Blow-up criteria and periodic peakons for a two-component extension of the μ-version modified Camassa–Holm equation**Supported by the National Nature Science Foundation of China (Grant Nos. 11871471, 11931017), Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (Grant No. 2019L0531), Shanxi Province Science Foundation for Youths (Grant No. 201901D211274), and the Fund for Shanxi ‘1331KIRT’.

Abstract

Some two-component extensions of the modified μ-Camassa–Holm equation are proposed. We show that these systems admit Lax pairs and bi-Hamiltonian structures. Furthermore, we consider the blow-up phenomena for one of these extensions (2μmCH), and the periodic peakons of this system are derived.