Complexity
Volume 2019 (2019), Article ID 9314693, 11 pages
https://doi.org/10.1155/2019/9314693

Research Article

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

Yongyi Gu¹ and Fanning Meng²

¹School of Statistics and Mathematics, Guangdong, University of Finance and Economics, Guangzhou, 510320, China
²School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

Correspondence should be addressed to Yongyi Gu; gdguyongyi@163.com and Fanning Meng; mfnfdbx@163.com

Received 29 January 2019; Revised 3 June 2019; Accepted 28 July 2019; Published 20 August 2019

Academic Editor: Irene Otero-Muras

Copyright © 2019 Yongyi Gu and Fanning Meng. 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.

Abstract.

In this paper, we derive analytical solutions of the (2+1)-dimensional Kadomtsev-Petviashvili (KP) equation by two different systematic methods. Using the -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.

1. Introduction

The (2+1)-dimensional KP equation [1] is given by

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 [2–4]. For example, 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 also appears in the study of nonlinear ion acoustic waves in magnetized dusty plasma [4]. Over the past few years, many research results for the (2+1)-dimensional KP equation have been generated. As to this equation, traveling wave solutions [5, 6], rogue wave, and a pair of resonance stripe solitons [7] are discovered. Symmetry reductions [8] and conservation laws [9] are also investigated. Using the Hirota bilinear form of the (2+1)-dimensional KP equation, mixed lump-kink solutions are presented under the help of Maple [10]. By the positive quadratic function and exponential function, rational lump solutions and line soliton pairs to the (2+1)-dimensional KP equation are established [11].

It is well known that nonlinear differential equations (NLDEs) are universally applied in plasma physics, solid state physics, nonlinear optics, fluid dynamics, biology, chemistry, etc. For instance, the singular behaviors [12, 13] and impulsive phenomena [14, 15] often show some blow-up properties [16, 17] which happen in lots of complex physical processes. In order to solve various differential equations, some analytical tools as well as symbolic calculation techniques were established, such as fixed-point theorems [18, 19], variational methods [20, 21], topological degree method [22–25], iterative techniques [26, 27], bilinear method [28–31], modified simple equation method [32], -expansion method [33–38], Lie group method [39, 40], and complex method [41–50].

The -expansion method is an efficient method for finding the exact solutions of NLDEs. Many researchers, such as Roshid, Khan, and Jafari, have used this method to study NLDEs [35–37]. The complex method, proposed by Yuan et al. [41, 42], is established via complex differential equations and complex analysis. It is a useful tool to find exact solutions of NLDEs which are Briot-Bouquet equations or satisfy condition [41]. 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 condition. The extended complex method can solve more differential equations in mathematical physics than the complex method. In this paper, two different systematic methods which are the -expansion method and extended complex method are employed to search analytical solutions of the (2+1)-dimensional KP equation. Computer simulations are given to illustrate our main results. Comparisons and conclusions are presented in the last section.

2. The -Expansion Method

Consider a nonlinear PDE as follows:

where is a polynomial consisting of the unknown function , the partial derivatives of , the highest-order partial derivatives of , and some nonlinear terms.

Step 1. Substitute traveling wave transformation,

into (2) to convert it to the ODE,

where is a polynomial of and its derivatives.

Step 2. Suppose that (4) has the following exact solutions:

where are constants to be determined later, such that and satisfies the ODE as follows:

Equation (6) has the solutions as follows.
When , ,

When , ,

When , , ,

When , , ,

When , , ,

where is an arbitrary constant and are constants in (7)- (13). We determine the positive integer through considering the homogeneous balance between the highest-order derivatives and nonlinear terms of (4).

Step 3. Inserting (5) into (4) and then considering the function yields a polynomial of . Let the coefficients of the same power about equal zero; then, we get a set of algebraic equations. We solve the algebraic equations to obtain the values of and then 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 -Expansion Method to the (2+1)-Dimensional KP Equation

Substitute

into (1), and we get

Take the homogeneous balance between and in (15) to yield

where and and are constants.

Substituting into (15) and equating the coefficients about to zero, we get

Solving the above algebraic equations yields

where and are arbitrary constants.

We substitute (18) into (16), and then

Using (7) to (13) into (19), respectively, we obtain exact solutions of the (2+1)-dimensional KP equation as follows.

When , ,

When , ,

When , , ,

When , , ,

When , , ,

The properties of the solutions are shown in Figures 1–5.

Figure 1: The 3D and 2D surfaces of by considering the values , , , , , , , and for the 2D graphic.

Figure 2: The 3D and 2D surfaces of by considering the values , , , , , , , and for the 2D graphic.

Figure 3: The 3D and 2D surfaces of by considering the values , , , , , , , and for the 2D graphic.

Figure 4: The 3D and 2D surfaces of by considering the values , , , , , , , and for the 2D graphic.

Figure 5: The 3D and 2D surfaces of by considering the values , , , , , , , and for the 2D graphic.

Remark. The -expansion method is an efficient method. Khan and Akbar [6] used this method to obtain the exact solutions of the KP equation. We still give the details of solving KP equation with the -expansion method in this paper for several reasons. First of all, we consider fourth-order ODE after the reduction instead of the second-order ODE of [6], so it is also a good example to show the use of the -expansion method. Secondly, we obtain more generalized results compared with [6]. If we take , , and , then the results of [6] can be obtained. Thirdly, we give some computer simulations to show the properties of the solutions.

4. The Extended Complex Method

Step 1. Substituting the transform , into the given PDE yields

Step 2. Determine the weak condition.
Let , and suppose that the meromorphic solutions of (25) have at least one pole. Substituting the Laurent series,

into (25), if it is determined distinct Laurent singular parts

then the weak condition of (25) holds.
Weierstrass elliptic function with double periods satisfies the equation as follows:

and has the following addition formula [51]:

Step 3. Substitute the indeterminate forms

into (25), respectively, to yield the systems of algebraic equations, and solve the algebraic equations to obtain elliptic function solutions, rational function solutions, and simply periodic solutions with the pole at , where are determined by (26), and , and , have distinct poles of multiplicity .

Step 4. Obtain the meromorphic solutions with arbitrary pole, and substitute the inverse transform into the meromorphic solutions to achieve the exact solutions to the original PDE.

5. Application of the Extended Complex Method to the (2+1)-Dimensional KP Equation

Inserting (26) into (15) yields , and then the weak condition of (15) holds.

By the weak condition and (30), we have the form of the elliptic solutions of (15):

with pole at .

Inserting into (15), we have

where

Equating the coefficients of all powers about in (34) to zero, we get a system of algebraic equations:

Solving the above equations, we obtain

and then

Therefore, the elliptic solutions of (15) with arbitrary pole are

where .

Apply the addition formula to , and then

where and and are arbitrary constants.

By the weak condition and (31), we have the indeterminate forms of rational solutions:

with pole at .

Inserting into (15), we have

where

Equating the coefficients of all powers about in (42) to zero, we get a system of algebraic equations:

Solving the above equations, we obtain

and then

Substitute into (15), and then

where ().

Substituting

into (47), we obtain that

where

Equating the coefficients of all powers about in (49) to zero, we get a system of algebraic equations:

Solving the above equations, we obtain

Therefore, simply periodic solutions to (15) with pole at are

Similar to , we substitute

into (47) to yield

and then

Substituting

into (47) to yield

then

By the above procedures, we collect meromorphic solutions of (15) with arbitrary pole as follows:

where and and are arbitrary constants.

The properties of the solutions are shown in Figures 6–8.

Figure 6: The 3D and 2D surfaces of by considering the values , , , , , and for the 2D graphic.

Figure 7: The 3D and 2D surfaces of by considering the values , , , , , , and for the 2D graphic.

Figure 8: The 3D and 2D surfaces of by considering the values , , , , , , and for the 2D graphic.

Remark. Based on the work of Yuan et al. [41, 42], we put forward the extended complex method for the first time. To our knowledge, the solutions obtained by this method have not been reported in former literature.

6. Comparisons and Conclusions

Borhanifar et al. [5] utilized the sine-cosine, the standard tanh, and the extended tanh methods to study the (2+1)-dimensional KP equation. We can observe that some important results to this equation have been obtained by using the above three methods. However, when we compare the results of this paper by applying two different systematic methods with the results of [5], the exponential function solutions and elliptic function solutions as a new aspect have been proposed to the literature. All these methods enrich the study of the (2+1)-dimensional KP equation.

The -expansion method allows us to express the exact solutions of NLDEs as a polynomial of , in which satisfies the ODE (6). We can determine the degree of the polynomial through the homogeneous balance and obtain the values of the undetermined coefficients of the polynomial via the calculations of computer software, and then we obtain the exact solutions. With this method, we obtained seven solutions to the mentioned equation in which rational solution is equivalent to if we consider .

With the extended complex method, we can derive meromorphic solutions of the differential equations which do not satisfy condition or are not Briot-Bouquet equation [41]. Therefore, more NLDEs in mathematical physics can be solved by the extended complex method. Using the indeterminate forms of the solutions, we are able to seek meromorphic solutions for the differential equation with a pole at , and then we can derive meromorphic solutions , , for the differential equation with an arbitrary pole.

In this article, we search for analytical solutions of the (2+1)-dimensional KP equation by two different systematic methods. By the -expansion method and extended complex method, we obtain five kinds of exact solutions. The results demonstrate that these two methods are efficient and direct methods allowing us to do complicated and tedious algebraic calculation.

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 competing interests.

Acknowledgments

This work is supported by the NSF of China (11901111, 11701111), the NSF of Guangdong Province (2016A030310257), and Visiting Scholar Program of Chern Institute of Mathematics.

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