Searching for Analytical Solutions of the (2+1)-Dimensional KP Equation by Two Different Systematic Methods
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.
The (2+1)-dimensional KP equation  is given bywhich 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 , and it also appears in the study of nonlinear ion acoustic waves in magnetized dusty plasma . 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  are discovered. Symmetry reductions  and conservation laws  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 . By the positive quadratic function and exponential function, rational lump solutions and line soliton pairs to the (2+1)-dimensional KP equation are established .
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 , -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 . 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
Substituteinto (1), and we get
Take the homogeneous balance between and in (15) to yieldwhere and and are constants.
Substituting into (15) and equating the coefficients about to zero, we get
Solving the above algebraic equations yieldswhere and are arbitrary constants.
When , ,
Remark. The -expansion method is an efficient method. Khan and Akbar  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 , so it is also a good example to show the use of the -expansion method. Secondly, we obtain more generalized results compared with . If we take , , and , then the results of  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 partsthen the weak condition of (25) holds.
Weierstrass elliptic function with double periods satisfies the equation as follows:and has the following addition formula :
Step 3. Substitute the indeterminate formsinto (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 into (15), we havewhere
Equating the coefficients of all powers about in (34) to zero, we get a system of algebraic equations:
Solving the above equations, we obtainand then
Therefore, the elliptic solutions of (15) with arbitrary pole arewhere .
Apply the addition formula to , and thenwhere 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 havewhere
Equating the coefficients of all powers about in (42) to zero, we get a system of algebraic equations:
Solving the above equations, we obtainand then
Substitute into (15), and thenwhere ().
Substitutinginto (47), we obtain thatwhere
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 substituteinto (47) to yieldand then
Substitutinginto (47) to yieldthen
By the above procedures, we collect meromorphic solutions of (15) with arbitrary pole as follows:where and and are arbitrary constants.
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.  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 , 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 . 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.
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.
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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