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May  2010, 13(3): 633-646. doi: 10.3934/dcdsb.2010.13.633

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

Rui Liu 1,

1. 

School of Mathematical Sciences, Peking University, Beijing 100871

Received  May 2009 Revised  December 2009 Published  February 2010

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.
Keywords: bifurcation method, $K(m, blow-up solutions., n)$ equation, new explicit solutions.
Mathematics Subject Classification: Primary: 35B65; Secondary: 34A20, 34C35, 58F05, 76B2.
Citation: Rui Liu. Some new results on explicit traveling wave solutions of $K(m, n)$ equation. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 633-646. doi: 10.3934/dcdsb.2010.13.633
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