On the singularity formation for a class of periodic higher-order Camassa–Holm equations

Abstract The primary goal of this paper is to understand how higher-order nonlinearities affect the dispersive dynamics. As a prototype, we study a class of higher-order Camassa–Holm equations which can be viewed as a generalization of the Camassa–Holm equation. We develop a delicate analysis to investigate the formation of singularities and provide sufficient conditions on initial data that lead to the finite time blow-up of the second-order derivative of the solution.