Blow-up solutions and peakons to a generalized $\mu$-Camassa-Holm integrable equation

Abstract

Consideration 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 a single peaked soliton and multi-peakon 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.

36 pages

References in corpus (5)

• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• Inverse Scattering Transform for the Camassa-Holm equation
• Stability of multipeakons
• Integrable evolution equations on spaces of tensor densities and their peakon solutions
• Generalised Fourier transform for the Camassa-Holm hierarchy

Cited by in corpus (3)

• Orbital stability of periodic peakons for a new higher-order $\mu$-Camassa-Holm equation
• Well-posedness and peakons for a higher-order $\mu$-Camassa-Holm equation
• Global conservative solution for the periodic $\mu$-Camassa-Holm equation