Peakon, loop and solitary travelling wave solutions for the general modified CH–DP equation

Travelling wave solutions for the general modified CH–DP equation ut − uxxt + αu2ux − βuxuxx = uuxxx are developed. By using the dynamical system method, a peakon and a dark soliton are found to coexist for the same wave speed. Exact explicit blow-up solutions are given. By using numerical simulation, a loop solution for a special case is discussed.