A family of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si542.svg" display="inline" id="d1e2752"> <mml:mi>μ</mml:mi> </mml:math> -Camassa–Holm-type equations with peaked solutions

In this paper, we present a general family of nonlinear dispersive wave equations, which can be regarded as a nonlocal counterpart of the f g -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 μ -Camassa–Holm and modified μ -Camassa–Holm equations are then proposed respectively, which preserve the Hamiltonian structure shared by the μ -Camassa–Holm and modified μ -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.