# Some Exact Solutions to Generalized Kadomtsev-Petviashvili Equation

Authors: Bao Wang; Zhiqiang Chen
Journal: Advances in Mathematical Physics
 (2022)

---

## Abstract

Most of the papers have explored the interactions between solitons with a zero background, while reports about exact solutions for nonzero background are rare. Hence, this paper is aimed at exploring the breather, lump, and interaction solutions with a small perturbation to (2+1)-dimensional generalized Kadomtsev-Petviashvili (gKP) equation. General high-order periodic breather solutions are obtained using Hirota’s bilinear method with a small perturbation. At the same time, combining the use of long wave limit methods and module resonance constraints, general lump solutions and mixed solutions to gKP equation are generated. Finally, the space-time structures of the breather solutions, lump solutions, and interaction solutions are investigated and discussed.


---

## Body

## 1. Introduction

The soliton, also known as a solitary wave, is a special form of ultrashort pulse, or a pulse-like traveling wave whose shape, amplitude, and velocity remain constant during its propagation [1]. So far, soliton phenomena have been discovered in many subject areas, for example, laser self-focusing in media, acoustic and electromagnetic waves in plasma, motion of domain walls in liquid crystals, vortex in fluid, dislocation of crystals, magnetic flux in superconductors, and signal transmission in the nervous system [2–4].In mathematics, the progress of soliton theory is embodied in the discovery of a large number of nonlinear partial differential equations with soliton solutions [5, 6] and has gradually established a more systematic mathematical and physical partial differential equations and the theory of soliton [7, 8]. In order to solve these nonlinear mathematical physical equations, scholars working in the field of solitons have developed a series of solution methods, such as the inverse scattering method [8], Hirota bilinear method [7], numerical method, and symbolic calculations [9].In this paper, we consider the2+1-dimensional generalized Kadomtsev-Petviashvili equation [10] as follows:
(1)ut+cuux+buxxxx+c02uyy=0,where u=ux,y,t denotes a scalar function of the space variables x,y and time variable t, the parameters c is the nonlinear term coefficient, b is the dispersion coefficient along the x-axis, c0 is the velocity of the linear wave, and c0/2 is the dispersion coefficient along the y-axis.Whenc=6, b=1, and c0/2=−1, Equation (1) is reduced to the KPI equation:
(2)uxt+6uuxx+uxxxx−uyy=0.And the exact solutions with a zero background includingN-soliton solution and lump solution to the standard Kadomtsev-Petviashvili equation has been studied systematically in Refs. [7, 11, 12]. In recent years, the research on the lump solution of Equation (3) has been very hot, mainly focusing on the normal scattering of lump waves [13], anomalous scattering between lump waves [14–18], and bound states of lump waves [19]. The reports on anomalous scattering focus on the diversity of scattering patterns, such as triangular patterns, and polygonal patterns [17]. At the same time, the resonance phenomenon between lump chains (we called breather waves in this paper) and lump waves has been fully studied [20–23].Whenc=6, b=1, and c0/2=1, Equation (1) is reduced to the KPII equation:
(3)uxt+6uuxx+uxxxx+uyy=0.This equation can be used to describe some nonlinear phenomena in shallow water [24]. For the KP2 system, Kodama has made a very outstanding contribution in the field about the resonance phenomena between line waves [25].Considering that the above-described solutions are all obtained on the zero background, the solutions with a nonzero background in the actual system are more general and can describe the objective world more accurately. In this paper, applying Hirota’s bilinear method with a perturbation parameteru0 to Equation (1), we obtain periodic breather wave solutions. Meanwhile, we generate lump and interaction solutions with a nonzero background to Equation (1) from solitons by taking long wave limits [12]. The arrangement of this paper is organized as follows: in Section 2, under the variable transformation, we construct bilinear formalism with a perturbation parameter. In Section 3, we mainly investigate general higher-order breather and lump solutions of Equation (1). In Section 4, we describe how to obtain mixed solutions and interaction solutions.


## 2. Bilinear Formalism with a Perturbation Parameter

Under the variable transformation,(4)ux,y,t=u0+12bclnfxx,where fx,y,t is a complex function and u0 is a free real number.Substituting Equation (4) into Equation (1), then Equation (1) becomes the following equation:
(5)lnfxxt+cu0lnfxxx+12blnfxxlnfxxx+blnfxxxx+c02lnfxyy=0.When Equation (5) integrates once with respect to x, we can obtain the following equation:
(6)lnfxt+cu0lnfxx+6blnfxx2+blnfxxxx+c02lnfyy=0.The bilinear form of Equation (1) with a small perturbation parameter u0 is generated as
(7)DxDt+cu0Dx2+bDx4+c02Dy2f·f=0.The operatorD is the Hirota’s bilinear differential operator defined by
(8)DxmDtnf·g=∂∂x−∂∂x′m∂∂t−∂∂t′nfx,y,t·gx′,y′,t′x′=x,t′=t.


## 3. General Higher-Order Breather and Lump Solutions

In this section, we mainly investigate general high-order breather and lump solutions of Equation (1).


### 3.1. First-Order Breather and Lump Solutions

We first assumef in Equation (4) as the following formal form to derive first-order breather solutions in Equation (1):
(9)f=1+εf1+ε2f2,with
(10)f1=eη1+eη2,f2=eη1+η2+A12,where
(11)ηs=wst+ksx+psy+ϕs,s=1,2,and ws, ks, ps, and ϕs are freely complex parameters. Substituting f defined in Equation (9) into Equation (7) and collecting the power order of ε, one can obtain the following equations at the ascending power order of ε:
(12)ε0:DxDt+cu0Dx2+bDx4+c02Dy21·1=0,ε1:DxDt+cu0Dx2+bDx4+c02Dy21·f1+f1·1=0,ε2:DxDt+cu0Dx2+bDx4+c02Dy21·f2+f1·f1+f2·1=0,namely,
(13)f1xt+cu0f1xx+bf1xxxx+c02f1yy=0,(14)cu0f1xxf1−f1x2+f2xx+bf1xxxxf1−4f1xxxf1x+3f1xx2+f2xxxx+f1xtf1−f1xf1t+f2xt=0.Substituting functionsf1 and f2 defined in Equation (10) into Equation (13) and Equation (14), we have
(15)2bk14+2ck12u0+c0p12+2k1w1eη1+2bk24+2ck22u0+c0p22+2k2w2eη2=0,and
(16)2w1−w2k1−k2+2cu0k1−k22+2bk1−k24+c0p1−p22eη1+η2+2k1+k2w1+w2+2cu0k1+k22+2bk1+k24+c0p1+p22eη1+η2+A12=0.By solving Equations (15) and (16), we can obtain the following formulas:
(17)eA12=−2k1−k24b+2u0k1−k22c+p1−p22c0+2w1−w2k1−k22k1+k24b+2u0k1+k22c+p1+p22c0+2w1+w2k1+k2,(18)2bks4+2cks2+c0ps2+2ksws=0,s=1,2.Under the constraints of Equations (17) and (18), when ε=1, f can be written as
(19)f=1+eη1+eη2+eη1+η2+A12,which corresponds to the two-soliton solution of Equation (1). To guarantee the corresponding breather solutions being real functions, there are two restrictions for a valid calculation: (1) η1=η2¯, so η1 and η2 are conjugates of each other. (2) Parameters ks,ws,ps,s=1,2 must satisfy the constraint of Equation (18).In particular, the following parameter constrains may be used to facilitate the calculation:(20)k1=−k2=i·k,p1=p2=p,ϕ1=ϕ2=ϕ0,where k, p, and ϕ0 are freely real parameters. Then, we can obtain η1=η2¯, and the function f in Equation (19) can be rewritten as
(21)f=HyMcoshθ+coswt+kx,where
(22)Hy=2epy+ϕ0,w=2bk4−2cu0k2+c0p22k,θ=py+ϕ0+lnM,M=−6bk4−c0p2c0p2.The first-order breather solutions in Equation (1) in the (x,y)-plane are shown in Figure 1. It is seen that there are dark-type and bright-type breather solutions in Equation (1).Figure 1
Two types of breather solutions in Equation (1) with parameters c0=1, b=−1, k=1, p=2, and ϕ0=0 at t=0. (a) Bright-type breather with u0=2 and c=−1. (b) Dark-type breather with u0=−2 and c=1.
(a)(b)To generate rational solution, we take a long wave limit with the provision in Equation (19).
(23)ps=Psε,ks=Ksε,ε↦0,eϕs=−1,s=1,2.Then, the expansions off in Equation (19) are given as follows:
(24)f=θ1θ2+a12ε2+oε2,where
(25)θs=−Kscu0t−Ps2c02Kst+Ksx+Psy,s=1,2,a12=24bK13K23c0K1P2−K2P12.In order to get rational solutions in Equation (1), divide both sides of Equation (24) byε2, and then, attempt to compute the limiting value asεapproaches0. For convenience, let us still call the limit that we just obtained f. Then, the f is given as follows:
(26)f=θ1θ2+a12.To guarantee the corresponding rational solutions being lump solutions, whereK1=K2¯ and P1=P2¯, then u in Equation (4) can be written as
(27)u=u0−12bθ12∂/∂xθ22+θ22∂/∂xθ12−2a12∂/∂xθ1∂/∂xθ2cθ1θ2+a122.This lump solutionu in Equation (27) possesses three critical points:
(28)A1=2cK1K2u0−P1P2c02K2K1t,K1P2+K2P1c0t2K2K1,A2=c0tK1P2−K2P12cK1K2u0−P1P2c0+122K12K22c0b2c0K1P2−K2P1K1K2,K1P2+K2P1c0t2K2K1,A3=c0tK1P2−K2P12cK1K2u0−P1P2c0−122K12K22c0b2c0K1P2−K2P1K1K2,K1P2+K2P1c0t2K2K1,which are derived by solving ux=0 and uy=0. Based on the analysis of these critical points at the second-order derivatives in
(29)uxxuyy−uxy2,to determine whether the aforementioned critical points are local maximum points or local minimum points.In order to describe the properties of lump solutions more clearly and facilitate discussion, let us setb=−1. Then, the lump solution can be classified into two patterns:
(a)
Bright lump. u0>0,c<0: u has one local maximum (point A1) and two minimum points (points A2 and A3) (see Figure 2(a))(b)
Dark lump. u0<0,c>0: u has two local maximum (points A2 and A3) and one minimum point (point A1) (see Figure 2(b))Figure 2
Two types of lump solutions in Equation (1) with parameter c0=1, b=−1, K1=1−i, K2=1+i, P1=−2, and P2=−2 at t=0. (a) Bright-type lump solution with u0=1 and c=−1. (b) Dark-type lump solution with u0=−1 and c=1.
(a)(b)Two different patterns of lump solutions, namely, bright-type and dark-type lump solutions, are shown in Figure2.


### 3.2. Second-Order Breather and Lump Solutions

In order to obtain the general high-order breather solutions and lump solutions in Equation (1), we assume that the auxiliary function f in Equation (7) has higher-order expansions in terms of ε:
(30)f=1+εf1+ε2f2+⋯+εnfn.Again, substituting Equation (30) into bilinear Equation (7) and then collecting the coefficient of ε, 2n+1 equations would be yielded corresponding to different orders of ε. Maybe it is tedious and troublesome to solve these 2n+1 equations. According to the work of Hirota, Kaur and Wazwaz, and Singh et al., [7, 26, 27], we calculate and verify that f has the following form:
(31)f=∑μ=0,1exp∑j<sNμjμsAjs+∑j=1Nμjηj,where the sum of μ is the sum over all the possibilities of μj=0,1,j=1,2⋯.The above coefficients and parameters are given explicitly as follows:(32)ηj=kjx+ωjt+pjy+ϕj,eAjs=−2bkj−ks4+2cu0kj−ks2+c0pj−ps2+2wj−wskj−ks2bkj+ks4+2cu0kj+ks2+c0pj+ps2+2wj+wskj+ks,(33)2bks4+2cks2u0+c0ps2+2ksws=0.In order to obtain the second-order breather solutions in Equation (1) and for a valid calculation, there are also some restrictions as the first-order breather solution: (1) η1=η2¯, and η3=η4¯, so η1 and η2 and η3and η4 are conjugates of each other; (2) ks, ws, and ps have to satisfy the constraint of Equation (33); and (3) N=4 in Equation (31). Then, the evolution of second-breather solutions in Equation (1) are shown in the Figure 3.Figure 3
Two types of second-order breather solutions in Equation (1) with parameters c0=1, b=−1, k1=−1/2, k2=i/2, p1=−1, p2=−1, k3=−3/5i, k4=3/5i, p3=−1, p4=−1, ϕ1=0, ϕ2=0, ϕ3=0, and ϕ4=0 at t=0. (a) Bright-type breather with u0=1 and c=−1. (b) Dark-type breather with u0=−1 and c=1.
(a)(b)The way to get second-order lump solutions is roughly the same as the way to get first-order lump solutions. We takeN=4 and expϕs=−1,s=1,2,3,4 in Equation (31) and take a long wave limit with the provision in Equation (31) and eliminate the oε4. Then, f is given as follows:
(34)f=∏jθj+∑j<sajs∏k≠jsθk+∑j<s∏ajs,with
(35)θs=−2Ks2cu0−Ps2c0t2Ks+Ksx+Psy,ajs=24bKj3Ks3c0KjPs−KsPj2,where j=1,2,3,4, s=1,2,3,4, and j<s,Kj,Pj are complex parameters. To guarantee the corresponding rational solutions being lump solutions, there is a restriction for a valid calculation: K1=K2¯, K3=K4¯, P1=P2¯, and P3=P4¯.Since there are too many parameters involved, in order to directly describe the properties of the second-order lump solutions, it is advisable to assign values to the following parameters:b=−1, c0=1, K1=−2−i, K2=−2+i, P1=2, P2=2, K3=−2+i, K4=−2−i, P3=−2, and P4=−2. If using parametersu0=1andc=−1, then we obtain thefwhich can lead to bright-type lump solutions in Equation (1); if using parametersu0=−1andc=1, then we can get dark-type lump solutions. After calculation, it can be found that f corresponding to bright-type lump solutions is the same as f corresponding to dark-type lump solutions. Then, f in Equation (34) can be written as
(36)f=289t425+1292t3x25+2294t2x225+104t2y225+76tx3−16txy2+25x4−24x2y2+16y4+70341t2100+3207tx2+4725x24+1791y2+248062564.According to Equation (4), the bright-type lump is explicitly as follows:
(37)u=1+12lnfxx,and the dark-type lump is explicitly as follows:
(38)u=−1−12lnfxx,where f is Equation (36). The second-order lump solutions in Equation (1) in the (x,y)-plane are shown in the Figure 4.Figure 4
Two types of second-order lump solutions in Equation (1) with parameters b=−1, c0=1, K1=−2−i, K2=−2+i, P1=2, P2=2, K3=−2+i, K4=−2−i, P3=−2, and P4=−2 at t=0. (a) Bright-type lump with u0=1 and c=−1. (b) Dark-type lump with u0=−1 and c=1.
(a)(b)


### 3.3. Higher-Order Breather and Lump Solutions

The similar procedures described previously could be generalized to the higher-order breather and lump solutions. To guarantee thenth-order breather solutions being real functions, there are two restrictions for a valid calculation: (1) take η1=η2¯,η3=η4¯⋯η2n−1=η2n¯ in Equation (31). (2) Parameters ks, ps, and ws must satisfy the constrain of Equation (33). Then, we obtain the nth-order breather solutions in Equation (1).For example, if we want to obtain the third-order breather solutions, according to the description in the previous paragraph, we will takeη1=η2¯, η3=η4¯, and η5=η6¯ in Equation (31). In order to describe the properties of the third-breather solution more clearly and facilitate the calculation, the parameters can be assigned as follows: c0=1, b=−1, k1=−i, k2=i, p1=2, p2=2, k3=i, k4=−i, p3=−7/4, p4=−7/4, k5=i, k6=−i, p5=−3/2, p6=−3/2, ϕs=0, and s=1,2,3,4,5,6. If using parameters u0=1 and c=−1, then we can derive bright-type breather solutions; if using parameters u0=−1 and c=1, we can derive dark-type breather solutions. Under the conditions of these parameters, bright-type breather solutions and dark-type breather solutions have the same f. The third-breather solutions in Equation (1) in the (x,y)-plane are shown in the Figure 5.Figure 5
Two types of breather solutions in Equation (1) with parameters c0=1, b=−1, k1=−i, k2=i, p1=2, p2=2, k3=i, k4=−i, p3=−7/4, p4=−7/4, k5=i, k6=−i, p5=−3/2, p6=−3/2, ϕs=0, and s=1,2,3,4,5,6 at t=0. (a) Bright-type breather solution with u0=1 and c=−1. (b) Dark-type breather solution with u0=−1 and c=1.
(a)(b)The process of obtainingnth-order lump solutions is roughly similar to that of obtaining first-order lump solutions and second-order lump solutions. In order to obtain nth-order lump solutions, we take N=2n in Equation (31). Then, we take a long wave limit with the provision in Equation (31):
(39)ps=Psε,ks=Ksε,ε↦0,eϕs=−1,s=1,2⋯N.And just like we did with Equations (24) and (26), we get rid of the higher-order terms of oεN, and then, we get a polynomial f. General higher-order rational solutions in Equation (1) can be presented in the following forms:
(40)u=uo+12bclnfNxx,where
(41)fN=∏s=1Nθs+12∑j,sNajs∏p≠j,sNθp+⋯+1M!2M∑l,s⋯m,nNalsajk⋯amn⏞M∏q≠l,s⋯m,nNθq+⋯,with
(42)θs=−2Ks2cu0−Ps2c0t2Ks+Ksx+Psy,ajs=24bKj3Ks3c0KjPs−KsPj2.To guarantee the corresponding rational solutions being lump solutions, there are some restrictions for a valid calculation:K1=K2¯,K3=K4¯,⋯K2n−1=K2n¯ and P1=P2¯,P3=P4¯,⋯P2n−1=P2n¯.For example, if we want to get third-lump solutions, we have to setN equal to 6 in Equation (41). In order to describe the properties of third-lump solution more clearly, we assign the following values to the following parameters: c0=1, b=−1, K1=3i, K2=−3i, P1=1, P2=1, K3=2i, K4=−2i, P3=−2, P4=−2, K5=3i, K6=−3i, P5=2, and P6=2. And if using u0=1 and c=−1, then we obtain bright-type lump solutions; if using u0=−1 and c=1, then we obtain dark-type lump solutions. Under the parameter constraints above, two types of lump solutions have the same f, as shown below:
(43)f=9t2+12tx+4x2+4y2121t2+198tx+81x2+36y2361t2+684tx+324x2+36y2324+4996352617t45400+222120217t3x75+88451544t2x225+2053073t2y215+46738512tx325+5841228txy225+9216018x425+2409264x2y225−828282y425+196632775107t21250+154136915604tx625+60286704219x2625+16102685466y2625+19763125578271250.And then from Equation (40), we can get two types of lump solutions. The third-lump solutions in Equation (1) in the (x,y)-plane are shown in Figure 6.Figure 6
Two types of third-lump solution in Equation (1) at t=0. (a) Bright-type lump with u0=1 andc=−1. (b) Dark-type lump with u0=−1 and c=1.
(a)(b)In addition, interaction solutions between breather solutions and lump solutions also can be obtained from solitons by taking long wave limits. If usingN=6, p1=P1ε, p2=P2ε, k1=K1ε, k2=K2ε, P1=P2¯, K1=K2¯, ϕ1=iπ, ϕ2=iπ, η3=η4¯, and η5=η6¯ in Equation (31), then we can obtain interactions between a lump and two breathers after taking a long wave limit. In order to better describe the structure of interactions, we assign the parameters as follows: c0=1, b=−1, K1=4i, K2=−4i, P1=10, P2=10, k3=i, k4=−i, p3=−5/3, p4=−5/3, k5=i, k6=−i, p5=11/6, p6=11/6, ϕ1=iπ, ϕ2=iπ, ϕ3=0, ϕ4=0, ϕ5=0, and ϕ6=0. If using u0=1 and c=−1, then we get the bright-type interactions; if using u0=−1 and c=1, then we can obtain dark-type interactions. Two types of interactions between a lump and two breathers in Equation (1) in the (x,y)-plane are shown in Figure 7.Figure 7
Two types of interactions between a lump and two breathers in Equation (1) at t=0. (a) Bright-type interaction solution with u0=1 andc=−1. (b) Dark-type interaction solution with u0=−1 and c=1.
(a)(b)The idea and process of obtaining interactions between two lumps and a breather are roughly the same as that of obtaining interaction between a lump and two breathers. If usingN=6, ps=Psε, ks=Ksε, ϕs=iπ, s=1,2,3,4, P2j−1=P2j¯, K2j−1=K2j¯, j=1,2, and η5=η6¯ in Equation (31), then we can obtain interactions between two lumps and a breather after taking a long wave limit. In order to better describe the structure of interactions between a breather and two lumps in Equation (1), we assign the parameters as follows: c0=1, b=−1, K1=3i, K2=−3i, P1=5, P2=5, K3=3i, K4=−3i, P3=6, P4=6, k5=i, k6=−i, p5=5/4, p6=5/4, ϕ1=iπ, ϕ2=iπ, ϕ3=iπ, ϕ4=iπ, ϕ5=0, and ϕ6=0, then we get the bright-type interactions; if using u0=−1 and c=1, then we can obtain dark-type interactions. Two types of interactions between a breather and two lumps in Equation (1) in the (x,y)-plane are shown in Figure 8.Figure 8
Two types of interactions between a lump and two breathers in Equation (1) at t=0. (a) Bright-type interaction solution with u0=1 andc=−1. (b) Dark-type interaction solution with u0=−1 andc=1.
(a)(b)


## 3.1. First-Order Breather and Lump Solutions

We first assumef in Equation (4) as the following formal form to derive first-order breather solutions in Equation (1):
(9)f=1+εf1+ε2f2,with
(10)f1=eη1+eη2,f2=eη1+η2+A12,where
(11)ηs=wst+ksx+psy+ϕs,s=1,2,and ws, ks, ps, and ϕs are freely complex parameters. Substituting f defined in Equation (9) into Equation (7) and collecting the power order of ε, one can obtain the following equations at the ascending power order of ε:
(12)ε0:DxDt+cu0Dx2+bDx4+c02Dy21·1=0,ε1:DxDt+cu0Dx2+bDx4+c02Dy21·f1+f1·1=0,ε2:DxDt+cu0Dx2+bDx4+c02Dy21·f2+f1·f1+f2·1=0,namely,
(13)f1xt+cu0f1xx+bf1xxxx+c02f1yy=0,(14)cu0f1xxf1−f1x2+f2xx+bf1xxxxf1−4f1xxxf1x+3f1xx2+f2xxxx+f1xtf1−f1xf1t+f2xt=0.Substituting functionsf1 and f2 defined in Equation (10) into Equation (13) and Equation (14), we have
(15)2bk14+2ck12u0+c0p12+2k1w1eη1+2bk24+2ck22u0+c0p22+2k2w2eη2=0,and
(16)2w1−w2k1−k2+2cu0k1−k22+2bk1−k24+c0p1−p22eη1+η2+2k1+k2w1+w2+2cu0k1+k22+2bk1+k24+c0p1+p22eη1+η2+A12=0.By solving Equations (15) and (16), we can obtain the following formulas:
(17)eA12=−2k1−k24b+2u0k1−k22c+p1−p22c0+2w1−w2k1−k22k1+k24b+2u0k1+k22c+p1+p22c0+2w1+w2k1+k2,(18)2bks4+2cks2+c0ps2+2ksws=0,s=1,2.Under the constraints of Equations (17) and (18), when ε=1, f can be written as
(19)f=1+eη1+eη2+eη1+η2+A12,which corresponds to the two-soliton solution of Equation (1). To guarantee the corresponding breather solutions being real functions, there are two restrictions for a valid calculation: (1) η1=η2¯, so η1 and η2 are conjugates of each other. (2) Parameters ks,ws,ps,s=1,2 must satisfy the constraint of Equation (18).In particular, the following parameter constrains may be used to facilitate the calculation:(20)k1=−k2=i·k,p1=p2=p,ϕ1=ϕ2=ϕ0,where k, p, and ϕ0 are freely real parameters. Then, we can obtain η1=η2¯, and the function f in Equation (19) can be rewritten as
(21)f=HyMcoshθ+coswt+kx,where
(22)Hy=2epy+ϕ0,w=2bk4−2cu0k2+c0p22k,θ=py+ϕ0+lnM,M=−6bk4−c0p2c0p2.The first-order breather solutions in Equation (1) in the (x,y)-plane are shown in Figure 1. It is seen that there are dark-type and bright-type breather solutions in Equation (1).Figure 1
Two types of breather solutions in Equation (1) with parameters c0=1, b=−1, k=1, p=2, and ϕ0=0 at t=0. (a) Bright-type breather with u0=2 and c=−1. (b) Dark-type breather with u0=−2 and c=1.
(a)(b)To generate rational solution, we take a long wave limit with the provision in Equation (19).
(23)ps=Psε,ks=Ksε,ε↦0,eϕs=−1,s=1,2.Then, the expansions off in Equation (19) are given as follows:
(24)f=θ1θ2+a12ε2+oε2,where
(25)θs=−Kscu0t−Ps2c02Kst+Ksx+Psy,s=1,2,a12=24bK13K23c0K1P2−K2P12.In order to get rational solutions in Equation (1), divide both sides of Equation (24) byε2, and then, attempt to compute the limiting value asεapproaches0. For convenience, let us still call the limit that we just obtained f. Then, the f is given as follows:
(26)f=θ1θ2+a12.To guarantee the corresponding rational solutions being lump solutions, whereK1=K2¯ and P1=P2¯, then u in Equation (4) can be written as
(27)u=u0−12bθ12∂/∂xθ22+θ22∂/∂xθ12−2a12∂/∂xθ1∂/∂xθ2cθ1θ2+a122.This lump solutionu in Equation (27) possesses three critical points:
(28)A1=2cK1K2u0−P1P2c02K2K1t,K1P2+K2P1c0t2K2K1,A2=c0tK1P2−K2P12cK1K2u0−P1P2c0+122K12K22c0b2c0K1P2−K2P1K1K2,K1P2+K2P1c0t2K2K1,A3=c0tK1P2−K2P12cK1K2u0−P1P2c0−122K12K22c0b2c0K1P2−K2P1K1K2,K1P2+K2P1c0t2K2K1,which are derived by solving ux=0 and uy=0. Based on the analysis of these critical points at the second-order derivatives in
(29)uxxuyy−uxy2,to determine whether the aforementioned critical points are local maximum points or local minimum points.In order to describe the properties of lump solutions more clearly and facilitate discussion, let us setb=−1. Then, the lump solution can be classified into two patterns:
(a)
Bright lump. u0>0,c<0: u has one local maximum (point A1) and two minimum points (points A2 and A3) (see Figure 2(a))(b)
Dark lump. u0<0,c>0: u has two local maximum (points A2 and A3) and one minimum point (point A1) (see Figure 2(b))Figure 2
Two types of lump solutions in Equation (1) with parameter c0=1, b=−1, K1=1−i, K2=1+i, P1=−2, and P2=−2 at t=0. (a) Bright-type lump solution with u0=1 and c=−1. (b) Dark-type lump solution with u0=−1 and c=1.
(a)(b)Two different patterns of lump solutions, namely, bright-type and dark-type lump solutions, are shown in Figure2.


## 3.2. Second-Order Breather and Lump Solutions

In order to obtain the general high-order breather solutions and lump solutions in Equation (1), we assume that the auxiliary function f in Equation (7) has higher-order expansions in terms of ε:
(30)f=1+εf1+ε2f2+⋯+εnfn.Again, substituting Equation (30) into bilinear Equation (7) and then collecting the coefficient of ε, 2n+1 equations would be yielded corresponding to different orders of ε. Maybe it is tedious and troublesome to solve these 2n+1 equations. According to the work of Hirota, Kaur and Wazwaz, and Singh et al., [7, 26, 27], we calculate and verify that f has the following form:
(31)f=∑μ=0,1exp∑j<sNμjμsAjs+∑j=1Nμjηj,where the sum of μ is the sum over all the possibilities of μj=0,1,j=1,2⋯.The above coefficients and parameters are given explicitly as follows:(32)ηj=kjx+ωjt+pjy+ϕj,eAjs=−2bkj−ks4+2cu0kj−ks2+c0pj−ps2+2wj−wskj−ks2bkj+ks4+2cu0kj+ks2+c0pj+ps2+2wj+wskj+ks,(33)2bks4+2cks2u0+c0ps2+2ksws=0.In order to obtain the second-order breather solutions in Equation (1) and for a valid calculation, there are also some restrictions as the first-order breather solution: (1) η1=η2¯, and η3=η4¯, so η1 and η2 and η3and η4 are conjugates of each other; (2) ks, ws, and ps have to satisfy the constraint of Equation (33); and (3) N=4 in Equation (31). Then, the evolution of second-breather solutions in Equation (1) are shown in the Figure 3.Figure 3
Two types of second-order breather solutions in Equation (1) with parameters c0=1, b=−1, k1=−1/2, k2=i/2, p1=−1, p2=−1, k3=−3/5i, k4=3/5i, p3=−1, p4=−1, ϕ1=0, ϕ2=0, ϕ3=0, and ϕ4=0 at t=0. (a) Bright-type breather with u0=1 and c=−1. (b) Dark-type breather with u0=−1 and c=1.
(a)(b)The way to get second-order lump solutions is roughly the same as the way to get first-order lump solutions. We takeN=4 and expϕs=−1,s=1,2,3,4 in Equation (31) and take a long wave limit with the provision in Equation (31) and eliminate the oε4. Then, f is given as follows:
(34)f=∏jθj+∑j<sajs∏k≠jsθk+∑j<s∏ajs,with
(35)θs=−2Ks2cu0−Ps2c0t2Ks+Ksx+Psy,ajs=24bKj3Ks3c0KjPs−KsPj2,where j=1,2,3,4, s=1,2,3,4, and j<s,Kj,Pj are complex parameters. To guarantee the corresponding rational solutions being lump solutions, there is a restriction for a valid calculation: K1=K2¯, K3=K4¯, P1=P2¯, and P3=P4¯.Since there are too many parameters involved, in order to directly describe the properties of the second-order lump solutions, it is advisable to assign values to the following parameters:b=−1, c0=1, K1=−2−i, K2=−2+i, P1=2, P2=2, K3=−2+i, K4=−2−i, P3=−2, and P4=−2. If using parametersu0=1andc=−1, then we obtain thefwhich can lead to bright-type lump solutions in Equation (1); if using parametersu0=−1andc=1, then we can get dark-type lump solutions. After calculation, it can be found that f corresponding to bright-type lump solutions is the same as f corresponding to dark-type lump solutions. Then, f in Equation (34) can be written as
(36)f=289t425+1292t3x25+2294t2x225+104t2y225+76tx3−16txy2+25x4−24x2y2+16y4+70341t2100+3207tx2+4725x24+1791y2+248062564.According to Equation (4), the bright-type lump is explicitly as follows:
(37)u=1+12lnfxx,and the dark-type lump is explicitly as follows:
(38)u=−1−12lnfxx,where f is Equation (36). The second-order lump solutions in Equation (1) in the (x,y)-plane are shown in the Figure 4.Figure 4
Two types of second-order lump solutions in Equation (1) with parameters b=−1, c0=1, K1=−2−i, K2=−2+i, P1=2, P2=2, K3=−2+i, K4=−2−i, P3=−2, and P4=−2 at t=0. (a) Bright-type lump with u0=1 and c=−1. (b) Dark-type lump with u0=−1 and c=1.
(a)(b)


## 3.3. Higher-Order Breather and Lump Solutions

The similar procedures described previously could be generalized to the higher-order breather and lump solutions. To guarantee thenth-order breather solutions being real functions, there are two restrictions for a valid calculation: (1) take η1=η2¯,η3=η4¯⋯η2n−1=η2n¯ in Equation (31). (2) Parameters ks, ps, and ws must satisfy the constrain of Equation (33). Then, we obtain the nth-order breather solutions in Equation (1).For example, if we want to obtain the third-order breather solutions, according to the description in the previous paragraph, we will takeη1=η2¯, η3=η4¯, and η5=η6¯ in Equation (31). In order to describe the properties of the third-breather solution more clearly and facilitate the calculation, the parameters can be assigned as follows: c0=1, b=−1, k1=−i, k2=i, p1=2, p2=2, k3=i, k4=−i, p3=−7/4, p4=−7/4, k5=i, k6=−i, p5=−3/2, p6=−3/2, ϕs=0, and s=1,2,3,4,5,6. If using parameters u0=1 and c=−1, then we can derive bright-type breather solutions; if using parameters u0=−1 and c=1, we can derive dark-type breather solutions. Under the conditions of these parameters, bright-type breather solutions and dark-type breather solutions have the same f. The third-breather solutions in Equation (1) in the (x,y)-plane are shown in the Figure 5.Figure 5
Two types of breather solutions in Equation (1) with parameters c0=1, b=−1, k1=−i, k2=i, p1=2, p2=2, k3=i, k4=−i, p3=−7/4, p4=−7/4, k5=i, k6=−i, p5=−3/2, p6=−3/2, ϕs=0, and s=1,2,3,4,5,6 at t=0. (a) Bright-type breather solution with u0=1 and c=−1. (b) Dark-type breather solution with u0=−1 and c=1.
(a)(b)The process of obtainingnth-order lump solutions is roughly similar to that of obtaining first-order lump solutions and second-order lump solutions. In order to obtain nth-order lump solutions, we take N=2n in Equation (31). Then, we take a long wave limit with the provision in Equation (31):
(39)ps=Psε,ks=Ksε,ε↦0,eϕs=−1,s=1,2⋯N.And just like we did with Equations (24) and (26), we get rid of the higher-order terms of oεN, and then, we get a polynomial f. General higher-order rational solutions in Equation (1) can be presented in the following forms:
(40)u=uo+12bclnfNxx,where
(41)fN=∏s=1Nθs+12∑j,sNajs∏p≠j,sNθp+⋯+1M!2M∑l,s⋯m,nNalsajk⋯amn⏞M∏q≠l,s⋯m,nNθq+⋯,with
(42)θs=−2Ks2cu0−Ps2c0t2Ks+Ksx+Psy,ajs=24bKj3Ks3c0KjPs−KsPj2.To guarantee the corresponding rational solutions being lump solutions, there are some restrictions for a valid calculation:K1=K2¯,K3=K4¯,⋯K2n−1=K2n¯ and P1=P2¯,P3=P4¯,⋯P2n−1=P2n¯.For example, if we want to get third-lump solutions, we have to setN equal to 6 in Equation (41). In order to describe the properties of third-lump solution more clearly, we assign the following values to the following parameters: c0=1, b=−1, K1=3i, K2=−3i, P1=1, P2=1, K3=2i, K4=−2i, P3=−2, P4=−2, K5=3i, K6=−3i, P5=2, and P6=2. And if using u0=1 and c=−1, then we obtain bright-type lump solutions; if using u0=−1 and c=1, then we obtain dark-type lump solutions. Under the parameter constraints above, two types of lump solutions have the same f, as shown below:
(43)f=9t2+12tx+4x2+4y2121t2+198tx+81x2+36y2361t2+684tx+324x2+36y2324+4996352617t45400+222120217t3x75+88451544t2x225+2053073t2y215+46738512tx325+5841228txy225+9216018x425+2409264x2y225−828282y425+196632775107t21250+154136915604tx625+60286704219x2625+16102685466y2625+19763125578271250.And then from Equation (40), we can get two types of lump solutions. The third-lump solutions in Equation (1) in the (x,y)-plane are shown in Figure 6.Figure 6
Two types of third-lump solution in Equation (1) at t=0. (a) Bright-type lump with u0=1 andc=−1. (b) Dark-type lump with u0=−1 and c=1.
(a)(b)In addition, interaction solutions between breather solutions and lump solutions also can be obtained from solitons by taking long wave limits. If usingN=6, p1=P1ε, p2=P2ε, k1=K1ε, k2=K2ε, P1=P2¯, K1=K2¯, ϕ1=iπ, ϕ2=iπ, η3=η4¯, and η5=η6¯ in Equation (31), then we can obtain interactions between a lump and two breathers after taking a long wave limit. In order to better describe the structure of interactions, we assign the parameters as follows: c0=1, b=−1, K1=4i, K2=−4i, P1=10, P2=10, k3=i, k4=−i, p3=−5/3, p4=−5/3, k5=i, k6=−i, p5=11/6, p6=11/6, ϕ1=iπ, ϕ2=iπ, ϕ3=0, ϕ4=0, ϕ5=0, and ϕ6=0. If using u0=1 and c=−1, then we get the bright-type interactions; if using u0=−1 and c=1, then we can obtain dark-type interactions. Two types of interactions between a lump and two breathers in Equation (1) in the (x,y)-plane are shown in Figure 7.Figure 7
Two types of interactions between a lump and two breathers in Equation (1) at t=0. (a) Bright-type interaction solution with u0=1 andc=−1. (b) Dark-type interaction solution with u0=−1 and c=1.
(a)(b)The idea and process of obtaining interactions between two lumps and a breather are roughly the same as that of obtaining interaction between a lump and two breathers. If usingN=6, ps=Psε, ks=Ksε, ϕs=iπ, s=1,2,3,4, P2j−1=P2j¯, K2j−1=K2j¯, j=1,2, and η5=η6¯ in Equation (31), then we can obtain interactions between two lumps and a breather after taking a long wave limit. In order to better describe the structure of interactions between a breather and two lumps in Equation (1), we assign the parameters as follows: c0=1, b=−1, K1=3i, K2=−3i, P1=5, P2=5, K3=3i, K4=−3i, P3=6, P4=6, k5=i, k6=−i, p5=5/4, p6=5/4, ϕ1=iπ, ϕ2=iπ, ϕ3=iπ, ϕ4=iπ, ϕ5=0, and ϕ6=0, then we get the bright-type interactions; if using u0=−1 and c=1, then we can obtain dark-type interactions. Two types of interactions between a breather and two lumps in Equation (1) in the (x,y)-plane are shown in Figure 8.Figure 8
Two types of interactions between a lump and two breathers in Equation (1) at t=0. (a) Bright-type interaction solution with u0=1 andc=−1. (b) Dark-type interaction solution with u0=−1 andc=1.
(a)(b)


## 4. Interaction between Lumps and a Stripe

The method of obtaining interaction solutions is roughly similar to that of obtaining rational solutions, but slightly different. In this part, we will describe how to obtain interaction solutions.


### 4.1. Interaction between a Lump and a Stripe

To obtain the interaction between a lump and a stripe of Equation (1), we substitute N=3, k1=K1ε, k2=K2ε, p1=P1ε, p2=P2ε, ϕ1=iπ, and ϕ2=iπ into Equation (31), and then, we expand the resulting f in terms of ε at ε=0. Similar to Equations (24) and (26), we also want to get rid of oε2. In other words, we divide the expansion by ε2, and then, we take the limit as ε is equal to 0. For convenience, the expression after we obtain the limit is still called f, as follows:
(44)f=θ1θ2+a12+eη3a13a23+θ2a13+θ1a23+θ1θ2+a12,with(45)ajs=24bKj3Ks3c0KjPs−KsPj2s<3,−24bKj3k336bKj2k34−c0Kj2p32+2c0Pjp3k3Kj−c0Pj2k32s=3,(46)θs=−2Ks2cu0−Ps2c0t2Ks+Ksx+Psys=1,2,3,2bk34+2ck32u0+c0p32+2k3w3=0.In order for the mixed solution in Equation (44) to become interaction between a lump and a stripe, there are some restrictions for a valid calculation: K1=K2¯, P1=P2¯, and parameters k3,p3,ϕ3 must be real parameters. In order to describe the properties of mixed solutions more directly and concretely, the parameters of Equation (45) are assigned as follows: K1=1+i, K2=1−i, P1=1, P2=1, k3=1/2, p3=2, ϕ3=0, c0=1, and b=−1. If using u0=1 and c=−1, we will get bright-type interaction solutions; if using u0=−1 and c=1, we are going to get dark-type interaction solutions. After calculation, two types of interaction solutions have the same f under the above parameters. The interactions between a lump and a stripe in the (x,y)-plane are shown in the Figure 9.
(47)f=4tx+2x2+y2+17t28+32ty+2xy+19248389−1212t389−1296x389−816y389e−27t/8+x/2+2y+4tx+2x2+y2+17t28+32ty+2xy+48.Figure 9
Two types of interactions between a lump and a stripe in Equation (1). (a) Bright-type interaction solutions with u0=1 and c=−1. (b) Dark-type interaction solutions with u0=−1 andc=1.
(a)(b)


### 4.2. Interaction between Two Lumps and a Stripe

The method and idea of obtaining interaction between two lumps and a stripe are roughly the same as the process of obtaining interaction between a lump and a stripe, but the calculation is more complicated. We substituteks=Ksε, ps=Psε, expϕs=−1, and s=1,2,3,4 into Equation (31), and then, we expand the expression at ε=0. And then, we are going to divide this by ε4, and we are going to take the limit as ε is equal to 0. For convenience, let us call this final result f.
(48)f=θ1θ2θ3θ4+a12θ3θ4+a13θ2θ4+a14θ2θ3+a23θ1θ4+a24θ1θ3+a34θ1θ2+a12a34+a13a24+a14a23+eη5a15a25a35a45+a15a25a35θ4+a15a25a45θ3+a15a25θ3θ4+a15a35a45θ2+a15a35θ2θ4+a15a45θ2θ3+a15θ2θ3θ4+a25a35a45θ1+a25a35θ1θ4+a25a45θ1θ3+a25θ1θ3θ4+a35a45θ1θ2+a35θ1θ2θ4+a45θ1θ2θ3+θ1θ2θ3θ4+a12a35a45+a12a35θ4+a12a45θ3+a12θ3θ4+a13a25a45+a13a25θ4+a13a45θ2+a13θ2θ4+a14a25a35+a14a25θ3+a14a35θ2+a14θ2θ3+a15a23a45+a15a23θ4+a15a24a35+a15a24θ3+a15a25a34+a15a34θ2+a23a45θ1+a23θ1θ4+a24a35θ1+a24θ1θ3+a25a34θ1+a34θ1θ2+a12a34+a13a24+a14a23,with
(49)ajs=24bKj3Ks3c0KjPs−KsPj2s<5,−24bKj3k536bKj2k54−Kj2c0p52+2KjPjc0k5p5−Pj2c0k52s=5,and
(50)θs=−2Ks2cu0−Ps2c0t2Ks+Ksx+Psys=1,2,3,4,2bk54+2ck52u0+c0p52+2k5w5=0.In order to ensure that mixed solution is an interaction solution between two lumps and a stripe, we make some restrictions in Equation (48). N=5, K1=K2¯, K3=K4¯, P1=P2¯, P3=P4¯, and parameters k5,p5,ϕ5 must be real parameters. To intuitively describe the properties of mixed solutions of five soliton, the parameters of Equation (48) are assigned as follows: K1=1+i, K2=1−i, P1=−2, P2=−2, K3=1−i, K4=1+i, P3=2, P4=2, k5=3/4, p5=2, ϕ5=0, c0=1, and b=−1. If using u0=1 and c=−1, we will obtain bright-type interaction solutions; if using u0=−1 and c=1,we will obtain dark-type interaction solutions. Under the above parameter constraints, two types of mixed solutions of five solitons correspond to the same f. The interaction solutions in the (x,y)-plane are shown in the Figure 10.Figure 10
Two types of interactions between two lumps and a stripe in Equation (1). (a) Bright-type interaction solution with u0=1 and c=−1. (b) Dark-type interaction solution with u0=−1 and c=1.
(a)(b)


## 4.1. Interaction between a Lump and a Stripe

To obtain the interaction between a lump and a stripe of Equation (1), we substitute N=3, k1=K1ε, k2=K2ε, p1=P1ε, p2=P2ε, ϕ1=iπ, and ϕ2=iπ into Equation (31), and then, we expand the resulting f in terms of ε at ε=0. Similar to Equations (24) and (26), we also want to get rid of oε2. In other words, we divide the expansion by ε2, and then, we take the limit as ε is equal to 0. For convenience, the expression after we obtain the limit is still called f, as follows:
(44)f=θ1θ2+a12+eη3a13a23+θ2a13+θ1a23+θ1θ2+a12,with(45)ajs=24bKj3Ks3c0KjPs−KsPj2s<3,−24bKj3k336bKj2k34−c0Kj2p32+2c0Pjp3k3Kj−c0Pj2k32s=3,(46)θs=−2Ks2cu0−Ps2c0t2Ks+Ksx+Psys=1,2,3,2bk34+2ck32u0+c0p32+2k3w3=0.In order for the mixed solution in Equation (44) to become interaction between a lump and a stripe, there are some restrictions for a valid calculation: K1=K2¯, P1=P2¯, and parameters k3,p3,ϕ3 must be real parameters. In order to describe the properties of mixed solutions more directly and concretely, the parameters of Equation (45) are assigned as follows: K1=1+i, K2=1−i, P1=1, P2=1, k3=1/2, p3=2, ϕ3=0, c0=1, and b=−1. If using u0=1 and c=−1, we will get bright-type interaction solutions; if using u0=−1 and c=1, we are going to get dark-type interaction solutions. After calculation, two types of interaction solutions have the same f under the above parameters. The interactions between a lump and a stripe in the (x,y)-plane are shown in the Figure 9.
(47)f=4tx+2x2+y2+17t28+32ty+2xy+19248389−1212t389−1296x389−816y389e−27t/8+x/2+2y+4tx+2x2+y2+17t28+32ty+2xy+48.Figure 9
Two types of interactions between a lump and a stripe in Equation (1). (a) Bright-type interaction solutions with u0=1 and c=−1. (b) Dark-type interaction solutions with u0=−1 andc=1.
(a)(b)


## 4.2. Interaction between Two Lumps and a Stripe

The method and idea of obtaining interaction between two lumps and a stripe are roughly the same as the process of obtaining interaction between a lump and a stripe, but the calculation is more complicated. We substituteks=Ksε, ps=Psε, expϕs=−1, and s=1,2,3,4 into Equation (31), and then, we expand the expression at ε=0. And then, we are going to divide this by ε4, and we are going to take the limit as ε is equal to 0. For convenience, let us call this final result f.
(48)f=θ1θ2θ3θ4+a12θ3θ4+a13θ2θ4+a14θ2θ3+a23θ1θ4+a24θ1θ3+a34θ1θ2+a12a34+a13a24+a14a23+eη5a15a25a35a45+a15a25a35θ4+a15a25a45θ3+a15a25θ3θ4+a15a35a45θ2+a15a35θ2θ4+a15a45θ2θ3+a15θ2θ3θ4+a25a35a45θ1+a25a35θ1θ4+a25a45θ1θ3+a25θ1θ3θ4+a35a45θ1θ2+a35θ1θ2θ4+a45θ1θ2θ3+θ1θ2θ3θ4+a12a35a45+a12a35θ4+a12a45θ3+a12θ3θ4+a13a25a45+a13a25θ4+a13a45θ2+a13θ2θ4+a14a25a35+a14a25θ3+a14a35θ2+a14θ2θ3+a15a23a45+a15a23θ4+a15a24a35+a15a24θ3+a15a25a34+a15a34θ2+a23a45θ1+a23θ1θ4+a24a35θ1+a24θ1θ3+a25a34θ1+a34θ1θ2+a12a34+a13a24+a14a23,with
(49)ajs=24bKj3Ks3c0KjPs−KsPj2s<5,−24bKj3k536bKj2k54−Kj2c0p52+2KjPjc0k5p5−Pj2c0k52s=5,and
(50)θs=−2Ks2cu0−Ps2c0t2Ks+Ksx+Psys=1,2,3,4,2bk54+2ck52u0+c0p52+2k5w5=0.In order to ensure that mixed solution is an interaction solution between two lumps and a stripe, we make some restrictions in Equation (48). N=5, K1=K2¯, K3=K4¯, P1=P2¯, P3=P4¯, and parameters k5,p5,ϕ5 must be real parameters. To intuitively describe the properties of mixed solutions of five soliton, the parameters of Equation (48) are assigned as follows: K1=1+i, K2=1−i, P1=−2, P2=−2, K3=1−i, K4=1+i, P3=2, P4=2, k5=3/4, p5=2, ϕ5=0, c0=1, and b=−1. If using u0=1 and c=−1, we will obtain bright-type interaction solutions; if using u0=−1 and c=1,we will obtain dark-type interaction solutions. Under the above parameter constraints, two types of mixed solutions of five solitons correspond to the same f. The interaction solutions in the (x,y)-plane are shown in the Figure 10.Figure 10
Two types of interactions between two lumps and a stripe in Equation (1). (a) Bright-type interaction solution with u0=1 and c=−1. (b) Dark-type interaction solution with u0=−1 and c=1.
(a)(b)


## 5. Conclusion

In this manuscript, applying Hirota’s bilinear method with a perturbation parameteru0 to generalized Kadomtsev-Petviashvili equation, we obtain a periodic breather wave solution. Meanwhile, lump solutions and interaction solutions are generated from solitons by taking long wave limits. The exact solutions contain some free parameters u0,b,c,c0, so some new and interesting space structures of breather, lump, and interaction solutions are found and investigated, which include structures of bright type and structures of dark type. Our results show the diversity of the spatial and space-time structures of solitary waves in nonlinear dynamic systems. Meanwhile, we also hope that our results will provide some valuable information in the field of nonlinear science.



---

*Source: 9882817-2022-11-28.xml*