Extended direct method and new similarity solutions of Kadomtsev–Petviashvili equation

In this study, we apply the extended direct method to seek new similarity solutions and reduction equations of the (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation. Through this method, we can reduce the KP equation to a system of ordinary differential equations, and we divide the discussion into four cases via calculation. In some cases, new similarity reductions and exact solutions are obtained, such as elliptic function solutions, polynomial solutions. In other cases, similarity solutions are consistent with the present solutions. Futhermore, a similarity solution we get has more clear and concise form than that obtained by the generalized Clarkson–Kruskal direct method. Therefore, it demonstrates that our study is correct and more efficient than before which can also be applied to the other nonlinear physical models.