Travelling waves and conservation laws of a (2+1)-dimensional coupling system with Korteweg-de Vries equation

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Abstract

In this paper we study a (2+1)-dimensional coupling system with the Korteweg-de Vries equation, which is associated with non-semisimple matrix Lie algebras. Its Lax-pair and bi-Hamiltonian formulation were obtained and presented in the literature. We utilize Lie symmetry analysis along with the (G′/G)–expansion method to obtain travelling wave solutions of this system. Furthermore, conservation laws are constructed using the multiplier method.

Abstract

In this paper we study a (2+1)-dimensional coupling system with the Korteweg-de Vries equation, which is associated with non-semisimple matrix Lie algebras. Its Lax-pair and bi-Hamiltonian formulation were obtained and presented in the literature. We utilize Lie symmetry analysis along with the (G′/G)–expansion method to obtain travelling wave solutions of this system. Furthermore, conservation laws are constructed using the multiplier method.

1 Introduction

It is well-known that nonlinear partial differential equations (NLPDEs) are extensively used to model many nonlinear physical phenomena of the real world, which can be seen from the number of research papers published in the literature. One such NLPDE is the celebrated Korteweg-de Vries (KdV) equation [1]

ut+6uux+uxxx=0,

which has applications in nonlinear dynamics, plasma physics and mathematical physics. It is an important equation in scientific fields and in the theory of integrable systems. It describes the unidirectional propagation of long waves of small amplitude and has a lot of applications in a number of physical contexts such as hydromagnetic waves, stratified internal waves, ion-acoustic waves, plasma physics and lattice dynamics [2]. Equation (1) has multiple-soliton solutions and an infinite number of conservation laws and many other physical properties. See for example [3,4,5] and references therein.

Recently focus has shifted to the study of coupled systems of Korteweg-de Vries equations because of their many applications in scientific fields. See for example [5,6,7,8,9].

However, in this work we study the (2+1)-dimensional coupling system with the Korteweg-de Vries equation [2], namely

ckdv11ut6uuxuxxx=0,

ckdv12vt6(uv)x6uuy3uxxyvxxx=0.

This system is a (2+1)-dimensional integrable coupling with the Korteweg-de Vries equation, which is associated with non-semisimple matrix Lie algebras. In the references [10] and [11], its Lax pair and bi-Hamiltonian formulation were presented respectively. It should be noted that its bi-Hamiltonian structure is the first example of local bi-Hamiltonian structures, which lead to hereditary recursion operators in (2+1)-dimensions.

Several methods have been developed to find exact solutions of the NLPDEs. Some of these are the homogeneous balance method [12], the ansatz method [13], the inverse scattering transform method [14], the Bäcklund transformation [15], the Darboux transformation [16], the Hirota bilinear method [17], the simplest equation method [18], the (G′/G)–expansion method[19,20], the Jacobi elliptic function expansion method [21], the Kudryashov method [22], the Lie symmetry method [23,24,25,26,27,28].

The outline of the paper is as follows. In Section 2 we determine the travelling wave solutions for the system (2a) using the Lie symmetry method along with the (G′/G)–expansion method. Conservation laws for (2a) are constructed in Section 3 by employing the multiplier approach [26,29,30,31,32,33,34,35,36,37]. Finally concluding remarks are presented in Section 4.

2 Travelling wave solutions of (2a)

In this section we use Lie symmetry analysis together with the (G′/G)–expansion method to obtain travelling wave solutions of (2a).

2.1 Lie point symmetries and symmetry reductions of (2a)

Lie symmetry analysis was introduced by Marius Sophus Lie (1842-1899), a Norwegian mathematician, in the later half of the nineteenth century. He developed the theory of continuous symmetry groups and applied it to the study of geometry and differential equations. This theory contains powerful methods which can be used to obtain exact analytical solutions of differential equations [23,24,25]. The theory is called symmetry groups theory or the classical Lie method of infinitesimal transformations. The symmetry group of a differential equation is the largest local Lie group of transformations of the independent and dependent variables of the differential equation that transforms solutions of the differential equation to other solutions. The symmetry group associated to a differential equation can be obtained by Lie’s infinitesimal criterion of invariance.

The (2+1)-dimensional coupling system with the Korteweg-de Vries equation (2a) is invariant under the symmetry group with the generator

Γ=ξ1(t,x,y,u,v)t+ξ2(t,x,y,u,v)x+ξ3(t,x,y,u,v)y+η1(t,x,y,u,v)u+η2(t,x,y,u,v)v

if and only if

ckdv21Γ[3](ut6uuxuxxx)(2a)=0,

ckdv22Γ[3](vt6(uv)x6uuy3uxxyvxxx)(2a)=0,

where Γ[3] denotes the third prolongation [23] of the generator (3) and the symbol |(2a) means it is evaluated on equations (2a). The third prolongation Γ[3] is given by

Γ[3]=Γ+ζt1ut+ζx1ux+ζy1uy+ζt2vt+ζx2vx+ζxxx1uxxx+ζxxy1uxxy+ζxxx2vxxx,

with

ζt1=Dt(η1)utDt(ξ1)uxDt(ξ2)uyDt(ξ3),ζx1=Dx(η1)utDx(ξ1)uxDx(ξ2)uyDx(ξ3),ζy1=Dy(η1)utDy(ξ1)uxDy(ξ2)uyDy(ξ3),ζt2=Dt(η2)vtDt(ξ1)vxDt(ξ2)vyDt(ξ3),ζx2=Dx(η2)vtDx(ξ1)vxDx(ξ2)vyDx(ξ3),ζxx1=Dx(ζx1)utxDx(ξ1)uxxDx(ξ2)uxyDx(ξ3),ζxx2=Dx(ζx2)vtxDx(ξ1)vxxDx(ξ2)vxyDx(ξ3),ζxxx1=Dx(ζxx1)utxxDx(ξ1)uxxxDx(ξ2)uxxyDx(ξ3),ζxxy1=Dx(ζxx1)utxyDx(ξ1)uxxyDx(ξ2)uyxyDx(ξ3),ζxxx2=Dx(ζxx2)vtxxDx(ξ1)vxxxDx(ξ2)vxxyDx(ξ3),

and Dt, Dx and Dy are the operators of total differentiation defined as

Dt=t+utu+vtv+,Dx=x+uxu+vxv+,Dy=y+uyu+vyv+,

respectively. Expanding (4a) and then splitting on the derivatives of u and v, we obtain the following overdetermined system of linear partial differential equations:

ξx1=0,ξy1=0,ξu1=0,ξv1=0,ξu2=0,ξv2=0,ξt3=0,ξx3=0,ξy2=0ξv3=0,ηv1=0,ηuu1=0,ηuu2=0,ηvv2=0,ηuv2=0,ηxu1ξxx2=0,2ηxu1ξxx2=0,ηxv2ξxx2=0,ηxu2+ηyu1=0,3ξx2+ξt1=0,4uξx2+2η1+ηxxu1=0,ηxxx1ηt1+6uηx1=0ηu1+ξy3+ηv2ξx2=0,2vξx2+2vξy3+2η2+ηxxu2+2ηxyu1=0,12uξx2+6η1ξxxx2+3ηxxu1+ξt2=0,ξt12ξx2ξy3ηv2+ηu1=0,6uηy1+6vηx1ηt2+3ηyxx1+ηxxx2+6uηx2=0,6uξx2+6uξy36uηu1+6η1ξxxx2+3ηxxv2+ξt2+6uηv2=0.

Solving the above system of partial differential equations, one obtains

ξ1=C1+3C3t,ξ2=C2+C3x,ξ3=C4{F(y)vF(y)},η1=2C3u,η2=C3v,

where C1, ⋯, C4 are arbitrary constants and F(y) is an arbitrary function of y. Thus the Lie algebra of infinitesimal symmetries of the system (2a) is spanned by the four vector fields

timetranslationΓ1=t,spacetranslationΓ2=x,scalingΓ3=3tt+xx2uuvv,Γ4=F(y)yvF(y)y.

We now use these Lie point symmetries to find exact solutions of (2a). The linear combination of the three symmetries Γ1, Γ2 and Γ4 with F(y) = 1 provides us with the three invariants

f=ty,g=xy,u=θ(f,g),v=ψ(f,g),

the system (2a) is reduced to a system of partial differential equations of two functions θ and ψ in two independent variables f and g;

ckdv41θf6θθgθggg=0,

ckdv42ψf6(θψ)g+6θθf+6θθg+3θggf+3θgggψggg=0.

System (7a) has the following symmetries

X1=f,X2=g,X3=6ff+(2g+3f)g(4θ+12)θ(8ψ+2θ+12)ψ,

Considering the symmetry X = X1 + α X2 given by the linear combination of X1 and X2 we get the invariants

z=gαf,θ=H(z),ψ=J(z).

This further reduces (2a) to a system of third-order ordinary differential equations in two functions H(z) and J(z).

ckdv51αH+6HH+H=0,

ckdv52αJ+6(HJ)+6(α1)HH+3(α1)H+J=0,

where the prime denotes derivative with respect to z.

2.2 Application of the (G′/G)–expansion method

In this section we employ the (G′/G)–expansion method to construct travelling wave solutions of the system of third order ordinary differential equatons (8a). This method was developed by the authors of [19] and has been extensively used by researchers. It assumes the solutions of the system (8a) to be of the form

H(z)=i=0MAi(G(z)G(z))i,J(z)=j=0NBj(G(z)G(z))j,

where G(z) satisfies the second-order ODE given by

G+λG+μG=0

with λ and μ being arbitrary constants. The homogeneous balance method between the highest order derivative and highest order nonlinear term appearing in (8a) determines the values of M and N. The parameters 𝓐i and 𝓑j, i = 0,1, ⋯, M and j = 0,1, ⋯, N need to be determined. In our case the balancing procedure yields M = 2 and N = 2, so the solutions of the system of ordinary differential equations (8a) are of the form

H(z)=A0+A1(G/G)+A2(G/G)2,J(z)=B0+B1(G/G)+B2(G/G)2.

Substituting (11) into (8a) and making use of (10), and then collecting all terms with same powers of (G′/G) and equating each coefficient to zero, yields a system of algebraic equations. Solving this system of algebraic equations, using Mathematica, we obtain the following set of values for the constants 𝓐i and 𝓑j, i,j = 0, 1, 2:

A0=16α+λ2+8μ,A1=2λ,A2=2,B0=16(α1)α2λ216μ,B1=4λ(α1),B2=4(α1).

Substituting these values of 𝓐i and 𝓑j into the corresponding solutions (11) of ordinary differential equations (5), we obtain the following three types of travelling wave solutions of equation (2a):

  • When λ2 – 4μ > 0, we obtain the hyperbolic function solutions

    u1(t,x,y)=16α+λ2+8μ2λ[λ2+δ1(C1sinhδ1z+C2coshδ1zC1coshδ1z+C2sinhδ1z)]2[λ2+δ1(C1sinhδ1z+C2coshδ1zC1coshδ1z+C2sinhδ1z)]2,

    v1(t,x,y)=16(α1)α2λ216μ+4λ(α1)[λ2+δ1(C1sinhδ1z+C2coshδ1zC1coshδ1z+C2sinhδ1z)]+4(α1)[λ2+δ1(C1sinhδ1z+C2coshδ1zC1coshδ1z+C2sinhδ1z)]2,

    where z = x +(α – 1)yα t, δ1=12λ24μ, C1 and C2 are arbitrary constants.

    Fig. 1

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    Fig. 1

    Profile of solution (12)

    Citation: Applied Mathematics and Nonlinear Sciences 3, 1; 10.21042/AMNS.2018.1.00018

    Fig. 2

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    Fig. 2

    Profile of solution (13)

    Citation: Applied Mathematics and Nonlinear Sciences 3, 1; 10.21042/AMNS.2018.1.00018

  • When λ2 – 4μ < 0, we obtain the trigonometric function solutions

    u2(t,x,y)=16α+λ2+8μ2λ(λ2+δ2C1sinδ2z+C2cosδ2zC1cosδ2z+C2sinδ2z)2(λ2+δ2C1sinδ2z+C2cosδ2zC1cosδ2z+C2sinδ2z)2,

    v2(t,x,y)=16(α1)α2λ216μ+4λ(α1)(λ2+δ2C1sinδ2z+C2cosδ2zC1cosδ2z+C2sinδ2z)+4(α1)(λ2+δ2C1sinδ2z+C2cosδ2zC1cosδ2z+C2sinδ2z)2,

    where z = x +(α – 1)yα t, δ2=124μλ2, C1 and C2 are arbitrary constants.

    Fig. 3

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    Fig. 3

    Profile of solution (14)

    Citation: Applied Mathematics and Nonlinear Sciences 3, 1; 10.21042/AMNS.2018.1.00018

    Fig. 4

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    Fig. 4

    Profile of solution (15)

    Citation: Applied Mathematics and Nonlinear Sciences 3, 1; 10.21042/AMNS.2018.1.00018

  • When λ2 – 4μ = 0, we obtain the rational solutions

    u3(t,x,y)=16α+λ2+8μ2λ(λ2+C2C1+C2z)2(λ2+C2C1+C2z)2,v3(t,x,y)=16(α1)α2λ216μ+4λ(α1)(λ2+C2C1+C2z)+4(α1)(λ2+C2C1+C2z)2.

    where z = x +( α – 1)yα t, C1 and C2 are arbitrary constants.

3 Conservation laws for (2a)

In this section we construct conservation laws for our (2+1)-dimensional coupling system with the Korteweg-de Vries equation (2a). Conservations laws are physical quantities such as mass, momentum, angular momentum, energy, electrical charge, that do not change in the course of time within a physical system. They play a vital role in the solution process of differential equations. They are significant for exploring integrability and for establishing existence, uniqueness and stability of solutions of differential equations. Also conservation laws play an essential role in the numerical integration of partial differential equations, for example, to control numerical errors and they can be used to construct solutions of partial differential equations.

Several methods have been developed by researchers for constructing conservation laws. These include the Noether’s theorem for variational problems, the Laplace’s direct method, the characteristic form method by Stuedel, the multiplier approach, Kara and Mahomed partial Noether approach. The computer software packages for computing conservation laws have also been developed over the past few decades.

Here we use the multiplier method to find conservation laws of the system (2a), namely

ckdv110E1ut6uuxuxxx=0,

ckdv120E2vt6(uv)x6uuy3uxxyvxxx=0.

A conservation law of the system (2a) is a space-time divergence such that

DtT+DxX+DyY=0

holds for all solutions (u(t, x,y); v(t, x,y)) of the system (2a). The vector (T,X,Y) is called the conserved vector of the system (2a).

We look for second-order multipliers Q1 and Q2, that is, Q1 and Q2 depend on t, x, y, u, v and first and second derivatives of u and v. The multipliers Q1 and Q2 of the system (2a) have the property that

Q1E1+Q2E2=DtT+DxX+DyY,

for all functions u(t,x,y) and v(t,x,y). The determining equations for the multipliers are obtained by solving the system

ckdv71δδu[Q1(ut6uuxuxxx)+Q2(vt6(uv)x6uuy3uxxyvxxx)]=0,

ckdv72δδv[Q1(ut6uuxuxxx)+Q2(vt6(uv)x6uuy3uxxyvxxx)]=0,

where δ/δ u and δ/δ v are the standard Euler-Lagrange operators given by

δδu=uDtutDxuxDyuyDx3uxxxDx2Dyuxxy+

and

δδv=vDtvtDxvxDyvyDx3vxxx+,

respectively. Expanding system (19a) using (20) and (21) yields an overdetermined system of partial differential equations, which after solving with the help of Maple [38], we obtain

Q1=x52u3+uuxx14(ux2utx)F2y6tux+x2F1y+vF4yx3u2+uxxF3yxuF4y+6tuF5y+152vu2+6tvF1y+148uxy+4vxxu+4uxxv+2uy2vxux+uty+vtxF2y+1410u3+4uuxxux2+utxF6y+6uv+2uxy+vxxF3y+3u2+uxxF7y+6tu+xF8y+uF9y+F10y,Q2=1410u3+4uuxxux2+utxF2y+6tu+xF1y+3u2+uxxF3y+F4yu+F5y,

where Fi, i = 1,⋯, 10 are arbitrary functions of y. As a result the ten conserved vectors are calculated via a homotopy formula [38] and are given by

Q2=1410u3+4uuxxux2+utxF2y+6tu+xF1y+3u2+uxxF3y+F4yu+F5y,

T1=6uvFyt3xFytu2ux2Fy+vFyx,X1=3u2x2FyxuxFy+uxxx2Fy2uxyFyxvxxFyx12uuxyFyt6uvxxFyt6vuxxFyt+6vxuxFyt6vuFyx3ux2xFyt+6uxuyFyt+6uuxxxFyt+Fyvx+uyFy36vFyu2t+12xFytu3,Y1=Fyux6uuxxFytuxxFyx12u3Fyt+3ux2Fyt3u2Fyx;

T2=116Fyvuxxxx+116Fyuuty+76Fyu2uxy+14Fyuuxxxy+116Fyuvtx+712Fyu2vxx+116Fyuvxxxx112Fyux2v+116Fyvutx58xFyu4+52vFyu3116Fyuxxxvx+116uyFyut116uyFyuxxx316uxFyuxxy+116uxFyvt116uxFyvxxx+116Fyutvx712Fyu2uxxx+112uux2Fyx16uFyuxuy116uxFyutx+116uxFyuxxxx116uutxFyx116uuxxxxFyx+76vFyuuxx16uFyuxvx,X2=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,Y2=712Fyu2utx116Fyuutt+18Fyuxutxx116Fyuutxxx52u3Fyuxx+34u2Fyux2+14ux2Fyuxx12uFyuxx2116utFyuxxx18uxxFyutx3u5Fy+116ut2Fy16uFyuxut;

T3=xFyu3+3Fyu2v+Fyuuxy+12Fyuvxx+12Fyuxxv12xFyuuxx,X3=12Fyuxx2x2uxxFyuxyuxxFyvxx12Fyuuty6Fyu2uxy12Fyuvtx3Fyu2vxx12Fyvutx+92xFyu418vFyu3+12uyFyut+12uxFyvt+12Fyutvx+3Fyu2uxxx12uxFyutx+12uutxFyx6vFyuuxx,Y3=3u2Fyuxx+12uxFyut12uFyutx92u4Fy12uxx2Fy;

T4=12u2xFyuvFy,X4=2xFyu36Fyu2v12xFyux22FyuuxyFyuvxx+Fyuxuy+FyuxvxFyuxxv+xFyuuxx,Y4=2u3Fy+12ux2FyuuxxFy;

T5=yv+3tu2Fy,X5=6uuxxtFy+uxFy2FyuxyFyvxx6Fyuv12tu3Fy+3ux2tFy,Y5=3u2FyuxxFy;

T6=112uFyux2+116uuxxxxFy116uxFyuxxx+712u2Fyuxx+116uxFyut+116uFyutx+58u4Fy,X6=712Fyu2utx116Fyuutt+18Fyuxutxx116Fyuutxxx52u3Fyuxx+34u2Fyux2+14ux2Fyuxx12uFyuxx2116utFyuxxx18uxxFyutx3u5Fy+116ut2Fy16uFyuxut,Y6=0;

T7=u3Fy+12uuxxFy,X7=3u2Fyuxx+12uxFyut12uFyutx92u4Fy12uxx2Fy,Y7=0;

T8=3tu2Fy+uxFy,X8=Fyux6uuxxFytuxxFyx12u3Fyt+3ux2Fyt3u2Fyx,Y8=0;

T9=12u2Fy,X9=2u3Fy+12ux2FyuuxxFy,Y9=0;

T10=Fyu,X10=3u2FyuxxFy,Y10=0.

Remark

Due to the arbitrary functions in the multipliers Q1 and Q2, infinitely many conserved vectors are obtained for the system (2a).

4 Conclusion

In this paper we studied a (2+1)-dimensional coupling system with the Korteweg-de Vries equation (2a). Lie point symmetries of (2a) were computed and used to reduce the system to a system of ordinary differential equations. This ordinary differential equations system was then solved by employing the (G′/G)–expansion method and as a result travelling wave solutions of (2a) were obtained. The solutions obtained were expressed in the form of hyperbolic functions, trigonometric functions and rational functions. Some of these solutions were plotted. Furthermore, conservation laws for the system (2a) were derived by using the multiplier approach. The significance of conservation laws was explained in the beginning of Section 3.

Communicated by Juan L.G. Guirao

Acknowledgments

CMK and IEM would like to thank T Motsepa for fruitful discussions. IEM thanks the Faculty Research Committee of FAST, North-West University, Mafikeng Campus for the financial support.

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M. L. Gandarias and M. S. Bruzón. Conservation laws for a Boussinesq equation. Applied Mathematics and Nonlinear Sciences, 2(2):465–472, 2017. 10.21042/AMNS.2017.2.00037

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T. Motsepa and C. M. Khalique. On the conservation laws and solutions of a (2+1) dimensional KdV-mKdV equation of mathematical physics. Open Phys., 16:211–214, 2018. 10.1515/phys-2018-0030

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A. F. Cheviakov. GeM software package for computation of symmetries and conservation laws of differential equations. Comput. Phys. Commun., 176:48–61, 2007. 10.1016/j.cpc.2006.08.001

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