Construction of invariant solutions and conservation laws to the (2+1)$(2+1)$-dimensional integrable coupling of the KdV equation

*Correspondence: benzi0116@163.com 1College of Mathematics, Taiyuan University of Technology, Taiyuan 030024, P.R. China Abstract Under investigation in this paper is the (2 + 1)-dimensional integrable coupling of the KdV equation which has applications in wave propagation on the surface of shallow water. Firstly, based on the Lie symmetry method, infinitesimal generators and an optimal system of the obtained symmetries are presented. At the same time, new analytical exact solutions are computed through the tanh method. In addition, based on Ibragimov’s approach, conservation laws are established. In the end, the objective figures of the solutions of the coupling of the KdV equation are performed.


Introduction
It is well known that the Korteweg-de Vries (KdV) equation describes the propagation of long waves on the surface of water with a small amplitude and is widely used to explain many complex science phenomena [1,2]. Various forms of the expansion for the KdV equation have been proposed because of its importance, such as the KdV-Burgers equation [3], the KdV-BBM equation [4], the Rosenau-KdV equation [5], the modified KdV equation [6], KdV-hierarchy [7], and the (2 + 1)-dimensional KdV equation [8]. In this research article, we consider the following (2 + 1)-dimensional integrable coupling of the KdV equation which has the bi-Hamiltonian structure for the (2 + 1)-dimensional perturbation equations of the KdV hierarchy [9]: where u = u(x, y, t), v = v(x, y, t) are the unknown real functions, the subscripts denote the partial derivatives, and the variable y is called a slow variable. Equation (1) plays an important role in many analyses of physical phenomena such as stratified internal waves and lattice dynamics [10,11], and it has aroused worldwide interest. The (2 + 1)-dimensional hereditary recursion operators were examined in [12], its integrability was verified by using Painlevé in [13], some traveling wave solutions were established in [14], the auto-Bäcklund transformation, doubly periodic solutions and new non-traveling wave solutions were analyzed in [15].
As is well known, some methods have been used to explore exact solutions for models of nonlinear partial differential equations (PDEs) [16][17][18], the Lie group method is considered to be one of the most important methods to study the properties of solutions of PDEs [19,20]. The main idea of the symmetry method is to construct an invariance condition and obtain reductions to differential equations [21][22][23]. Once reduction equations have been given, one can get a large number of corresponding exact solutions. In order to obtain the classification of all reduction equations, we require an optimal system of the one-dimensional subalgebra of the Lie algebra constructed by the Lie group method [19]. Using a symmetry analysis, we will get an optimal system of (1), from which the fascinating special solutions are inferred. Another important area is the conservation laws of PDEs which have an important impact on constructing solutions of PDEs [24][25][26]. We will obtain conservation laws of Eq. (1) by using Ibragimov's approach [27].
The rest of this paper is organized as follows. In Sect. 2, symmetries of the (2 + 1)dimensional integrable coupling of the KdV equation are discussed; Sect. 3 considers the reduced equations by means of similar variables; in Sect. 4, some new explicit solutions are presented with the help of the tanh method, and some objective features of the solutions are presented; in Sect. 5, the nonlinearly self-adjointness of Eq. (1) is proved and its conservation laws are established by using Ibragimov's method. Finally, concluding remarks are given at the end of the paper.

Lie point symmetry
In this section, we apply Lie's theory of symmetries for Eq. (1), and get its infinitesimal generators, commutator of Lie algebra.
First, let us consider a Lie algebra of infinitesimal symmetries of Eq. (1) of the form According to the invariance conditions for Eq. (1) with respect to the transformation (2), we have [19,28] where pr (3) X is the third-order prolongation of X [19,28] and 1 = u tu xxx -6uu x , 2 = v tv xxx -3u xxy -6(uv) x -6uu y , on this condition, and D x , D y , D t stand for the operators of the total differentiation, for instance, Next, we get a system of over-determined linear equations of ξ 1 , ξ 2 , ξ 3 , φ and ϕ, Solving these equations, one can get where c 1 , c 2 , c 3 are real constants, F(y) is an arbitrary function. To obtain physically crucial solutions, we take F 1 (z) = c 4 y + c 5 , then on substituting the above obtaining Therefore, the Lie algebra L 5 of the transformations of Eq. (1) is spanned by the following generators: In order to classify all the group-invariant solutions, we need an optimal system of one-dimensional subalgebras. In this section, the optimal system of subgroups for Eq. (1) is constructed by only using the commutator table [29]. First, using the commutator [X m , X n ] = X m X n -X n X m , we attained the commutation relations of X 1 , X 2 , X 3 , X 4 , X 5 listed in Table 1.
The establishment of the optimal system requires a simplification of the vector by applying the transformations T 1 -T 5 . Our task is to construct a simplest representative of each class of similar vectors (4). Two cases will be considered separately. Case 2.1. l 1 = 0 By making a 2 = -l 2 l 1 and a 3 = -3l 3 l 1 in T 2 and T 3 , we enablel 2 ,l 3 = 0. The vector (4) becomes (l 1 , 0, 0, l 4 , l 5 ). We get the following representatives:
We get the following representatives: Making all the possible combinations, we get the following representatives: The vector (9) becomes (0, l 2 , l 3 , 0, l 5 ).
We get the following representatives: Finally, by gathering the operators (6, 8, 10 and 11), we obtain the following theorem.

Similarity reductions of the (2 + 1)-dimensional integrable coupling of the KdV equation
In this section, based on Theorem 2.1, we will find some reduced equations of Eq. (1) by using similarity variables. Case 3.1. Reduction by X 2 + X 4 .
Integrating the characteristic equation for X 2 + X 4 , we get the invariancẽ Case 3.4. For the generator X 2 +X 5 , we havex = y . Equation (1) can be reduced to

The exact solutions of reduced equations
In the previous section, we have dealt with the similarity reductions and derived the corresponding reduced equations. In this section, we use the tanh method on reduced equations, obtaining some exact solutions of Eq. (1). With the help of exact solutions, we can understand some motion rules of waves of the (2 + 1)-dimensional integrable coupling of KdV equation.
The main steps of the tanh method [24,25] are expressed as follows: 1. Consider the following nonlinear differential equations: . . , u y , v y , . . . , u t , v t , . . .) = 0, where F 1 , F 2 are polynomials of the u, v and their derivatives.

By using the wave transformations
where ξ = lx + ky + ct, and l, k, c are unknown constants, and substituting (21) into Eq. (20), we obtain the following nonlinear ordinary differential equations: 3. Next, we introduce the independent variable which leads to the following changes: 4. We assume that the solution of Eq. (22) is written as the following form: where n, m are positive integers, which are decided by balancing the highest order nonlinear terms with the derivative terms in the resulting equations. After deciding n, m, taking (23) and (24) (12), we get the following equations: Concerning ( Then, substituting Eq. (23) and Eq. (26) into Eq. (25), we collect all terms of Y i and obtain the algebraic equations including unknown numbers a i , b i (i = 0, 1, 2), l and k. By solving these equations, we have the following solutions: Putting (27) into Eq. (12), we obtain the exact solution as follows: where l = 0, k are arbitrary constants. Figures 1 and 2 depict the exact solution of Eq. (12), which is obtained by taking l = 1, k = 1 at t = 1.
Then, balancing (3) and , (3) and for (29), we have n = m = 2. Therefore, on the basis of Eq. (24), the solution of Eq. (13) can be assumed to be Next, substituting Eq. (23) and Eq. (30) into Eq. (29), we make all coefficients of Y i vanish and obtain the algebraic equations including the unknown numbers a i , b i (i = 0, 1, 2), l and k. Solving these equations, we have the following solutions: So, the exact solution of Eq. (13) is ⎧ ⎨ ⎩ u(x, y, t) = 4l 2 3 -2l 2 tanh 2 (lx + ky), v(x, y, t) = 8lk 3 -4lk tanh 2 (lx + ky),  (14), we get the following ordinary differential equations: Furthermore, balancing (3) and , (3) and for (33), we have n = m = 2. Therefore, based on Eq. (24), the solution of Eq. (14) can be assumed to be Next, substituting Eq. (23) and Eq. (34) into Eq. (33), we make all coefficients of Y i vanish and obtain the algebraic equations including unknown numbers a i , b i (i = 0, 1, 2), k and c. Solving these equations, we have the following solutions: So, the exact solution of Eq. (14) is where c = 0 and k are arbitrary constants. Figures 5 and 6 portray the solution of Eq. (14), which is obtained by taking k = -1, c = 1 at t = 1.  (15), we have the following ordinary differential equations: Then, balancing (3) and , (3) and for (33), we have n = m = 2. Therefore, based on Eq. (24), the solution of Eq. (15) can be assumed to be  Solving these equations, we have the following solutions: So, the exact solution of Eq. (15) is where l = 0, k are arbitrary constants.
When we take l = 1, k = -1 at t = 0, the values of u, v are illustrated in Figs. 7 and 8.  (16), we get the following ordinary differential equations: (41) Figure 9 u(x, y, t) for l = 1, c = 1 at t = 0 Then, balancing (3) and , (3) and for (41), we have n = m = 2. Therefore, based on Eq. (24), the solution of Eq. (16) can be assumed to be Next, substituting Eq. (23) and Eq. (42) into Eq. (41), we make all coefficients of Y i vanish and obtain the algebraic equations including unknown numbers a i , b i (i = 0, 1, 2), l and c. Solving these equations, we have the following solutions: So, the exact solution of Eq. (16) is where l = 0, c are arbitrary constants. Figures 9 and 10 depict the exact solution of Eq. (16), which is obtained by taking l = 1, c = 1 at t = 0.

Construction of conservation laws
In this section, we chiefly construct conservation laws of Eq. (1) using Ibragimov's method [27,30]. First, we prove that Eq. (1) is nonlinear self-adjoint.

Theorem 5.1 ([32])
The determining system of the multiplier (x, u) of system (47) is identical to the system of nonlinearly self-adjoint substitution.

Construction of conservation laws
admitted by the system of Eq. (47), gives rise to a conservation law, where the components C i of the conserved vector C = (C 1 , . . . , C n ) are determined by and W α = η αξ j u α j . The formal Lagrangian L should be written in the symmetric form concerning all mixed derivatives u α ij , u α ijk , . . . .

Conclusions
In this paper, Lie group analysis is applied to the (2 + 1)-dimensional integrable coupling of the KdV equation. The optimal system of the obtained symmetries and reduced equations are obtained based on symmetry method. Moreover, explicit solutions of the reduced equations are constructed by using the tanh method. Through the figures related to solutions, we can show the rules of the wave propagation corresponding to Eq. (1). Finally, nonlinearly self-adjointness of Eq. (1) is manifested and its conservation laws are derived by using Ibragimov's method.