Blow-Up Solutions and Peakons to a Generalized μ-Camassa-Holm Integrable Equation

Considered here is a generalized mu-type integrable equation, which can be regarded as a generalization to both the mu-Camassa-Holm and modified mu-Camassa-Holm equations. It is shown that the proposed equation is formally integrable with the Lax-pair and the bi-Hamiltonian structure and its scale limit is an integrable model of hydrodynamical systems describing short capillary-gravity waves. Local well-posedness of the Cauchy problem in the suitable Sobolev space is established by the viscosity method. Existence of peaked traveling wave solutions and formation of singularities of solutions for the equation are investigated. It is found that the equation admits single and multi-peaked traveling wave solutions. The effects of varying mu-Camassa-Holm and modified mu-Camassa-Holm nonlocal nonlinearities on blow-up criteria and wave breaking are illustrated in detail. Our analysis relies on the method of characteristics and conserved quantities and is proceeded with a priori differential estimates.