Well-posedness, wave breaking and peakons for a modified μ-Camassa-Holm equation

Considered herein is a modified periodic Camassa-Holm equation with cubic nonlinearity which is called the modified μ-Camassa-Holm equation. The proposed equation is shown to be formally integrable with the Lax pair and bi-Hamiltonian structure. Local well-posedness of the initial-value problem to the modified μ-Camassa-Holm equation in the Besov space is established. Existence of peaked traveling-wave solutions and formation of singularities of solutions for the equation are then investigated. It is shown that the equation admits a single peaked soliton and multi-peakon solutions with a similar character of the μ-Camassa-Holm equation. Singularities of the solutions can occur only in the form of wave-breaking, and several wave-breaking mechanisms for solutions with certain initial profiles are described in detail.