# Weakly Coupled B-Type Kadomtsev-Petviashvili Equation: Lump and Rational Solutions

Authors: Na Xiong; Wen-Tao Li; Biao Li
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

Through the method ofZN-KP hierarchy, we propose a new (3+1)-dimensional weakly coupled B-KP equation. Based on the bilinear form, we obtain the lump and rational solutions to the dimensionally reduced cases by constructing a symmetric positive semidefinite matrix. Then, we do numerical analysis on the rational solutions and fit the trajectory equation of the crest. Furthermore, we verify the accuracy of the trajectory equation by numerical analysis. This method of solving the lump and rational solutions can also be applied to other nonlinear evolution equations.


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## Body

## 1. Introduction

During the recent years, in the field of nonlinear evolution equations, the exact solution is also widely concerned, such as the rational solutions, lump solutions, hybrid solutions, and rogue wave solutions [1–5]. And it has been applied to many fields: solid-state physics, fluid dynamics, plasma physics, and other nonlinear engineering problems. Furthermore, constructing exact solutions for nonlinear evolution equations is important in many scientific or engineering applications. Therefore, a series of systematic approaches, the Lie group method [6], Bäcklund transformation method, Darboux transformation method [7–10], Hirota bilinear method [11–13], inverse scattering transform method [14], and other methods [15], have been rapidly developed. It is meaningful and interesting to search for solutions to nonlinear evolution equations.In this paper, the (3+1)-dimensional B-KP equation is expressed as(1)uxxxy+αuxuyx+ux+uy+uzt−uxx+uzz=0.The B-KP equation describes the evolution of quasi-one-dimensional shallow water waves, when the effects of surface tension and viscosity are negligible and these are widely used in various physics fields such as the surface and internal oceanic waves, nonlinear optics, plasma physics, ferromagnetics, Bose-Einstein condensation, and string theory. Because of its importance, many properties have been studied. Wazwaz [16] established one and two soliton solutions of this equation by using the simplified Hereman-Nuseir form. Huang et al. [17] derived the bilinear Bäcklund transformations and soliton solutions to Equation (1); Bright-Dark lump wave solutions for a new form of the (3+1)-dimensional B-KP-Boussinesq equation have been explored in [18]. Lump, breather, and solitary wave solutions to the new reduced form of the generalized B-KP equation have been found in [19], and the dynamical analysis of lump solutions is exhibited in [20].Based on bilinear derivatives, the B-KP equation can be transformed into the following bilinear equation:(2)Dx3Dy+DxDt+DyDt+DzDt−Dx2−Dz2f·f=0,under a dependent variable transformation(3)ux,y,z,t=6αlnfx.The operatorD is the Hirota bilinear differential operator, which is defined by(4)DxmDtnf·g=∂∂x−∂∂x′m∂∂t−∂∂t′nfx,y,z,t·gx,y,z,tx′=x,t′=t.By taking theZN-KP hierarchy method [21], a new (3+1)-dimensional weakly coupled B-KP equation can be derived in the following form:(5)uxxxy+αuxuyx+ux+uy+uzt−uxx+uzz=0,vxxxy+αvxuyx+αuxvyx+vx+vy+vzt−vxx+vzz=0.Equation (5) will be reduced to the following (2+1)-dimensional weakly coupled B-KP equation under z=x:(6)uxxxy+αuxuyx+2ux+uyt=0,vxxxy+αvxuyx+αuxvyx+2vx+vyt=0.In this work, we mainly explore the lump and rational solutions to weakly coupled B-KP Equation (6). The rest of the paper is arranged as follows. In Section 2, the bilinear formalism of (2+1)-dimensional weakly coupled B-KP equation is derived. In Section 3, we obtain the lump and rational solutions to this equation by constructing symmetric positive semidefinite matrices, and some properties of the lump solutions are analyzed and discussed. Furthermore, we get the trajectory equation of the crest by using numerical analysis methods. Finally, our conclusion is presented in Section 4.


## 2. Bilinear Formalism

We start from a potential fieldf to construct bilinear transformation(7)u=6αlnfx,v=6αgfx.By substitutingu and v into Equation (6), the result is presented as(8)2fxxxy−2fxxxfy−6fxxyfx+6fxxfxy+4ftxf−4fxft+2fty−2fyft−4fxxf+4fx2=0,fxxxyg−fxxxgy−3fxxygx+3fxxgxy+3fxygxx−3fxgxxy−fygxxx+fgxxxy+2ftxg−2fxgt−2ftgx+2fgtx+ftyg−fygt−ftgy+fgty−2fxxg+4fxgx−2fgxx=0.Equation (6) becomes the Hirota bilinear equation:(9)Dx3Dy+2DxDt+DyDt−2Dx2f·f=0,Dx3Dy+2DxDt+DyDt−2Dx2f·g=0,where f and g are real functions with respect to variables x, y, and t and the operator D is the Hirota bilinear operate.


## 3. Lump Solutions and Rational Solutions

### 3.1. Lump Solutions

In order to obtain rational solutions to Equation (6), we set(10)f=XTAX+c1,g=XTBX+c2,with(11)A=a11a12a13a12a22a23a13a23a33,B=b11b12b13b12b22b23b13b23b33,X=xyt,where aij, bij, and ci are the real parameters that need to be determined; the matrices A and B are symmetric matrices.After directly utilizing Maple symbolic calculation withf, a series of constraint equations for parameters are as follows:(12)a11=a11,a12=2a11c1a11−a133a112+a13c1,a13=a13,a22=4a11a11−a133a112+a11c1−a13c1a133a112+a13c1,a23=2a11−a13−3a112+a13c13a112+a13c1,a33=a133a112+a13c1a113a11+c1,c1=c1.By combining Equations (10)–(12), the f is presented as follows:(13)f=1a11a133a11+c13a112+a13c136a116y2+a115−36yt+9x2−36y2a13+24y2c1+a1149tt+2x+4ya132−12c1a133y2+t−xy−x24−34+4y2c12+62y2+4t−2xy+xt+x2a13+2c13xy−2y2+34a13c1a113+6a132c1a112ta13t+x−2y+2c13y2+t−xy+x24+34+2a132c12a11a13tx−2y+c12+a134c12t2.To obtain lump solutionsu to Equation (6), the matrix A and parameter c1 should satisfy the following conditions: (1) the matrix A must be a positive semidefinite matrix; (2) the real parameter c1 must be greater than 0; (3) the elements of matrix A must also satisfy the following constraints a11>0 and a11a22−a122>0. Among the above conditions, (1) and (2) are to ensure that f is always greater than 0. Condition (3) is to ensure that f has only one minimum at any time.Under the above conditions, we found that we can take the following special parameter values:(14)a11=3,a13=1,c1=9.Substituting Equation (14) into f in Equation (7), we find(15)f=2xt+6xy−4yt+3x2+30y2+9+23t2.Therefore, we obtain a lump solution of the2+1-dimensional B-KP Equation (6):(16)u=62t+6y+6xα2xt+6xy−4yt+3x2+30y2+9+2/3t2.Figure1 shows the contour plot, x-curve, and y-curve of the lump solution. And the lump solution u has a maxium point −4/9t+3,1/9t and a minimum point −4/9t−3,1/9t. Besides, the lump solution reaches the maximum value of 23/3 and the minimum value of −23/3.Figure 1
(Color online) The lump solutionu to Equation (6) at α=3. (a) Contour plot. (b) x-curve. (c) y-curve.
(a)
t=0(b)(c)We find a class of lump solutions to Equation (6). As mentioned above, since u=6/αlnfx is a lump solution to Equation (6), then we make g=fx, so v=6/αg/fx is also a lump solution to Equation (6). The correlative solutions are presented in the following form:(17)u=62t+6y+6xα2xt+6xy−4yt+3x2+30y2+9+2/3t2,v=−3242xt+8yt+3x2+6xy−24y2−9α2t2+6xt−12yt+9x2+18xy+90y2+272.As can be seen in Figure2, when α=3, the lump solution has a maximum point and two minimum points. The maximum value is obtained at −4/9t,1/9t, and the maximum value is 4/3; the minimum value is obtained at −4/9t+3,1/9t, −4/9t−3,1/9t, and the minimum value is −1/6.Figure 2
(Color online) The lump solutionsv to Equation (6) at α=3. (b) Density plot of lump solution v. (c) Contour plot of lump solution v.
(a)
t=0(b)
t=0(c)
t=0


### 3.2. Rational Solutions

In order to get the value ofg and make the result clear, we substitute Equation (14) directly into Equation (8). And the related constraint equations are presented in the following form:(18)b11=27b332−9b13+b12,b12=b12,b13=b13,b22=28b12−288b13+351b33,b23=28b13−36b33−2b12,b33=b33,c2=45b13−54b33.Equation (18) contains three arbitrary parameters b12, b13, and b33; we assign values to parameters as follows:(19)b12=1,b13=2,b33=2.The rational solution can be presented as(20)vp=6αgfx,with(21)f=9+6xy+2xt−4yt+3x2+30y2+23t2,g=2t2+4xt−36yt+10x2+2xy+154y2−18.Therefore, it can be seen from the above thatvp is the rational solution to Equation (6), and Equations (16) and (20) are a set of solutions to weakly coupled Equation (6).After a detailed analysis, we find that rational solutionvp has some fascinating properties. This property is that vp is an odd function, which means vp−x,−y,−t=−vpx,y,t. Therefore, when we know the trajectory of the crest at t<0 at a certain period of time, we can know the corresponding trajectory of the crest at t>0. We find that the solutions of vp,x=0 and vp,y=0 have extremely complex algebraic structures, so we do not know what curve of the crest is over time. Therefore, we investigate the motion trajectory of the rational solution by taking the numerical analysis method.Whenα=3, the position of the crest at different times can be seen in Table 1. Here, h is the height of the crest and the vp,x and vp,y are used to measure the error of numerical calculation. When vp,x and vp,y are closer to 0, the position of the crest will be more accurate.Table 1
The position of the crest of rational solutionvp.
txyhvp,xvp,y70-31.165930527.784650053-72.77660922-0.0099917450.00672053880-35.603531488.894905304-83.12405064-0.0069676610.00468702890-40.0426514510.00535012-93.47655836-0.0049851070.003357235100-44.4828351911.11592780-103.8325833-0.0036296980.002450331110-48.9237928912.22660221-114.1912831-0.0026799830.001809572120-53.3653311913.33734922-124.5519253-0.0019959000.001336341130-57.8073162414.44815212-134.9139212-0.0014868560.000991834140-62.2496523615.55899896-145.2777821-0.0011182180.000753025150-66.6922694116.66988097-155.6419954-0.0008371700.000580849160-71.1351147217.78079155-166.0064119-0.0006103800.000408621From the above description, we construct a nonlinear fit to the data in Table1. During a certain period of time t>0, the motion trajectory of this crest can be fitted to the following equation:(22)x=−3.84308585121464619t+0.104467282340778067−49t,y=0.483098011069225342t+0.296660717750461933+19t.We find from Equation (22) that as the time increases gradually, the trajectory of the crest will fit more linearly. And, an interesting phenomenon that can be seen from Figure 2 is that the strange wave of vp is very similar to the lump wave at every moment, and its amplitude and time are proportional. Since vp is an odd function, we obtain the trajectory equation of vp at t<0 by combining Equation (23). So the result is presented in the following form:(23)x=−3.84308585121464619t−0.104467282340778067−49t,y=0.483098011069225342t−0.296660717750461933+19t.Thus, Equations (22) and (23) can accurately express the trajectory of vp at every moment. Next, we do some numerical tests on the equations in Equations (22) and (23). The results are presented in Table 2.Table 2
Test of the fitted equation.
txyhvp,xvp,y-700311.1166020-77.77846821725.9264744-0.0017268010.001024956-600266.6730730-66.66747223622.2330861-0.0020948500.001361426-500222.2299100-55.55652233518.5607668-0.0024830380.001668119500-222.229906855.55652118-518.52674850.000035034-0.000062859600-266.673070766.66747144-622.2417001-0.0002083890.000347771700-311.116600477.77846763-725.9318515-0.000131428-0.000185396As can be seen from Table2, when t∈−700,700, vp,x and vp,y are very close to 0, so it can be explained that Equations (22) and (23) can accurately represent the position of the crest.


## 3.1. Lump Solutions

In order to obtain rational solutions to Equation (6), we set(10)f=XTAX+c1,g=XTBX+c2,with(11)A=a11a12a13a12a22a23a13a23a33,B=b11b12b13b12b22b23b13b23b33,X=xyt,where aij, bij, and ci are the real parameters that need to be determined; the matrices A and B are symmetric matrices.After directly utilizing Maple symbolic calculation withf, a series of constraint equations for parameters are as follows:(12)a11=a11,a12=2a11c1a11−a133a112+a13c1,a13=a13,a22=4a11a11−a133a112+a11c1−a13c1a133a112+a13c1,a23=2a11−a13−3a112+a13c13a112+a13c1,a33=a133a112+a13c1a113a11+c1,c1=c1.By combining Equations (10)–(12), the f is presented as follows:(13)f=1a11a133a11+c13a112+a13c136a116y2+a115−36yt+9x2−36y2a13+24y2c1+a1149tt+2x+4ya132−12c1a133y2+t−xy−x24−34+4y2c12+62y2+4t−2xy+xt+x2a13+2c13xy−2y2+34a13c1a113+6a132c1a112ta13t+x−2y+2c13y2+t−xy+x24+34+2a132c12a11a13tx−2y+c12+a134c12t2.To obtain lump solutionsu to Equation (6), the matrix A and parameter c1 should satisfy the following conditions: (1) the matrix A must be a positive semidefinite matrix; (2) the real parameter c1 must be greater than 0; (3) the elements of matrix A must also satisfy the following constraints a11>0 and a11a22−a122>0. Among the above conditions, (1) and (2) are to ensure that f is always greater than 0. Condition (3) is to ensure that f has only one minimum at any time.Under the above conditions, we found that we can take the following special parameter values:(14)a11=3,a13=1,c1=9.Substituting Equation (14) into f in Equation (7), we find(15)f=2xt+6xy−4yt+3x2+30y2+9+23t2.Therefore, we obtain a lump solution of the2+1-dimensional B-KP Equation (6):(16)u=62t+6y+6xα2xt+6xy−4yt+3x2+30y2+9+2/3t2.Figure1 shows the contour plot, x-curve, and y-curve of the lump solution. And the lump solution u has a maxium point −4/9t+3,1/9t and a minimum point −4/9t−3,1/9t. Besides, the lump solution reaches the maximum value of 23/3 and the minimum value of −23/3.Figure 1
(Color online) The lump solutionu to Equation (6) at α=3. (a) Contour plot. (b) x-curve. (c) y-curve.
(a)
t=0(b)(c)We find a class of lump solutions to Equation (6). As mentioned above, since u=6/αlnfx is a lump solution to Equation (6), then we make g=fx, so v=6/αg/fx is also a lump solution to Equation (6). The correlative solutions are presented in the following form:(17)u=62t+6y+6xα2xt+6xy−4yt+3x2+30y2+9+2/3t2,v=−3242xt+8yt+3x2+6xy−24y2−9α2t2+6xt−12yt+9x2+18xy+90y2+272.As can be seen in Figure2, when α=3, the lump solution has a maximum point and two minimum points. The maximum value is obtained at −4/9t,1/9t, and the maximum value is 4/3; the minimum value is obtained at −4/9t+3,1/9t, −4/9t−3,1/9t, and the minimum value is −1/6.Figure 2
(Color online) The lump solutionsv to Equation (6) at α=3. (b) Density plot of lump solution v. (c) Contour plot of lump solution v.
(a)
t=0(b)
t=0(c)
t=0


## 3.2. Rational Solutions

In order to get the value ofg and make the result clear, we substitute Equation (14) directly into Equation (8). And the related constraint equations are presented in the following form:(18)b11=27b332−9b13+b12,b12=b12,b13=b13,b22=28b12−288b13+351b33,b23=28b13−36b33−2b12,b33=b33,c2=45b13−54b33.Equation (18) contains three arbitrary parameters b12, b13, and b33; we assign values to parameters as follows:(19)b12=1,b13=2,b33=2.The rational solution can be presented as(20)vp=6αgfx,with(21)f=9+6xy+2xt−4yt+3x2+30y2+23t2,g=2t2+4xt−36yt+10x2+2xy+154y2−18.Therefore, it can be seen from the above thatvp is the rational solution to Equation (6), and Equations (16) and (20) are a set of solutions to weakly coupled Equation (6).After a detailed analysis, we find that rational solutionvp has some fascinating properties. This property is that vp is an odd function, which means vp−x,−y,−t=−vpx,y,t. Therefore, when we know the trajectory of the crest at t<0 at a certain period of time, we can know the corresponding trajectory of the crest at t>0. We find that the solutions of vp,x=0 and vp,y=0 have extremely complex algebraic structures, so we do not know what curve of the crest is over time. Therefore, we investigate the motion trajectory of the rational solution by taking the numerical analysis method.Whenα=3, the position of the crest at different times can be seen in Table 1. Here, h is the height of the crest and the vp,x and vp,y are used to measure the error of numerical calculation. When vp,x and vp,y are closer to 0, the position of the crest will be more accurate.Table 1
The position of the crest of rational solutionvp.
txyhvp,xvp,y70-31.165930527.784650053-72.77660922-0.0099917450.00672053880-35.603531488.894905304-83.12405064-0.0069676610.00468702890-40.0426514510.00535012-93.47655836-0.0049851070.003357235100-44.4828351911.11592780-103.8325833-0.0036296980.002450331110-48.9237928912.22660221-114.1912831-0.0026799830.001809572120-53.3653311913.33734922-124.5519253-0.0019959000.001336341130-57.8073162414.44815212-134.9139212-0.0014868560.000991834140-62.2496523615.55899896-145.2777821-0.0011182180.000753025150-66.6922694116.66988097-155.6419954-0.0008371700.000580849160-71.1351147217.78079155-166.0064119-0.0006103800.000408621From the above description, we construct a nonlinear fit to the data in Table1. During a certain period of time t>0, the motion trajectory of this crest can be fitted to the following equation:(22)x=−3.84308585121464619t+0.104467282340778067−49t,y=0.483098011069225342t+0.296660717750461933+19t.We find from Equation (22) that as the time increases gradually, the trajectory of the crest will fit more linearly. And, an interesting phenomenon that can be seen from Figure 2 is that the strange wave of vp is very similar to the lump wave at every moment, and its amplitude and time are proportional. Since vp is an odd function, we obtain the trajectory equation of vp at t<0 by combining Equation (23). So the result is presented in the following form:(23)x=−3.84308585121464619t−0.104467282340778067−49t,y=0.483098011069225342t−0.296660717750461933+19t.Thus, Equations (22) and (23) can accurately express the trajectory of vp at every moment. Next, we do some numerical tests on the equations in Equations (22) and (23). The results are presented in Table 2.Table 2
Test of the fitted equation.
txyhvp,xvp,y-700311.1166020-77.77846821725.9264744-0.0017268010.001024956-600266.6730730-66.66747223622.2330861-0.0020948500.001361426-500222.2299100-55.55652233518.5607668-0.0024830380.001668119500-222.229906855.55652118-518.52674850.000035034-0.000062859600-266.673070766.66747144-622.2417001-0.0002083890.000347771700-311.116600477.77846763-725.9318515-0.000131428-0.000185396As can be seen from Table2, when t∈−700,700, vp,x and vp,y are very close to 0, so it can be explained that Equations (22) and (23) can accurately represent the position of the crest.


## 4. Conclusion

In this paper, by using theZm-KP hierarchy, a new (3+1)-dimensional weakly coupled B-KP equation has been constructed. And its dimensionally reduced form has been investigated. We obtain the lump solutions to Equation (6) by taking the Hirota bilinear method and constructing the symmetric positive semidefinite matrix technique. The lump solutions are displayed in three-dimensional graphics (see Figures 1 and 2). We also found an interesting set of rational solutions u,vp to Equation (6) by using the same method. And the height of vp will also increase proportionally over time. Time-varying rational solutions are presented in graphics (see Figure 3). By taking a numerical analysis of the rational solution vp, the motion trajectory of vp is approximately linear motion over time (see Table 1). And we also test the equations of the moving trajectory by using numerical analysis methods. The position and fitting results of the trajectory equation are presented (see Table 2). In this paper, the method of finding the lump and rational solutions to weakly coupled B-KP equations by constructing a symmetric positive semidefinite matrix can also be applied to other nonlinear evolution equations. We hope that our results can provide valuable help in the field of nonlinear science.Figure 3
(Color online) The lump solutionsv to Equation (6) at α=3.
(a)
t=50(b)
t=100(c)
t=200(d)
t=400(e)
t=50(f)
t=100(g)
t=200(h)
t=400



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