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+ au~2ux=2uxuxx+uuxxxUnder 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 infinity 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.