Exact special solutions with solitary patterns for the nonlinear dispersive K(m,n) equations

In this paper, we study the genuinely nonlinear dispersive K(m,n) equation, ut−(um)x+(un)xxx=0, which exhibits solutions with solitary patterns. Exact solutions that create solitary patterns having cusps or infinite slopes are developed. The nonlinear equation K(m,n) is addressed for two different cases, namely when m=n=odd integer and when m=n=even integer. General formulas for the solutions of these cases of the K(m,n) equations are established.