laws with global maximum and minimum of the solution.

► A new integrable system which is a μμ-version of the modified Camassa–Holm equation is introduced.
► It is shown that a scale limit of the modified μμ-Camassa–Holm is the so-called short pulse equation.
► Lyapunov functionals are obtained corresponding to the peakons.
► Peakons are orbitally stable under small perturbations in the energy space.