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

Authors: Yongyi Gu; Fanning Meng
Journal: Complexity
 (2019)

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## Abstract

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


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## Body

## 1. Introduction

The (2+1)-dimensional KP equation [1] is given by(1)uxt-6ux2-6uuxx+uxxxx+3uyy=0,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], exp⁡(-ψ(z))-expansion method [33–38], Lie group method [39, 40], and complex method [41–50].Theexp⁡(-ψ(z))-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 Yuanet 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 p,q condition [41]. Based on the work of Yuanet 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. 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 exp⁡(-ψ(z))-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. Theexp⁡⁡(-ψ(z))-Expansion Method

Consider a nonlinear PDE as follows:(2)Fu,ux,uy,ut,uxx,uyy,utt,…=0,where F is a polynomial consisting of the unknown function u(x,y,t), the partial derivatives of u(x,y,t), the highest-order partial derivatives of u(x,y,t), and some nonlinear terms.Step 1.
Substitute traveling wave transformation,(3)ux,y,t=uz,z=kx+Ly+λt,into (2) to convert it to the ODE,(4)Pu,u′,u′′,u′′′,…=0,where P is a polynomial of u and its derivatives.Step 2.
Suppose that (4) has the following exact solutions:(5)uz=∑j=0nBjexp⁡-ψzj,where Bj(0≤j≤n) are constants to be determined later, such that Bn≠0 and ψ=ψ(z) satisfies the ODE as follows:(6)ψ′z=γ+exp⁡-ψz+μexp⁡ψz.Equation (6) has the solutions as follows.
Whenγ2-4μ>0, μ≠0,(7)ψz=ln⁡-γ2-4μtanh⁡γ2-4μ/2z+a-γ2μ,(8)ψz=ln⁡-γ2-4μcoth⁡γ2-4μ/2z+a-γ2μ.When γ2-4μ<0, μ≠0,(9)ψz=ln⁡4μ-γ2tan⁡4μ-γ2/2z+a-γ2μ,(10)ψz=ln⁡4μ-γ2cot⁡4μ-γ2/2z+a-γ2μ.When γ2-4μ>0, γ≠0, μ=0,(11)ψz=-ln⁡γexp⁡γz+a-1.When γ2-4μ=0, γ≠0, μ≠0,(12)ψz=ln⁡-2γz+a+2γ2z+a.When γ2-4μ=0, γ=0, μ=0, (13)ψz=ln⁡z+a,where a is an arbitrary constant and Bn≠0,γ,μ are constants in (7)- (13). We determine the positive integer n 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 exp⁡(-ψ(z)) yields a polynomial of exp⁡(-ψ(z)). Let the coefficients of the same power about exp⁡(-ψ(z)) equal zero; then, we get a set of algebraic equations. We solve the algebraic equations to obtain the values of Bn≠0,γ,μ 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 theexp⁡⁡(-ψ(z))-Expansion Method to the (2+1)-Dimensional KP Equation

Substitute(14)ux,y,t=uz,z=kx+Ly+λtinto (1), and we get(15)k4u′′′′-6k2uu′′-6k2u′2+kλ+3L2u′′=0.Take the homogeneous balance betweenu′′′′ and (u′)2 in (15) to yield(16)uz=B0+B1exp⁡-ψz+B2exp⁡-ψz2,where B2≠0 and B1 and B0 are constants.Substitutingu′′′′,uu′′,(u′)2,u′′ into (15) and equating the coefficients about exp⁡(-ψ(z)) to zero, we get(17)e0-ψz:k4B1γ3μ+14k4B2μ2γ2+8k4B1γμ2+16k4B2μ3-6k2B0B1μγ-12k2B0B2μ2-6k2B12μ2+3B1L2μγ+B1kλμγ+6B2L2μ2+2B2kλμ2=0,e1-ψz:B1γ4k4+30B2γ3k4μ+22B1γ2k4μ+120B2γk4μ2+16B1k4μ2-6B0B1γ2k2-36B0B2γk2μ-18B12γk2μ-36B1B2k2μ2-12k2B0B1μ+3B1L2γ2+B1γ2kλ+18B2L2γμ+6B2γkλμ+6B1L2μ+2B1kλμ=0,e2-ψz:16B2γ4k4+15k4B1γ3+232B2γ2k4μ+60B1γk4μ+136k4B2μ2-24B0B2γ2k2-12B12γ2k2-90B1B2γk2μ-36B22k2μ2-18B0B1γk2-48B0B2k2μ-24B12k2μ+12B2L2γ2+4B2γ2kλ+9B1L2γ+3B1γkλ+24B2L2μ+8B2kλμ=0,e3-ψz:130B2γ3k4+50B1γ2k4+440B2γk4μ-54B1B2γ2k2+40B1k4μ-84B22γk2μ-60B0B2γk2-30B12γk2-108B1B2k2μ-12B0B1k2+30B2L2γ+10B2γkλ+6B1L2+2B1kλ=0,e4-ψz:330B2γ2k4+60B1γk4-48B22γ2k2+240B2k4μ-126B1B2γk2-96B22k2μ-36B0B2k2-18B12k2+18B2L2+6B2kλ=0,e5-ψz:336B2γk4+24B1k4-108B22γk2-72B1B2k2=0,e6-ψz:120B2k4-60B22k2=0.Solving the above algebraic equations yields(18)B2=2k2,B1=2k2γ,B0=γ2k4+8k4μ+3L2+kλ6k2,where γ and μ are arbitrary constants.We substitute (18) into (16), and then(19)uz=γ2k4+8k4μ+3L2+kλ6k2+2k2γexp⁡-ψz+2k2exp⁡-ψz2.Using (7) to (13) into (19), respectively, we obtain exact solutions of the (2+1)-dimensional KP equation as follows.Whenγ2-4μ>0, μ≠0,(20)u1z=γ2k4+8k4μ+3L2+kλ6k2-4k2γμγ2-4μtanh⁡γ2-4μ/2z+a+γ+8k2μ2γ2-4μtanh⁡γ2-4μ/2z+a+γ2,u2z=γ2k4+8k4μ+3L2+kλ6k2-4k2γμγ2-4μcoth⁡γ2-4μ/2z+a+γ+8k2μ2γ2-4μcoth⁡γ2-4μ/2z+a+γ2.Whenγ2-4μ<0, μ≠0,(21)u3z=γ2k4+8k4μ+3L2+kλ6k2+4k2γμ4μ-γ2tan⁡4μ-γ2/2z+a-γ+8k2μ24μ-γ2tan⁡4μ-γ2/2z+a-γ2,u4z=γ2k4+8k4μ+3L2+kλ6k2+4k2γμ4μ-γ2cot⁡4μ-γ2/2z+a-γ+8k2μ24μ-γ2cot⁡4μ-γ2/2z+a-γ2.When γ2-4μ>0, γ≠0, μ=0,(22)u5z=γ2k4+3L2+kλ6k2+2k2γ2exp⁡γz+a-1+2k2γ2exp⁡γz+a-12.When γ2-4μ=0, γ≠0, μ≠0,(23)u6z=4k4μ+3L2+kλ6k2-k2γ3z+aγz+a+2+k2γ4z+a22γz+a+22.When γ2-4μ=0, γ=0, μ=0,(24)u7z=3L2+kλ6k2+2k2z+a2.The properties of the solutions are shown in Figures 1–5.Figure 1
The 3D and 2D surfaces ofu1(z) by considering the values γ=4, μ=3, k=1, L=1, λ=-43, y=0, a=1, and t=0 for the 2D graphic.Figure 2
The 3D and 2D surfaces ofu2(z) by considering the values γ=4, μ=3, k=1, L=1, λ=-43, y=0, a=1, and t=0 for the 2D graphic.Figure 3
The 3D and 2D surfaces ofu3(z) by considering the values γ=3, μ=2.5, k=1, L=1, λ=-32, y=0, a=1, and t=0 for the 2D graphic.Figure 4
The 3D and 2D surfaces ofu4(z) by considering the values γ=3, μ=2.5, k=1, L=1, λ=-32, y=0, a=1, and t=0 for the 2D graphic.Figure 5
The 3D and 2D surfaces ofu5(z) by considering the values γ=1, μ=0, k=1, L=1, λ=-4, y=0, a=1, and t=0 for the 2D graphic.Remark. The exp⁡-ψz-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 exp⁡(-ψ(z))-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 exp⁡(-ψ(z))-expansion method. Secondly, we obtain more generalized results compared with [6]. If we take k=1, L=1, and λ=-ω, 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 transformT:u(x,y,t)→U(z), (x,y,t)→z into the given PDE yields(25)WU,U′,U′′,U′′′,…=0.Step 2.
Determine the weakp,q condition.
Letp,q∈N, and suppose that the meromorphic solutions U of (25) have at least one pole. Substituting the Laurent series,(26)Uz=∑k=-q∞Akzk,q>0,A-q≠0,into (25), if it is determined p distinct Laurent singular parts(27)∑k=-q-1Akzk,then the weak p,q condition of (25) holds.
Weierstrass elliptic function℘(z)≔℘(z,g2,g3) with double periods satisfies the equation as follows:(28)℘′z2=4℘z3-g2℘z-g3and has the following addition formula [51]:(29)℘z-z0=-℘z+14℘′z+℘′z0℘z-℘z02-℘z0.Step 3.
Substitute the indeterminate forms(30)Uz=∑i=1s-1∑j=2q-1jβ-ijj-1!dj-2dzj-214℘′z+Di℘z-Bi2-℘z+∑i=1s-1β-i12℘′z+Di℘z-Bi+∑j=2q-1jβ-sjj-1!dj-2dzj-2℘z+β0,(31)Uz=∑i=1s∑j=1qβijz-zij+β0,(32)Ueαz=∑i=1s∑j=1qβijeαz-eαzij+β0into (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 z=0, where β-ij are determined by (26), Di2=4Bi3-g2Bi-g3 and ∑i=1sβ-i1=0, and R(z), R(eαz)(α∈C) have s(≤p) distinct poles of multiplicity q.Step 4.
Obtain the meromorphic solutions with arbitrary pole, and substitute the inverse transformT-1 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 p=1,q=2, and then the weak 1,2 condition of (15) holds.By the weak1,2 condition and (30), we have the form of the elliptic solutions of (15):(33)U10z=β-12℘z+β0,with pole at z=0.InsertingU10(z) into (15), we have(34)∑i=03c1i℘iz=0,where(35)c10=-12k4β-12g3+3k2β-12β0g2+6k2β-122g3-32β-12L2g2-12β-12kλg2,c11=-18k4β-12g2+9k2β-122g2,c12=-36k2β0β-12+18L2β-12+6kλβ-12,c13=120k4β-12-60k2β-122.Equating the coefficients of all powers about℘(z) in (34) to zero, we get a system of algebraic equations:(36)c10=0,c11=0,c12=0,c13=0.Solving the above equations, we obtain(37)β-12=2k2,β0=3L2+kλ6k2,and then(38)U10z=2k2℘z+3L2+kλ6k2.Therefore, the elliptic solutions of (15) with arbitrary pole are(39)U1z=2k2℘z-z0+3L2+kλ6k2,where z0∈C.Apply the addition formula toU1(z), and then(40)U1z=-2k2℘z+k22℘′z+D℘z-C2+3L2+kλ-12k4C6k2,where C2=4D3-g2D-g3 and g2 and g3 are arbitrary constants.By the weak1,2 condition and (31), we have the indeterminate forms of rational solutions:(41)U20z=β12z-12+β11z-1+β10,with pole at z=0.InsertingU20(z) into (15), we have(42)∑i=14c2iz-i-2=0,where(43)c21=-12k2β20β11+6L2β11+2kλβ11,c22=-36k2β20β12-18k2β112+18L2β12+6kλβ12,c23=24k4β11-72k2β12β11,c24=120k4β12-60k2β122.Equating the coefficients of all powers aboutz in (42) to zero, we get a system of algebraic equations:(44)c21=0,c22=0,c23=0,c24=0.Solving the above equations, we obtain(45)β12=2k2,β11=0,β10=3L2+kλ6k2,and then(46)U20z=2k2z-12+3L2+kλ6k2.SubstituteU(z)=R(η) into (15), and then(47)k4α4R4η4+6R′′′η3+7R′′η2+R′η-6k2α2RηR′+η2R′′-6k2αR′η2+kλ+3L2α2ηR′+η2R′′=0,where η=eαz (α∈C).Substituting(48)U30z=b12eαz-12+b11eαz-1+b10into (47), we obtain that(49)∑i=15c3iα2e6-iαzeαz-1-6=0,where(50)c31=k4α2b11-6k2b0b11+3b11L2+b11k,c32=10k4α2b11+16k4α2b12+12k2b0b11-24k2b0b12-12k2b112-6b11L2+12b12L2-2b11kλ+4b12kλ,c33=66k4α2b12+36k2b0b12+18k2b112-54k2b12b11-18b12L2-6b12kλ,c34=-10k4α2b11+36k4α2b12-12k2b0b11+36k2b12b11-48k2b122+6b11L2+2b11kλ,c35=-k4α2b11+2k4α2b12+6k2b0b11-12k2b0b12-6k2b112+18k2b12b11-12k2b122-3b11L2+6b12L2-b11kλ+2b12kλ.Equating the coefficients of all powers abouteαz in (49) to zero, we get a system of algebraic equations:(51)c31=0,c32=0,c33=0,c34=0,c35=0.Solving the above equations, we obtain(52)b12=2k2α2,b11=2k2α2,b10=k4α2+3L2+kλ6k2.Therefore, simply periodic solutions to (15) with pole at z=0 are(53)U30z=2k2α2eαz-12+2k2α2eαz-1+k4α2+3L2+kλ6k2=2k2α2eαzeαz-12+k4α2+3L2+kλ6k2=k2α22coth2⁡αz2+3L2+kλ-2k4α26k2.Similar toU30(z), we substitute(54)U40z=b12eαz+12+b11eαz+1+b10into (47) to yield(55)b12=2k2α2,b11=-2k2α2,b10=k4α2+3L2+kλ6k2,and then(56)U40z=2k2α2eαz+12-2k2α2eαz+1+k4α2+3L2+kλ6k2=-2k2α2eαzeαz+12+k4α2+3L2+kλ6k2=k2α22tanh2⁡αz2-2k4α2-3L2-kλ6k2.Substituting(57)U50z=b14eαz-12+b13eαz-12+b12eαz-1+b11eαz+1+b10into (47) to yield(58)b14=2k2α2,b13=2k2α2,b12=2k2α2,b11=-2k2α2,b10=k4α2+3L2+kλ3k2,then(59)U50z=2k2α2eαz-12+2k2α2eαz+12+2k2α2eαz-1-2k2α2eαz+1+k4α2+3L2+kλ3k2=2k2α2eαzeαz-12-2k2α2eαzeαz+12+k4α2+3L2+kλ3k2=k2α22coth2⁡αz2+k2α22tanh2⁡αz2+3L2+kλ-2k4α23k2.By the above procedures, we collect meromorphic solutions of (15) with arbitrary pole as follows:(60)U1z=-2k2℘z+k22℘′z+D℘z-C2+3L2+kλ-12k4C6k2,U2z=2k2z-z0-12+3L2+kλ6k2,U3z=k2α22coth2⁡αz-z02+3L2+kλ-2k4α26k2,U4z=k2α22tanh2⁡αz-z02-2k4α2-3L2-kλ6k2,U5z=k2α22coth2⁡αz-z02+k2α22tanh2⁡αz-z02+3L2+kλ-2k4α23k2.where z0∈C,C2=4D3-g2D-g3 and g2 and g3 are arbitrary constants.The properties of the solutions are shown in Figures6–8.Figure 6
The 3D and 2D surfaces ofU2(z) by considering the values k=1, L=1, λ=1, y=0, z0=1, and t=0 for the 2D graphic.Figure 7
The 3D and 2D surfaces ofU3(z) by considering the values k=1, L=1, λ=1, α=1, y=0, z0=1, and t=0 for the 2D graphic.Figure 8
The 3D and 2D surfaces ofU4(z) by considering the values k=1, L=1, λ=1, α=1, y=0, z0=1, and t=0 for the 2D graphic.Remark. Based on the work of Yuanet 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

Borhanifaret 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.Theexp⁡(-ψ(z))-expansion method allows us to express the exact solutions of NLDEs as a polynomial of exp⁡(-ψ(z)), in which ψ(z) 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 u7(z) is equivalent to U2(z) if we consider z0=-a-1.With the extended complex method, we can derive meromorphic solutions of the differential equations which do not satisfyp,q 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 U(z) for the differential equation with a pole at z=0, and then we can derive meromorphic solutions U(z-z0), z0∈C, 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 theexp⁡(-ψ(z))-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.



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