On the study of solutions of Bogoyavlenskii equation via improved $G^{\prime }/G^2$ method and simplified $\tan (\varphi (\xi )/2)$ method

The Bogoyavlenskii equation is used to describe some kinds of waves on the sea surface and discussed by many researchers. Recently, the $G^{\prime }/G^2$ method and simplified $\tan (\frac{\varphi (\xi )}{2})$ method are introduced to find novel solutions to differential equations. To the best of our knowledge, the Bogoyavlenskii equation has not been investigated by these two methods. In this article, we applied these two methods to the Bogoyavlenskii equation in order to obtain the novel exact traveling wave solutions. Consequently, we found that some new rational functions, trigonometric functions, and hyperbolic functions can be the traveling wave solutions of this equation. Some of these solutions we obtained have not been reported in the former literature. Through comparison, we see that the two methods are more effective than the previous methods for this equation. In order to make these solutions more obvious, we draw some 3D and 2D plots of them.