AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 271960 10.1155/2014/271960 271960 Research Article Exact Solutions and Conservation Laws of the Drinfel’d-Sokolov-Wilson System Matjila Catherine http://orcid.org/0000-0002-1586-7307 Muatjetjeja Ben http://orcid.org/0000-0002-1986-4859 Khalique Chaudry Masood Hong Baojian International Institute for Symmetry Analysis and Mathematical Modelling Department of Mathematical Sciences, North-West University Mafikeng Campus Private Bag X 2046, Mmabatho 2735 South Africa nwu.ac.za 2014 2732014 2014 22 01 2014 04 03 2014 27 3 2014 2014 Copyright © 2014 Catherine Matjila et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the Drinfel'd-Sokolov-Wilson system, which was introduced as a model of water waves. Firstly we obtain exact solutions of this system using the (G/G)-expansion method. In addition to exact solutions we also construct conservation laws for the underlying system using Noether's approach.

1. Introduction

The classical Drinfel’d-Sokolov-Wilson (DSW) system given by (1)ut+pvvx=0,vt+qvxxx+svux+ruvx=0, where p, q, r, and s are nonzero constants, has been studied by . The authors obtained various types of explicit solutions for (1) by using the bifurcation method and qualitative theory of dynamical systems. Also, Yao and Li  and C. Liu and X. Liu  obtained some exact solutions for the DSW system (1) by using a direct algebra method. A special case of the classical DSW system, namely, (2)ut+3vvx=0,vt+2vxxx+vux+2uvx=0, was studied by several authors . Hirota et al.  investigated the soliton structure of (2) and by employing an algebraic method, Fan  constructed some exact solutions. By using the improved generalized Jacobi elliptic function method, Yao  obtained some traveling wave solutions of (2), whereas by applying the Adomian decomposition method, Inc  obtained approximate doubly periodic wave solutions of (2). Zhao and Zhi  constructed exact doubly periodic solutions of (2) by using an improved F-expansion method.

In this paper, we study the Drinfel’d-Sokolov-Wilson (DSW) system given by(3a)ut+βvvx=0,(3b)vt+αvxxx+βvux+βuvx=0,which can be derived from (1) by taking p=r=s=β and q=α. We obtain exact solutions and construct conservation laws of the DSW systems (3a) and (3b).

Nonlinear partial differential equations (PDEs) model diverse nonlinear phenomena in natural and applied sciences such as mechanics, fluid dynamics, biology, plasma physics, and mathematical finance. Therefore finding exact solutions of nonlinear PDEs is very important. Unfortunately, this is a very difficult task and there are no systematic methods that can be used to find exact solutions of the nonlinear PDEs. However, in the past few decades a number of new methods have been developed to obtain exact solutions to nonlinear PDEs. Some of these methods include the exp-function method, the homogeneous balance method, the sine-cosine method, the hyperbolic tangent function expansion method, and the (G/G)-expansion function method .

We recall that conservation laws are mathematical expressions of the physical laws, such as conservation of energy, mass, and momentum. They are of great significance in the solution process and reduction of PDEs. In the literature, one finds that the conservation laws have been widely used in studying the existence, uniqueness, and stability of solutions of nonlinear partial differential equations , as well as in the development and use of numerical methods [19, 20]. Also, conserved vectors associated with Lie point symmetries have been employed to find exact solutions of partial differential equations . There are various methods of constructing conservation laws. One of the methods for variational problems is by means of Noether’s theorem . In order to apply Noether’s theorem, the knowledge of a suitable Lagrangian is necessary. For nonlinear differential equations that do not have a Lagrangian, several methods have been developed (see, e.g., ).

This paper is structured as follows. In Section 2, exact solutions of (3a) and (3b) are obtained using the (G/G)-expansion function method. In Section 3, we construct Noether’s symmetries and the conserved vectors for the DSW system (3a) and (3b). Concluding remarks are presented in Section 4.

2. Exact Solutions of the DSW System (<xref ref-type="disp-formula" rid="EEq1.3a">3a</xref>) and (<xref ref-type="disp-formula" rid="EEq1.3b">3b</xref>)

In this section, we obtain exact solutions of the DSW system (3a) and (3b). We first transform the system (3a) and (3b) into a system of ordinary differential equations by using the substitutions (4)u(x,t)=U(z),v(x,t)=V(z), where z=x-ct. Substituting (4) into the system (3a) and (3b) and integrating with respect to z, we obtain the following ordinary differential equations (ODEs): (5)-cU+β2V2=0,-cV+αV′′+βVU=0, where integration constants are taken to be zero. From the first equation, we obtain U=βV2/(2c). Substituting this value of U in the second equation of the system, we obtain (6)-cV+αV′′+β22cV3=0. Now multiplying the above equation by V and integrating while taking the constant of integration to be zero, we arrive at a first-order variables separable equation. Integrating this equation and reverting back to our original variables, we obtain (7)v(x,t)=4c2exp[c/α(x-ct+A)]1+βc2exp[2c/α(x-ct+A)], where A is an arbitrary constant of integration. Since U=βV2/(2c), we have (8)u(x,t)=β2c{4c2exp[c/α(x-ct+A)]1+βc2exp[2c/α(x-ct+A)]}2. Thus, we have obtained one exact solution of the DSW system (3a) and (3b).

To obtain more exact solutions of the DSW system (3a) and (3b), we employ the (G/G)-expansion function method . We assume that the solutions of the ODE (6) can be expressed as a polynomial in (G/G) by (9)V(z)=i=0nϕi(GG)i, where n is the balancing number to be determined and the function G(z) satisfies the second-order linear ODE given by (10)G′′(z)+λG(z)+μG(z)=0 with λ and μ being arbitrary constants. In our case, the balancing procedure gives n=1. Thus (11)V(z)=ϕ0+ϕ1(GG). Substituting (11) into (6) and making use of (10) and then equating all the terms with the same powers of (G/G) to zero yield the following system of algebraic equations: (12)αϕ1λ2+2αϕ1μ+3β2ϕ02ϕ12c-cϕ1=0,3αϕ1λ+3β2ϕ0ϕ122c=0,2αϕ1+β2ϕ132c=0,αϕ1λμ+β2ϕ032c-cϕ0=0. Solving the above equations, with the aid of Mathematica, we obtain (13)α=-2c(λ2-4μ),ϕ0=-2c2-2αcμβ,ϕ0=2c2-2αcμβ,ϕ1=2ϕ0λ. Consequently, we obtain the following two types of travelling wave solutions of the DSW system.

For λ2-4μ>0, we obtain the hyperbolic functions travelling wave solutions (14)v1(t,x)=±2(c2-2αcμ)β±22(c2-2αcμ)λβ×[×sinh(λ2-4μ2)(x-ct)×sinh(λ2-4μ2)(x-ct))-1))-1)-λ2+λ2-4μ2×(×sinh(λ2-4μ2)(x-ct)×sinh(λ2-4μ2)(x-ct))-1))-1)((λ2-4μ2)c1sinh(λ2-4μ2)(x-ct)+c2cosh(λ2-4μ2)(x-ct))×(×sinh(λ2-4μ2)(x-ct))-1)c1cosh(λ2-4μ2)(x-ct)+c2×sinh(λ2-4μ2)(x-ct)×sinh(λ2-4μ2)(x-ct))-1))-1)],u1(t,x)=β2c{×sinh(λ2-4μ2)(x-ct)+c2sinh(λ2-4μ2)(x-ct))-1)])-1)]±2(c2-2αcμ)β±22(c2-2αcμ)λβ×[×sinh(λ2-4μ2)(x-ct)+c2sinh(λ2-4μ2)(x-ct))-1)])-1)-λ2+λ2-4μ2×(×sinh(λ2-4μ2)(x-ct)+c2sinh(λ2-4μ2)(x-ct))-1)])-1(+c2sinh(λ2-4μ2)(x-ct))-1)]c1sinh(λ2-4μ2)(x-ct)+c2×cosh(λ2-4μ2)(x-ct)+c2sinh(λ2-4μ2)(x-ct))-1)])×(+c2sinh(λ2-4μ2)(x-ct))-1)]c1cosh(λ2-4μ2)(x-ct)+c2×sinh(λ2-4μ2)(x-ct)+c2sinh(λ2-4μ2)(x-ct))-1)])-1)]}2. For λ2-4μ<0, we obtain the trigonometric function travelling solutions (15)v2(t,x)=±2(c2-2αcμ)β±22(c2-2αcμ)λβ×[+c2sin(4μ-λ22)(x-ct))-1)-λ2+4μ-λ22×(+c2sin(4μ-λ22)(x-ct))-1)(-c1sin(4μ-λ22)(x-ct)+c2cos(4μ-λ22)(x-ct))×(+c2sin(4μ-λ22)(x-ct))-1)c1cos(4μ-λ22)(x-ct)+c2×sin(4μ-λ22)(x-ct))-1)],u2(t,x)=β2c{+c2sin(4μ-λ22)(x-ct))-1)]±2(c2-2αcμ)β±22(c2-2αcμ)λβ×[+c2sin(4μ-λ22)(x-ct))-1)]-λ2+4μ-λ22×(+c2sin(4μ-λ22)(x-ct))-1)](-c1sin(4μ-λ22)(x-ct)+c2×cos(4μ-λ22)(x-ct))×(+c2sin(4μ-λ22)(x-ct))-1)]c1cos(4μ-λ22)(x-ct)+c2×sin(4μ-λ22)(x-ct))-1)]}2.

3. Conservation Laws of the DSW Equations (<xref ref-type="disp-formula" rid="EEq1.3a">3a</xref>) and (<xref ref-type="disp-formula" rid="EEq1.3b">3b</xref>)

In this section, we construct the conservation laws of the DSW system (3a) and (3b). Since the third-order DSW system (3a) and (3b) does not have a Lagrangian, we cannot apply the Noether theorem. However, if we transform the third-order DSW system (3a) and (3b) to a fourth-order with the aid of the transformation u=Ux, v=Vx , we obtain(16a)Utx+βVxVxx  =  0,(16b)Vtx+αVxxxx+βVxUxx+βUxVxx=0.This system has a Lagrangian given by (17)L=12(αVxx2-βUxVx2-UxUt-VxVt) and it satisfies the Euler-Lagrange equations (18)δLδU=0,δLδV=0, where δ/δU and δ/δV are defined by (19)δδU=U-DtUt-DxUxδδU=+Dt2Utt+Dx2Uxx+DxDtUtx-,δδV=V-DtVt-DxVx+Dt2VttδδU=+Dx2Vxx+DxDtVtx-, respectively.

Let us now consider the vector field (20)X=ξ1(t,x,U,V)t+ξ2(t,x,U,V)x+η1(t,x,U,V)U+η2(t,x,U,V)V.

The second prolongation operator, X of X, is given by (21)X=ξ1(t,x,U,V)t+ξ2(t,x,U,V)x+η1(t,x,U,V)U+η2(t,x,U,V)V+ζt1Ut+ζt2Vt+ζx1Ux+ζx2Vx+, where (22)ζt1=Dt(η1)-UtDt(ξ1)-UxDt(ξ2),ζt2=Dt(η2)-VtDt(ξ1)-VxDt(ξ2),ζx1=Dx(η1)-UtDx(ξ1)-UxDx(ξ2),ζx2=Dx(η2)-VtDx(ξ1)-VxDx(ξ2),Dt=t+UtU+VtV+UttUtDt=+VttVt+UtxUx+VtxVx+,Dx=x+UxU+VxV+UxxUxDt=+VxxVx+UtxUt+VtxVt+. We recall that X, given by (20), is a Noether symmetry of (16a) and (16b), if it satisfies (23)X(L)+L[Dt(ξ1)+Dx(ξ2)]=Dt(B1)+Dx(B2), where B1(t,x,U,V) and B2(t,x,U,V) are the gauge functions. Expanding the above equation gives (24)-12Ux[ηt1+UtηU1+VtηV1-Utξt1-12Ux-Ut2ξU1-UtVtξV1-12Ux-Uxξt2-UtUtξU2-UxVtξV2]-12Vx[ηt2+UtηU2+VtηV2-Vtξt1-12Ux-UtVtξU1-Vt2ξV1-Vxξt2-12Ux-UtVxξU2-VtVxξV2]-12(βVx2+Ut)×[ηx1+UxηU1+VxηV1-12Ux-Utξx1-UtUxξU1-UtVxξV1-12Ux-Uxξx2-Ux2ξU2-UxVxξV2]-(12Vt+βUxVx)×[ηx2+UxηU2+VxηV2-Vtξx1-UxVtξU1-12Ux-VtVxξV1-Vxξx2-UxVxξU2-Vx2ξV2]+αVxx[Dx2η2-VtDx2ξ1-VxDx2ξ2-2Vtx(ξx1+UxξU1+VxξV1)-2Vxx(ξx2+UxξU2+VxξV2)]+12[αVxx2-βUxVx2-UxUt-VxVt]×[ξt1+UtξU1+VtξV1+ξx2+UxξU2+VxξV2]=Bt1+UtBu1+VtBv1+Bx2+UxBu2+VxBv2. This leads to an overdetermined system of PDEs for the functions ξ1,ξ2,η1,η2,B1, and B2. Solving the system of PDEs gives (25)ξ1=c1,ξ2=c2,η1=f(t),η2=g(t),B1=A(t,x),B2=-12f(t)-12g(t)+H(t,x),At+Hx=0. We may choose H=0 and A=0 as they contribute to the trivial part of the conserved vector. The conserved vector for the second-order Lagrangian L is given by [24, 31] (26)T1=-B1+ξ1LT1=+W1[LUt-DtLUtt-DxLUtx]  T1=+W2[LVt-DtLVxt-DxLVtt]T1=+Dt(W1)LUtt+Dt(W2)LVtt,T2=-B2+ξ2LT1=+W1[LUx-DtLUxt-DxLUxx]T1=+W2[LVx-DtLVxt-DxLVxx]T1=+Dx(W1)LUxx+Dx(W2)LVxx, where W1=η1-Utξ1-Uxξ2 and W2=η2-Vtξ1-Vxξ2 are the characteristic functions. Now using (26) in conjunction with (25), u=Ux, and v=Vx, we obtain the following independent conserved vectors for system (3a) and (3b): (27)T11=12(αvx2-βuv2),T12=-12βutdx+12utdxutdxT12=+βuvvtdx+12vtdxvtdxT12=+αuxxvtdx-αvtvx,(28)T21=12(u2+v2),T22=-12αvx2+αvvxx+βuv2. We note that (27) is a nonlocal conserved vector, whereas (28) is a local conserved vectors. Also, for the arbitrary functions f(t) and g(t), we obtain the following conserved vectors: (29)T(f,g)1=-12uf(t)-12vg(t),T(f,g)2=12f(t)udx+12g(t)vdxT(f,g)2=+f(t)[-12βv2-12utdx]T(f,g)2=+g(t)[-12βuv-12vtdx-αvxx], which gives us infinitely many nonlocal conservation laws.

4. Conclusion

The third-order DSW system (3a) and (3b) was studied. Exact solutions of the DSW system were obtained using direct integration and the (G/G)-expansion function method. The solutions obtained were hyperbolic and trigonometric solutions. In addition conservation laws were also derived. This system does not have a Lagrangian. In order to invoke Noether’s theorem we used the transformations u=Ux and v=Vx to convert the DSW system to a fourth-order system, which has a Lagrangian. The conservation laws were then obtained and consisted of a local and infinite number of nonlocal conserved vectors.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

All authors would like to thank NRF and North-West University, Mafikeng Campus, for financial support.

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