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.
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 
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 . 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.
However, in this work we study the (2+1)-dimensional coupling system with the Korteweg-de Vries equation , namely
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  and , 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 , the ansatz method , the inverse scattering transform method , the Bäcklund transformation , the Darboux transformation , the Hirota bilinear method , the simplest equation method , the (G′/G)–expansion method[19,20], the Jacobi elliptic function expansion method , the Kudryashov method , 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
if and only if
and Dt, Dx and Dy are the operators of total differentiation defined as
respectively. Expanding (4a) and then splitting on the derivatives of u and v, we obtain the following overdetermined system of linear partial differential equations:
Solving the above system of partial differential equations, one obtains
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
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
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;
System (7a) has the following symmetries
Considering the symmetry X = X1 + α X2 given by the linear combination of X1 and X2 we get the invariants
This further reduces (2a) to a system of third-order ordinary differential equations in two functions H(z) and J(z).
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  and has been extensively used by researchers. It assumes the solutions of the system (8a) to be of the form
where G(z) satisfies the second-order ODE given by
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
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:
When λ2 – 4μ > 0, we obtain the hyperbolic function solutions
where z = x +(α – 1)y – α t,
C1 and C2 are arbitrary constants.
When λ2 – 4μ < 0, we obtain the trigonometric function solutions
where z = x +(α – 1)y – α t,
C1 and C2 are arbitrary constants.
When λ2 – 4μ = 0, we obtain the rational solutions
where z = x +( α – 1)y – α 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
A conservation law of the system (2a) is a space-time divergence such that
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
for all functions u(t,x,y) and v(t,x,y). The determining equations for the multipliers are obtained by solving the system
where δ/δ u and δ/δ v are the standard Euler-Lagrange operators given by
where Fi, i = 1,⋯, 10 are arbitrary functions of y. As a result the ten conserved vectors are calculated via a homotopy formula  and are given by
Due to the arbitrary functions in the multipliers Q1 and Q2, infinitely many conserved vectors are obtained for the system (2a).
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
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.
D. J. Korteweg and G. de Vries. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil. Mag., 39:422–443, 1895. 10.1080/14786449508620739
W. X. Ma and B. Fuchssteiner. The bi-Hamiltonian structure of the perturbation equations of the KdV hierarchy. Phys. Lett. A, 213:49–55, 1996. 10.1016/0375-9601(96)00112-0
A. M. Wazwaz. Integrability of coupled KdV equations. Cent. Eur. J. Phy., 9:835–840, 2011. 10.2478/s11534-010-0084-y
A. M. Wazwaz. Partial Differential Equations and Solitary Waves Theory. Springer, New York, 2009. 10.1007/978-3-642-00251-9
A. R. Adem and C. M. Khalique. On the solutions and conservation laws of a coupled KdV system. Appl. Math. Comput., 219:959–969, 2012. 10.1016/j.amc.2012.06.076
D. S. Wang. Integrability of a coupled KdV system: Painlevé property, Lax pair and Bäcklund transformation. Appl. Math. Comput., 216:1349–1354, 2010. 10.1016/j.amc.2010.02.030
C. X. Li. A Hierarchy of Coupled Korteweg-de Vries Equations and the Corresponding Finite-Dimensional Integrable System. J. Phys. Soc. Jpn., 73:327–331, 2004. 10.1143/JPSJ.73.327
Z. Qin. A finite-dimensional integrable system related to a new coupled KdV hierarchy. Phys. Lett. A, 355:452–459, 2006. 10.1016/j.physleta.2005.09.089
A. M. Wazwaz. Solitons and periodic wave solutions for coupled nonlinear equations. Int. J. Nonlinear Sci., 14:266–277, 2012.
W. X. Ma and B. Fuchssteiner. Integrable theory of the perturbation equations. Chaos Solitons Fractals, 7(8):1227–1250, 1996. 10.1016/0960-0779(95)00104-2
W. X. Ma. A bi-Hamiltonian formulation for triangular systems by perturbations. J. Math. Phys., 43:1408–1421, 2002. 10.1063/1.1432775
M. Wang, Y. Zhou, and Z. Li. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys. Lett. A, 216:67–75, 1996. 10.1016/0375-9601(96)00283-6
J. Hu and H. Zhang. A new method for finding exact traveling wave solutions to nonlinear partial differential equations. Phys. Lett. A, 286:175–179, 2001.
M. J. Ablowitz and P. A. Clarkson. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge, 1991. 10.1017/CBO9780511623998
C. H. Gu. Soliton Theory and its Application. Zhejiang Science and Technology Press, Zhejiang, 1990.
V. B. Matveev and M. A. Salle. Darboux Transformations and Solitons. Springer-Verlag, Berlin, 1991. 10.1007/978-3-662-00922-2
R. Hirota. The Direct Method in Soliton Theory. Cambridge University Press, Cambridge, 2004.
N. A. Kudryashov. Exact solitary waves of the Fisher equation. Phys. Lett. A, 342:99-106, 2005. 10.1016/j.physleta.2005.05.025
M. Wang, X. Li, and J. Zhang. The (G’/G)–expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A, 372:417-423, 2008. 10.1016/j.physleta.2007.07.051
T. Aziz, T. Motsepa, A. Aziz, A. Fatima, and C. M. Khalique. Classical model of Prandtl’s boundary layer theory for radial viscous flow: application of (G’/G)–expansion method. J. Comput. Anal. Appl., 23:31–41, 2017.
Z. Zhang. Jacobi elliptic function expansion method for the modified Korteweg-de Vries-Zakharov-Kuznetsov and the Hirota equations. Phys. Lett. A, 289:69–74, 2001.
N. A. Kudryashov. One method for finding exact solutions of nonlinear differential equations. Commun Nonlinear Sci. Numer. Simulat., 17:2248–2253, 2012. 10.1016/j.cnsns.2011.10.016
P. J. Olver. Applications of Lie Groups to Differential Equations. Springer-Verlag, Berlin, 1993. 10.1007/978-1-4612-4350-2
G. W. Bluman and S. Kumei. Symmetries and Differential Equations. Springer-Verlag, New York, 1989. 10.1007/978-1-4757-4307-4
N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations. Vols. 1–3, CRC Press, Boca Raton, FL, 1994–1996.
T. Motsepa, C. M. Khalique, and M. L. Gandarias. Symmetry analysis and conservation laws of the Zoomeron equation. Symmetry, 9, 2017, 11 pages.
J. C. Camacho, M. Rosa, M. L. Gandarias, and M. S. Bruzón. Classical symmetries, travelling wave solutions and conservation laws of a generalized Fornberg-Whitham equation. J. Comput. Appl. Math., 318:149–155, 2017. 10.1016/j.cam.2016.11.017
M. Rosa, J. C. Camacho, M. S. Bruzón, and M.L. Gandarias. Classical and potential symmetries for a generalized Fisher equation. J. Comput. Appl. Math., 318:181–188, 2017. 10.1016/j.cam.2016.10.028
H. Steudel. Uber die Zuordnung zwischen invarianzeigenschaften und Erhaltungssatzen. Z Naturforsch, 17A:129–132, 1962.
S. Anco and G. Bluman. Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications. European J. Appl. Math., 13:545–566, 2002.
T. Motsepa and C. M. Khalique. Conservation laws and solutions of a generalized coupled (2+1)-dimensional Burgers system. Comput. Math. Appl., 74:1333–1339, 2017. 10.1016/j.camwa.2017.06.015
A. Fatima, F. M. Mahomed, and C. M. Khalique. Noether symmetries and exact solutions of an Euler-Bernoulli beam model. Int. J. of Mod Phys B, 30:1640011, 2016. 10.1142/S0217979216400117
I. Simbanefayi and C. M. Khalique. Travelling wave solutions and conservation laws for the Korteweg-de Vries- Bejamin-Bona-Mahony equation. Results in Physics, 8:57–63, 2018. 10.1016/j.rinp.2017.10.041
R. de la Rosa and M. S. Bruzón. On the classical and nonclassical symmetries of a generalized Gardner equation. Applied Mathematics and Nonlinear Sciences, 1(1):263–272, 2016. 10.21042/AMNS.2016.1.00021
M. Rosa and M. L. Gandarias. Multiplier method and exact solutions for a density dependent reaction-diffusion equation. Applied Mathematics and Nonlinear Sciences, 1(2):311–320, 2016. 10.21042/AMNS.2016.2.00026
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
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
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