# Lie Symmetry Analysis, Traveling Wave Solutions, and Conservation Laws to the (3 + 1)-Dimensional Generalized B-Type Kadomtsev-Petviashvili Equation

Authors: Huizhang Yang; Wei Liu; Yunmei Zhao
Journal: Complexity
 (2020)

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## Abstract

In this paper, the (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili(BKP) equation is studied applying Lie symmetry analysis. We apply the Lie symmetry method to the (3 + 1)-dimensional generalized BKP equation and derive its symmetry reductions. Based on these symmetry reductions, some exact traveling wave solutions are obtained by using the tanh method and Kudryashov method. Finally, the conservation laws to the (3 + 1)-dimensional generalized BKP equation are presented by invoking the multiplier method.


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## Body

## 1. Introduction

Nonlinear partial differential equations (NPDEs) are widely used as models to describe a great number of complex nonlinear phenomena which appear in many fields, such as hydrodynamics, biology, plasma physics, fluid dynamics, solid state physics, optics, and applied mathematics. To really understand such phenomena describing in nature, searching for exact solutions of NPDEs plays an important role in the study of nonlinear science. In recent years, more and more methods have been proposed, such as the inverse scattering method [1], the Bäcklund transform method [2], the Darboux transform method [3], the Hirota bilinear transformation method [4], the Exp-function method [5], the tanh-function method [6], the sine-Gordon equation expansion technique [7], and the Kudryashov method [8].The Lie symmetry method presented by Lie [9] is one of the well-known methods for obtaining exact solutions of nonlinear PDEs. Up to now, the Lie symmetry method has been applied to a number of mathematical and physical models, see [10–14] and references therein. This method is effective to get similarity solutions and solitary wave solutions of NPDEs.The study of different BKP equations has attracted a considerable size of research. Different forms of BKP equations were studied by using the Hirota method, the multiple exp-function algorithm, the Pfaffian technique, the Wronskian technique, and the Bäcklund transformation [15–21]. In this paper, by means of the Lie symmetry group method, we consider the following (3 + 1)-dimensional generalized BKP equation:(1)uyt−uxxxy−3uxuyx+3uxx+3uzz=0,where u is a real differentiable function of the scaled spatial coordinates x,y,andz and temporal coordinate t, while the subscripts denote the partial derivatives. This equation was first presented by Ma et al. [15]. When z=0, the (3 + 1)-dimensional generalized BKP equation can be reduced to the following (2 + 1)-dimensional BKP equation:(2)uyt−uxxxy−3uxuyx+3uxx=0.Equation (1) has been investigated by different methods. Following the linear superposition principle of exponential waves, Ma et al. obtained an N-wave solution for equation (1) in [15]. A bilinear Bäcklund transformation and a class of exact Pfaffian solutions for equation (1) were established using the Pfaffian technique and Hirotas bilinear operator identities in [16]. Moreover, Wazwaz [17] established two sets of distinct kinds of multiple-soliton solutions under specific conditions for equation (1) using the simplified form of the Hirota method. Soliton solutions in Wronskian form of (1) were also presented in [18, 19]. Multiple wave solutions and auto-Bäcklund transformation for the (1) were obtained in [20]. The lump solutions, periodic waves, and rogue waves as well as interaction solutions of (1) were obtained in [21].As far as we know, the Lie symmetry analysis and conservation laws to the (3 + 1)-dimensional generalized BKP equation (1) have not been discussed. The main purpose of this paper is to study the Lie symmetry analysis method [22–24], exact traveling wave solutions, and conservation laws of equation (1). The rest paper is arranged as follows: Section 2 is devoted to describe the Lie symmetry vectors and symmetry reductions using Lie symmetry analysis. In Section 3, various exact traveling wave solutions are attained after reductions process by using the tanh method and Kudryashov method. Section 4 is devoted to find the conservation laws of equation (1) by utilizing the multiplier method. Some conclusions are made in Section 5.


## 2. Lie Symmetry Analysis and Symmetry Reductions for Equation (1)

We consider the one-parameter Lie group of infinitesimal transformations inx,t,y,z,andu given by(3)x⟶x+εξx,t,y,z,u,t⟶t+ετx,t,y,z,u,y⟶y+εηx,t,y,z,u,z⟶z+εμx,t,y,z,u,u⟶u+εϕx,t,y,z,u,where ϵ is a group parameter. The associated vector field of equation (1) can be written as(4)V=ξx,t,y,z,u∂∂x+τx,t,y,z,u∂∂t+ηx,t,y,z,u∂∂y+μx,t,y,z,u∂∂z+ϕx,t,y,z,u∂∂u,where the coefficient functions ξx,t,y,z,u, τx,t,y,z,u,ηx,t,y,z,u, μx,t,y,z,u, and ϕx,t,y,z,u are to be determined later.The vector field (4) is a symmetry of equation (1) if and only if(5)pr4VΔ|Δ=0=0,where Δ=uyt−uxxxy−3uxxuy−3uxuxy+3uxx+3uzz and pr4V denote the fourth prolongation of V. Based on the Lie theory, we obtain an equivalent condition for (5) as(6)pr4VΔ=ϕyt−ϕxxxy−3uyϕxx−3uxxϕy−3uxyϕx−3uxϕxy+3ϕxx+3ϕzz=0,with(7)ϕi=Diϕ−ξux−ηuy−μuz−τut+ξuix+ηuiy+μuiz+τuit,i=x,y,ty,xx,xy,zz,xxx,xxxy,where Dt,Dx,Dy,andDz are total derivative operators.Substituting (7) into (6), we can obtain the determining equations for the symmetry group of equation (1). Solving these determining equations, we conclude that the general infinitesimal symmetry of (1) has the following forms of the coefficient functions:(8)ξ=−c1x−3h1t+c3,τ=−3c1t+c4,η=c1+c2y−c5z+c6,μ=−c1+12c2z+6c5t+c7,ϕ=c1u+h1′tx+c2y+h2tz+h3t,where cii=1,…,7 are arbitrary constants, h1t,h2t,andh3t are arbitrary functions of t, and “′” is the derivative with respect to t. Therefore, the infinitesimal symmetries of equation (1) are(9)V1=∂∂x,V2=∂∂t,V3=∂∂y,V4=∂∂z,V5=h2tz∂∂u,V6=h3t∂∂u,V7=−z∂∂y+6t∂∂z,V8=−3h1t∂∂x+h1′tx∂∂u,V9=2y∂∂y+z∂∂z+2y∂∂u,V10=−x∂∂x−3t∂∂t+y∂∂y−z∂∂z+u∂∂u.It can be verified thatV1,…,V9 is closed under the Lie bracket. The commutator table is shown in Tables 1 and 2.Table 1
Commutator table.
Vi,VjV1V2V3V4V5V100000V20000V5h2′tV300000V40000V6h2tV50−V5h2′t0−V6h2t0V60−V6h3′t000V70−6V40V36tV6h2tV8−V6h1′t−V8h1′t000V900−2V3+V6/h3t−V4V5V10V13V2−V3V4−2V5−3tV5h2′tTable 2
Commutator table.
Vi,VjV6V7V8V9V10V100V6h1′0−V1V2V6h3′t6V4V8h1′0−3V2V30002V3+V6/h3tV3V40−V30V4−V4V50−6tV6h2t0−V52V5+3tV5h2′tV60000V6+3tV6h3′tV7000V7−2V5/h2t2V7V80000V8−h1t+3th1′tV90−V7+2V5/h2t000V10−V6−3tV6h3′t−2V7−V8−h1t+3th1′t00In order to derive the traveling wave solutions of (1), we consider a linear combination of the translation symmetries V1,V2,V3, and V4, namely, V=αV1+V2+V3+V4, where α is a constant. Solving the corresponding characteristic equation,(10)dxα=dy1=dz1=dt1=du0,we obtain the following invariants:(11)f=x−αt,g=x−αy,h=y−z,u=vf,g,h.Substituting (11) into (1), we can reduce the equation (1) to(12)α2+6vfg−αvfh+αvfffg+3αvffgg+3αvfggg+αvgggg−vfffh−3vffgh−3vfggh−vgggh+3αvfvfg+3αvfvgg+9αvgvfg+6αvgvgg+3αvgvff−3vfvfh−3vfvgh−3vgvfh−3vgvgh−6vhvfg−3vhvgg−3vhvff+3vff+3vgg+3vhh=0.We write the vector field of (12) as(13)X=ξ1f,g,h,v∂∂f+ξ2f,g,h,v∂∂g+ξ3f,g,h,v∂∂h+ρf,g,h,v∂∂v,where the coefficient functions ξ1f,g,h,v, ξ2f,g,h,v,ξ3f,g,h,v, and ρf,g,h,v of the vector field are(14)ξ1=−C1f+C2,ξ2=2C1f−3C1g+C3,ξ3=−3C1h+C4,ρ=2α3−4αC1−C5αf+4C1+C5αg+C5h+C1v+C6,and Cii=1,…,6 are arbitrary constants.We perform further symmetry reductions by applying Lie symmetries to (12), then (12) has the following six Lie symmetries:(15)X1=∂∂f,X2=∂∂g,X3=∂∂h,X4=∂∂v,X5=−fα+gα+h∂∂v,X6=−f∂∂f+2f∂∂g−3h∂∂h+2α3−4αC1f+4αg+v∂∂v.We choose the symmetryX=X1+βX2+X3, where β is a constant. Three invariants are derived by solving the corresponding characteristic equation, namely,(16)r=f−h,s=g−βh,vf,g,h=ψr,s.Substituting (16) into (12), we reduce equation (12) to(17)α+6ψrr+α2+αβ+6β+6ψrs+3β2+1ψss+3α+β+1ψrψss+3α+β+1ψsψrr+6ψrψrr+33α+3β+1ψsψrs+6α+βψsψss+3α+β+3ψrψrs+α+βψssss+3α+3β+1ψrsss+3α+β+1ψrrss+α+β+3ψrrrs+ψrrrr=0.Further symmetry reduction will transform (17) into an ODE. The symmetries of (17) include two translation symmetries Γ1=∂/∂r and Γ2=∂/∂s. We consider the symmetry Γ=Γ1+ωΓ2, where ω is a constant. We obtain two invariants:(18)ξ=s−ωr,ψr,s=Uξ.Substituting (18) into (17), equation (17) is reduced to the following ODE.(19)AU4+BU′U″+CU″=0,where A=−ω−13α+β−ω, B=6ω−12α+β−ω, C=α+6ω2−α2+αβ+6β+6ω+3β2+1, and ξ=αωt+1−ωx−α+β−ωy+β−ωz.


## 3. Traveling Wave Solutions to Equation (1)

### 3.1. Exact Solutions of Equation (19) Using Tanh Method

In this subsection, we apply the tanh method to obtain solutions of equation (19). Suppose that the solution of equation (19) can be expressed as(20)Uξ=∑i=0naiZi,where n is a positive integer and aii=0,1,2,…,n are all constants. Z=tanhξ is a new independent variable, the derivatives of U with respect to ξ can be written as(21)U′=1−Z2dUdZ,U″=1−Z22d2UdZ2−2Z1−Z2dUdZ,U‴=1−Z23d3UdZ3−6Z1−Z22d2UdZ2+2Z1−Z23Z2−1dUdZ,U4=1−Z24d4UdZ4−12Z1−Z23d3UdZ3+41−Z229Z2−2d2UdZ2−8Z1−Z23Z2−2dUdZ.Balancing the linear term of the highest order with the highest order nonlinear term, we can easily obtainn=1 and thus (20) becomes(22)U=a0+a1Z.Substituting (21) into (19) and using (22), we have an algebraic equation, which on splitting with respect to the powers of Z gives the following system:(23)2a1−Ba1+12A=0,−2a1−2Ba1+20A−C=0,2a1−Ba1+8A−C=0,whose solution is(24)a0is an arbitrary constant,a1=12AB=21−ω,C=−4A.Substituting (24) into (22), we obtain the traveling wave solution to equation (19)(25)U=a0+12ABtanhξ.Thus, the traveling wave solution of the (3 + 1)-dimensional generalized BKP equation (1) is(26)ux,y,z,t=a0+21−ωtanhαωt+1−ωx−α+β−ωy+β−ωz,where a0,α,β,andω are constants and satisfy the algebraic equation 4A+C=41−ω3α+β−ω+α+6ω2−α2+αβ+6β+6ω+3β2+1=0.


### 3.2. Exact Solutions of Equation (19) Using Kudryashov Method

Suppose that the solution of equation (19) is presented as a finite series:(27)U=∑i=0naiQiξ,where n is a positive integer and aii=0,1,2,…,n are constants, which can be determined later. Here,(28)Qξ=11+daξ,which satisfies the Bernoulli equation:(29)Q′ξ=QξQξ−1lna.Applying the homogeneous balance method similar as the tanh method, we have n=1. Thus, the solution (27) becomes(30)U=a0+a1Qξ.Substituting (30) into equation (19) together with (29), we have a system of nonlinear algebraic equations:(31)2lna3a1Ba1+12Alna=0,−5lna3a1Ba1+12Alna=0,2lna2a125lna2A+2a1Blna+C=0,−lna2a115lna2A+a1Blna+3C=0,lna2a1lna2A+C=0.Solving this system with the help of Maple, we obtain(32)a0=a0,a1=−12ABlna=−21−ωlna,C=B12lna.Substituting (32) into (30) together with (29), we can obtain the traveling wave solution to equation (19):(33)U=a0−12ABlna1+daξ.Thus, the traveling wave solution of the (3 + 1)-dimensional generalized BKP equation (1) is(34)ux,y,z,t=a0−21−ωlna1+daξ,where ξ=αωt+1−ωx−α+β−ωy+β−ωz and a0,a,α,β,andω are constants and satisfy the algebraic equation 12C−Blna=12α+6ω2−12α2+αβ+6β+6ω+36β2+1−61−ω2α+β−ωlna=0.


## 3.1. Exact Solutions of Equation (19) Using Tanh Method

In this subsection, we apply the tanh method to obtain solutions of equation (19). Suppose that the solution of equation (19) can be expressed as(20)Uξ=∑i=0naiZi,where n is a positive integer and aii=0,1,2,…,n are all constants. Z=tanhξ is a new independent variable, the derivatives of U with respect to ξ can be written as(21)U′=1−Z2dUdZ,U″=1−Z22d2UdZ2−2Z1−Z2dUdZ,U‴=1−Z23d3UdZ3−6Z1−Z22d2UdZ2+2Z1−Z23Z2−1dUdZ,U4=1−Z24d4UdZ4−12Z1−Z23d3UdZ3+41−Z229Z2−2d2UdZ2−8Z1−Z23Z2−2dUdZ.Balancing the linear term of the highest order with the highest order nonlinear term, we can easily obtainn=1 and thus (20) becomes(22)U=a0+a1Z.Substituting (21) into (19) and using (22), we have an algebraic equation, which on splitting with respect to the powers of Z gives the following system:(23)2a1−Ba1+12A=0,−2a1−2Ba1+20A−C=0,2a1−Ba1+8A−C=0,whose solution is(24)a0is an arbitrary constant,a1=12AB=21−ω,C=−4A.Substituting (24) into (22), we obtain the traveling wave solution to equation (19)(25)U=a0+12ABtanhξ.Thus, the traveling wave solution of the (3 + 1)-dimensional generalized BKP equation (1) is(26)ux,y,z,t=a0+21−ωtanhαωt+1−ωx−α+β−ωy+β−ωz,where a0,α,β,andω are constants and satisfy the algebraic equation 4A+C=41−ω3α+β−ω+α+6ω2−α2+αβ+6β+6ω+3β2+1=0.


## 3.2. Exact Solutions of Equation (19) Using Kudryashov Method

Suppose that the solution of equation (19) is presented as a finite series:(27)U=∑i=0naiQiξ,where n is a positive integer and aii=0,1,2,…,n are constants, which can be determined later. Here,(28)Qξ=11+daξ,which satisfies the Bernoulli equation:(29)Q′ξ=QξQξ−1lna.Applying the homogeneous balance method similar as the tanh method, we have n=1. Thus, the solution (27) becomes(30)U=a0+a1Qξ.Substituting (30) into equation (19) together with (29), we have a system of nonlinear algebraic equations:(31)2lna3a1Ba1+12Alna=0,−5lna3a1Ba1+12Alna=0,2lna2a125lna2A+2a1Blna+C=0,−lna2a115lna2A+a1Blna+3C=0,lna2a1lna2A+C=0.Solving this system with the help of Maple, we obtain(32)a0=a0,a1=−12ABlna=−21−ωlna,C=B12lna.Substituting (32) into (30) together with (29), we can obtain the traveling wave solution to equation (19):(33)U=a0−12ABlna1+daξ.Thus, the traveling wave solution of the (3 + 1)-dimensional generalized BKP equation (1) is(34)ux,y,z,t=a0−21−ωlna1+daξ,where ξ=αωt+1−ωx−α+β−ωy+β−ωz and a0,a,α,β,andω are constants and satisfy the algebraic equation 12C−Blna=12α+6ω2−12α2+αβ+6β+6ω+36β2+1−61−ω2α+β−ωlna=0.


## 4. Conservation Laws of Equation (1)

In this section, we study the conservation laws of the generalized (3 + 1)-dimensional BKP equation (1) using the direct multiplier method [25]. We find all the first-order multipliers Λt,x,y,z,u,ut,ux,uy,uz for the equation (1) from the determined equation:(35)δδuΛuyt−uxxxy−3uxuyx+3uxx+3uzz=0,where(36)δδu=∂∂u−Dt∂∂ut−Dx∂∂ux−Dy∂∂uy−Dz∂∂uz+DtDy∂∂uty+DxDy∂∂uxy+Dx2∂∂uxx+Dz2∂∂uzz+Dx3Dy∂∂uxxxy,denotes the Euler-Lagrange operator.In general, for a given PDE (1), if the multiplier Λ yields a divergence expression,(37)Λuyt−uxxxy−3uxuyx+3uxx+3uzz=DtTt+DxTx+DyTy+DzTz,for every u, then PDE (1) has a local conservation law in the form of DtTt+DxTx+DyTy+DzTz=0, where T=Tt,Tx,Ty,Tz is a conserved vector corresponding to the conservation law.In order to obtain the multipliers of the equation (1), expanding (35) and splitting with respect to derivative of u, we obtain 10 overdetermined equations as follows:(38)Λu=0,Λut=0,Λuy=0,Λuz=0,Λxux=0,Λyux=0,Λzux=0,Λuu=0,−6Λx+Λtux=0,Λty+3Λzz=0,Using the software Maple, we obtain the solution of the above overdetermined equations:(39)Λt,x,y,z,u,ut,ux,uy,uz=6F1t+Cux+F1′tx+F2y−t−z3,where C is an arbitrary constant, F1t is an arbitrary function with respect to t, and F2ξ is an arbitrary function with respect to ξ with ξ=y−t−z/3. We further derive all the first-order multipliers in the following form:(40)Λ1=ux,Λ2=6F1t+F1′tx,Λ3=F2ξ,ξ=y−t−z3.Corresponding to the multiplierΛ1=ux, we obtain the conservation law of equation (1) as(41)DtT1t+DxT1x+DyT1y+DzT1z=0,with the following components:(42)T1t=−14uuxy+14uxuy,T1x=12uuty−18uuxxxy+38uxxuxxy−18uxxxuy−58uxuxxy+32uxx2−32ux2uy−32uz2,T1y=−14uutx+18uuxxxx−12ux3−14uxuxxx+18uxx2+14utux,T1z=3uxuz.For the multiplierΛ2=6F1t+F1′tx, we obtain the conservation law of equation (1) as(43)DtT2t+DxT2x+DyT2y+DzT2z=0,with the following components:(44)T2t=3F1tuxuy,T2x=−6F1tuxuxxy−9F1tux2uy−3F1′tuxuxy−3F1tutuy+3xF1′tux−3F1′tu+9F1tux2−9F1tuz2,T2y=xF1′tut+3F1tuxx2−xF1′tuxxx−3F1tux3+3xF1tutux,T2z=18F1tuxuz+3xF1′tuz.For the multiplierΛ3=F2ξ, the conservation law of equation (1) is(45)DtT3t+DxT3x+DyT3y+DzT3z=0,and the components are(46)T3t=−12F2′ξu+12F2ξuy,T3x=−F2ξuxxy−3F2ξuxuy+3F2ξux,T3y=12F2ξut+12F2′ξu,T3z=32uz+3F2′ξu,ξ=y−t−z3.


## 5. Conclusion

The exact solutions of the (3 + 1)-dimensional generalized BKP equation (1) have been constructed using different methods by many authors. The Pfaffian technique, the Hirotas bilinear method, and the bilinear Bäcklund transformation were applied by many researchers, and various exact solutions such as Pfaffian solutions, lump wave solutions, periodic wave solutions, rogue wave solutions, and interaction solutions were derived. Based on the Hirota bilinear form, the Wronskian method, and the multiple Exp-function method, the Grammian and Pfaffian techniques were used to construct multiple wave solutions to the (3 + 1)-dimensional generalized BKP equation. Different from these existing results, we choose another approach to study equation (1). In this paper, the combination of Lie symmetry method and the symbolic computation is applied to equation (1). The geometric vector fields of this equation are presented for the first time in the literature. With the aid of Lie symmetry reductions, some exact traveling wave solutions are obtained by using the tanh method and Kudryashov method to the reduced equation of (1). The conservation laws of equation (1) are presented by using the multiplier method at the end of the paper. We can find that these are different approaches to find exact solutions of the (3 + 1)-dimensional generalized BKP equation, and we hope that some more interesting solutions of equation (1) are shown in the future.



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