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

In this paper, we consider the generalized Camassa–Holm equation ut + 2kux - uxxt + au2ux=2uxuxx + uuxxx. Under substitution ξ 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 in?nity on ξ- 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.