Periodic blow-up solutions and their limit forms for the generalized Camassa–Holm equation

In this paper, we consider the generalized Camassa–Holm equationut+2kux-uxxt+au2ux=2uxuxx+uuxxx.ut+2kux-uxxt+au2ux=2uxuxx+uuxxx.Under substitution ξ=x-ctξ=x-ct, some new explicit periodic wave solutions and their limit forms are presented through some special phase orbits. These periodic wave solutions tend to infinity on ξ-uξ-u plane periodically. Thus we call them periodic blow-up solutions. To our knowledge, such periodic blow-up solutions have not been found in any other equations.