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

Meng, Fanning; Gu, Yongyi

doi:https://doi.org/10.1155/2020/5740310

Journal of Function Spaces

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2020

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Research Article | Open Access

Volume 2020 | Article ID 5740310 | https://doi.org/10.1155/2020/5740310

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

Fanning Meng¹and Yongyi Gu^2,3

Guest Editor: Chuanjun Chen

Received12 May 2020

Accepted05 Jun 2020

Published22 Jun 2020

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.

1. 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 [1–8]. The investigation of NLDEs is helpful for well understanding of nonlinear physical phenomena [9–16]. Numerous methods have been developed for seeking traveling wave exact solutions to NLDEs, such as sine-Gordon 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].

In recent years, Yuan et al. [27] introduced an efficient method named complex method to get exact solutions for NLDEs. The complex method is developed by complex analysis and complex differential equations. More details about the complex method can be found in [28–34]. In this work, we will utilize the complex method to achieve exact solutions of the following two (3+1)-dimensional NLDEs.

The (3+1)-dimensional BKP equation [35] is given by where is a constant.

The (3+1)-dimensional gSW equation [36] is given by

Class W consists of elliptic function or their degeneration. Substituting traveling wave transform into Eq. (1), and then integrating it we get where is the integration constant.

Theorem 1. If , then meromorphic solutions of Eq. (4) belong to class W and Eq. (4) has the following solutions where are the integral constants. (i)The rational function solutionswhere . (ii)The simply periodic solutionswhere , . (iii)The elliptic function solutionswhere , , and are arbitrary.

Substituting traveling wave transform into Eq. (2), and then integrating it we get where λ is the integration constant.

Theorem 2. If , then meromorphic solutions of Eq. (9) belong to the class W and Eq. (9) has the following solutions where are the integral constants. (i)The rational function solutionswhere . (ii)The simply periodic solutionswhere , .. (iii)The elliptic function solutionswhere , , and is arbitrary.

2. Preliminaries

Set , , , , and

The degree of is defined as . The differential polynomial is given by where is a finite index set, then is the degree of .

Considering the following equation: where , and are constants.

Assume that meromorphic solutions w of Eq. (13) have at least one pole and let p, q ∈ N. Substitute the Laurent series into Eq. (15) to determine distinct Laurent principal parts then, Eq. (15) is said to satisfy weak condition.

It is know that Weierstrass elliptic function has double periods and satisfies:

Weierstrass zeta function is a meromorphic function which satisfies

These two Weierstrass functions have the following addition formulas:

Eremenko et al. [37] had investigated the following -order Briot-Bouquet equation (BBEq) in which m ∈ N, and P_j(U) are constant coefficient polynomials.

Lemma 1 [38–40]. Let m, n, p, s ∈ N, degP(U,U^(m)) < n. If the m-order BBEq satisfies weak condition; then, meromorphic solutions belong to class W. Assume that some values of parameters such solutions exist; then, other meromorphic solutions should form 1 parameter family . In addition, each elliptic solution with a pole at is. where are determined by (16), and .
Each rational function solution is which contains distinct poles of multiplicity .
Each simply periodic solution is a rational function of , that is which contains distinct poles of multiplicity q.

3. Proofs of Main Results

Proof of Theorem 1. Let , then Eq. (4) becomes Substituting (16) into Eq. (4), we have and is an arbitrary constant. Thus, Eq. (26) is a second-order BBEq as well as satisfies weak condition. Therefore, by Lemma 1, we know that the meromorphic solutions of Eq. (26) belong to class W.

From (23) of Lemma 1, we have the form of elliptic solutions of Eq. (26) with pole at .

Put into Eq. (26) to yield where and g₃ is arbitrary.

Therefore, the elliptic solutions of Eq. (26) with arbitrary pole are where .

Therefore, the solutions of Eq. (4) are where , , is the integral constant, and and are arbitrary.

By (24), we infer that the indeterminant rational solutions of Eq.(26) are with pole at .

Substitute into Eq. (26) to yield where .

So the rational solutions of Eq. (26) with arbitrary pole are

Therefore, the solutions of Eq. (4) are where is the integral constant, .

Let . To obtain simply periodic solutions, we insert into Eq. (26) and get

Substituting into the Eq.(35), we obtain that where .

Substituting into Eq. (36) and Eq. (37) yields simply periodic solutions to Eq. (26) with pole at where .

So simply periodic solutions of Eq. (4) with arbitrary pole are and where .

Therefore, the solutions of Eq. (4) are and where and are the integral constants, , .

Proof of Theorem 2. Let , then Eq. (9) becomes Substituting (16) into Eq.(9), we have , and is an arbitrary constant. Thus, Eq. (43) is a second-order BBEq as well as satisfies weak condition. Therefore, by Lemma 1, we know that the meromorphic solutions of Eq. (43) belong to class W.

From (23) of Lemma 1, we have the form of elliptic solutions of Eq. (43) with pole at .

Put into Eq. (43) to yield where and g₃ is arbitrary.

Therefore, the elliptic solutions of Eq. (43) with arbitrary pole are where .

Therefore, the solutions of Eq. (9) are where , , is the integral constant, and and are arbitrary.

By (24), we infer that the indeterminant rational solutions of Eq. (43) are with pole at .

Substitute into Eq. (43) to yield where .

So the rational solutions of Eq. (43) with arbitrary pole are

Therefore, the solutions of Eq. (9) are where is the integral constant, .

Let . To obtain simply periodic solutions, we insert into Eq. (43) and get

Substituting into the Eq. (35), we obtain that where .

Substituting into Eq. (53) and Eq. (54) yields simply periodic solutions to Eq. (43) with pole at where .

So simply periodic solutions of Eq. (9) with arbitrary pole are and where .

Therefore, the solutions of Eq. (9) are and where and are the integral constants, , .

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflict of interest.

Acknowledgments

This research is supported by the NSFC (11701111, 11901111) and Visiting Scholar Program of Chern Institute of Mathematics.

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Copyright

Copyright © 2020 Fanning Meng and Yongyi Gu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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