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 uₜ+a(uᵐ)ₓ+(uⁿ)x x x=0 by using thebifurcation method and numerical simulation approach of dynamicalsystems. We obtain some new results as follows: (i) For$K(2, 2)$ equation, we extend the expressions of the smoothperiodic wave solutions and obtain a new solution, theperiodic-cusp wave solution. Further, we demonstrate that theperiodic-cusp wave solution may become the peakon wave solution. (ii) For $K(3, 2)$ equation, we extend the expression of theelliptic smooth periodic wave solution and obtain a new solution,the elliptic periodic-blow-up solution. From the limit forms ofthe two solutions, we get other three types of new solutions, thesmooth solitary wave solutions, the hyperbolic 1-blow-up solutionsand the trigonometric periodic-blow-up solutions. (iii) For$K(4, 2)$ equation, we construct two new solutions, the 1-blow-upand 2-blow-up solutions.