Abstract
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 w r , 2 ( z ) and simply periodic solutions w s , 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. MSC: Primary 30D35; secondary 34A05. PACS Codes: 02.30.Jr; 02.70.Wz; 02.30.-f.
Highlights
Introduction and main resultsStudies of various physical structures of nonlinear evolution equations (NLEEqs) have attracted much attention in connection with the important problems that arise in scientific applications
It is shown that the complex method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics
The idea introduced in this paper can be applied to other nonlinear evolution equations
Summary
Introduction and main resultsStudies of various physical structures of nonlinear evolution equations (NLEEqs) have attracted much attention in connection with the important problems that arise in scientific applications. We employ the complex method to obtain all meromorphic exact solutions of the complex equation ( ) first, combining the transform (TWT) to find all exact traveling wave solutions of the KdV-mKdV equation. Substituting the transform (TWT) into all meromorphic solutions w(z) of ( ) gives all exact traveling wave solutions of the KdV-mKdV equations.
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