On the Exact Solutions of Two (3+1)-Dimensional Nonlinear Differential Equations

Fanning Meng,Yongyi Gu

doi:10.1155/2020/5740310

Abstract

In this article, exact solutions of two (3+1)-dimensional nonlinear differential equations are derived by using the complex method. We change the (3+1)-dimensional B-type Kadomtsev-Petviashvili (BKP) equation and generalized shallow water (gSW) equation into the complex differential equations by applying traveling wave transform and show that meromorphic solutions of these complex differential equations belong to class W, and then, we get exact solutions of these two (3+1)-dimensional equations.

Highlights

Nonlinear differential equations (NLDEs) play an important role in the research of nonlinear science, which has attracted a lot of attentions of the researchers [1,2,3,4,5,6,7,8]
The investigation of NLDEs is helpful for well understanding of nonlinear physical phenomena [9,10,11,12,13,14,15,16]
Numerous methods have been developed for seeking traveling wave exact solutions to NLDEs, such as sineGordon expansion method [17], Kudryashov method [18], modified simple equation method [19], Jacobi elliptic function expansion [20], exp(−ψ(z))-expansion method [21, 22], modified extended tanh method [23, 24], generalized (G'/G) expansion method [25], and improved F-expansion method [26]

Summary

Introduction and Main Results

Nonlinear differential equations (NLDEs) play an important role in the research of nonlinear science, which has attracted a lot of attentions of the researchers [1,2,3,4,5,6,7,8]. We will utilize the complex method to achieve exact solutions of the following two (3+1)-dimensional NLDEs. x ux + uy + us t − ðuxx + ussÞ = 0, ð1Þ where θ is a constant. If θn1n2 ≠ 0, meromorphic solutions w of Eq (4) belong to class W and Eq (4) has the following solutions where ciði = 1, 2, 3, 4Þ are the integral constants. If m1m2 ≠ 0, meromorphic solutions w of Eq (9) belong to the class W and Eq (9) has the following solutions where ciði = 1, 2, 3, 4Þ are the integral constants.

Preliminaries

Proofs of Main Results