Exact Traveling Wave Solutions of Explicit Type, Implicit Type, and Parametric Type for K ( m , n ) Equation

Abstract

By using the integral bifurcation method, we study the nonlinear $K\left(m,n\right)$ equation for all possible values of $m$ and $n$. Some new exact traveling wave solutions of explicit type, implicit type, and parametric type are obtained. These exact solutions include peculiar compacton solutions, singular periodic wave solutions, compacton-like periodic wave solutions, periodic blowup solutions, smooth soliton solutions, and kink and antikink wave solutions. The great parts of them are different from the results in existing references. In order to show their dynamic profiles intuitively, the solutions of $K\left(n,n\right)$, $K\left(2n-1,n\right)$, $K\left(3n-2,n\right)$, $K\left(4n-3,n\right)$, and $K\left(m,1\right)$ equations are chosen to illustrate with the concrete features.