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

TL;DR

This paper introduces a generalized $bla$-type integrable equation extending the $bla$-Camassa-Holm family, demonstrating its integrability, well-posedness, and the existence of peakon solutions, with implications for hydrodynamical wave models.

Contribution

It presents a new integrable equation generalizing the $bla$-Camassa-Holm models, analyzes its solutions, and explores wave-breaking phenomena and peakon solutions.

Findings

The equation is formally integrable with Lax pair and bi-Hamiltonian structure.

Existence of single and multi-peakon solutions is established.

Wave-breaking and blow-up criteria depend on the nonlocal nonlinearities.

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.