A family of μ-Camassa–Holm-type equations with peaked solutions

Abstract

In this paper, we present a general family of nonlinear dispersive wave equations, which can be regarded as a nonlocal counterpart of the $fg$-family. We first show that the family of equations admits multi-peaked and single peaked solutions under certain conditions on two arbitrary functions. As typical subfamilies of the equations with peaked solutions, two generalized versions of the $\mu$-Camassa–Holm and modified $\mu$-Camassa–Holm equations are then proposed respectively, which preserve the Hamiltonian structure shared by the $\mu$-Camassa–Holm and modified $\mu$-Camassa–Holm equations. The peaked solutions and higher-order conserved densities are derived from these generalized equations. Furthermore, the interactions of two-peaked solutions are exhibited. It demonstrates that the higher-order nonlinearities have an impact on interactions of peaked solutions.