Some new results on explicit traveling wave solutions of $K(m, n)$ equation

In this paper, we investigate the traveling wave solutions of $K(m, n)$ equation $ u_t+a(u^m)_{x}+(u^n)_{x x x}=0$ by using the bifurcation method and numerical simulation approach of dynamical systems. We obtain some new results as follows: (i) For $K(2, 2)$ equation, we extend the expressions of the smooth periodic wave solutions and obtain a new solution, the periodic-cusp wave solution. Further, we demonstrate that the periodic-cusp wave solution may become the peakon wave solution. (ii) For $K(3, 2)$ equation, we extend the expression of the elliptic smooth periodic wave solution and obtain a new solution, the elliptic periodic-blow-up solution. From the limit forms of the two solutions, we get other three types of new solutions, the smooth solitary wave solutions, the hyperbolic 1-blow-up solutions and the trigonometric periodic-blow-up solutions. (iii) For $K(4, 2)$ equation, we construct two new solutions, the 1-blow-up and 2-blow-up solutions.