Searching for Analytical Solutions of the (2+1)-Dimensional KP Equation by Two Different Systematic Methods

Fanning Meng,Yongyi Gu

doi:10.1155/2019/9314693

Abstract

In this paper, we derive analytical solutions of the (2+1)-dimensional Kadomtsev-Petviashvili (KP) equation by two different systematic methods. Using the exp⁡(-ψ(z))-expansion method, exact solutions of the mentioned equation including hyperbolic, exponential, trigonometric, and rational function solutions have been obtained. Based on the work of Yuan et al., we proposed the extended complex method to seek exact solutions of the (2+1)-dimensional KP equation. The results demonstrate that the applied methods are efficient and direct methods to solve the complex nonlinear systems.

Highlights

The (2+1)-dimensional KP equation [1] is given by uxt − 6u2x − 6uuxx + uxxxx + 3uyy = 0, (1)which is a universal nonlinear integrable system in two spatial and one temporal coordinates and can be utilized to describe the law of motion of water waves in (2+1)-dimensional spaces and plasmas in magnetic fields [2,3,4]
Based on the work of Yuan et al, we proposed the extended complex method to seek exact solutions of the (2+1)-dimensional KP equation
In the study of water waves, this equation appears in the description of a tsunami wave travelling in the inhomogeneous zone on the bottom of the ocean [2], and it appears in the study of nonlinear ion acoustic waves in magnetized dusty plasma [4]

Summary

Introduction

Based on the work of Yuan et al, we introduce the extended complex method to seek exact solutions of NLDEs which are not Briot-Bouquet equations or do not satisfy ⟨p, q⟩ condition. Two different systematic methods which are the exp(−ψ(z))-expansion method and extended complex method are employed to search analytical solutions of the (2+1)-dimensional KP equation. We solve the algebraic equations to obtain the values of Bn ≠ 0, γ, μ and we put these values into (5) along with (7)-(13) to finish the determination of the solutions for the given PDE. 3. Application of the exp (−ψ(z))-Expansion Method to the (2+1)-Dimensional KP Equation. Using (7) to (13) into (19), respectively, we obtain exact solutions of the (2+1)-dimensional KP equation as follows. We give some computer simulations to show the properties of the solutions

The Extended Complex Method

Comparisons and Conclusions