All Traveling Wave Exact Solutions of the Kawahara Equation Using the Complex Method

Abstract

In this article, we prove that the ⟨p,q⟩ condition holds, first by using the Fuchs index of the complex Kawahara equation, and then proving that all meromorphic solutions of complex Kawahara equations belong to the class W. Moreover, the complex method is employed to get all meromorphic solutions of complex Kawahara equation and all traveling wave exact solutions of Kawahara equation. Our results reveal that all rational solutions ur(x+νt) and simply periodic solutions us,1(x+νt) of Kawahara equation are solitary wave solutions, while simply periodic solutions us,2(x+νt) are not real-valued. Finally, computer simulations are given to demonstrate the main results of this paper. At the same time, we believe that this method is a very effective and powerful method of looking for exact solutions to the mathematical physics equations, and the search process is simpler than other methods.

Keywords

• Kawahara equation
• complex method
• exact solution
• elliptic function