All exact traveling wave solutions of the combined KdV-mKdV equation

Huang, Y. and Wu, Y.H. and Meng, F. and Yuan, W. 2014. All exact traveling wave solutions of the combined KdV-mKdV equation. Advances in Difference Equations. 2014: Article ID 261.

Source Title

Advances in Difference Equations

DOI

10.1186/1687-1847-2014-261

ISSN

1687-1839

School

Department of Mathematics and Statistics

Remarks

This open access article is distributed under the Creative Commons license http://creativecommons.org/licenses/by/4.0/

URI

http://hdl.handle.net/20.500.11937/7341

Collection

Curtin Research Publications

Abstract

In this article, we employ the complex method to obtain all meromorphic solutions of complex combined Korteweg-de Vries-modified Korteweg-de Vries equation (KdV-mKdV equation) at first, then we find all exact traveling wave solutions of the combined KdV-mKdV equation. The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic exact traveling wave solutions of the combined KdV-mKdV equation are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions wr,2(z)wr,2(z) and simply periodic solutions ws,2(z)ws,2(z) such that they are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role in finding exact solutions in mathematical physics. We also give some computer simulations to illustrate our main results.