Lie symmetry reductions and conservation laws for fractional order coupled KdV system

Lie symmetry analysis is achieved on a new system of coupled KdV equations with fractional order, which arise in the analysis of several problems in theoretical physics and numerous scientific phenomena. We determine the reduced fractional ODE system corresponding to the governing factional PDE system. In addition, we develop the conservation laws for the system of fractional order coupled KdV equations.


Introduction
Fractional partial differential equations (FPDEs) have a significant role to play in various fields such as chemistry, physics, fluid dynamics and biology, therefore obtaining solutions of such FPDEs is unavoidable [1,2]. There are many numerical and theoretical methods for solving fractional order differential equations [1][2][3][4].
The Lie symmetry technique is one of the most useful techniques to conclude to solutions of nonlinear FPDEs, generally, Lie symmetries might be used to reduce the order of the original equation (system of equations) as well as the number of independent variables [5][6][7][8][9][10][11].
In this paper, we consider the new coupled KdV system (1) of fractional order given by where α ∈ (0, 2). The article is organized as follows. In Sect. 2, some definitions and properties of Lie group scheme to analysis of (2) are given. In Sect. 3, we find Lie point symmetries of system(2) and reduced system of this system. The conservation laws of (2) are obtained in Sect. 4. Discussion and conclusions are summarized in Sect. 5.

The symmetry group analysis of (2)
In this section, we briefly review some key definitions and properties of the fractional Lie group scheme to obtain infinitesimal function of the FPDE system.

Lie symmetries and similarity reductions for (2)
We apply the α-prolongation of X (α) to Eq. (2). It gives the following claim. (2) is spanned by the following vector fields:

Theorem 1 Lie symmetry group of
Proof Let us consider the one parameter Lie group of infinitesimal transformation in x, t, u, v, w given by By applying the X (α) to both sides of (2),we have We obtain the Lie point symmetries by expanding (8), and solving the resulting system using Maple as follows: ξ (x,ū) = c 1 + c 2 αx, τ (x,ū) = 3c 2 t, η u (x,ū) = -2c 2 αu, η v (x,ū) = -2c 2 αv, η w (x,ū) = -2c 2 αw, here c 1 and c 2 are arbitrary constants. Thus, the corresponding vector fields are Here we want to obtain symmetry reductions of (2), then the system (2) transforms into a system of fractional ODE.
In order to solve the following associated Lagrange equations: We consider the following cases.
• Case 1: X 1 = ∂ ∂x . In this case the symmetry X 1 gives rise to the group-invariant solution: substituting (9) into (2) results in the fact that F(r), G(r) and H(r) fulfill the following differential equations: By using a Laplace transformation we get where k 1 , k 2 and k 3 are constant; therefore • Case 2: X 2 = αx ∂ ∂x + 3t ∂ ∂t -2αu ∂ ∂u -2αv ∂ ∂v -2αw ∂ ∂w . In this case, the group-invariant solution is substituting (10) into (2) leads to the following fractional ODE system:

Conservation laws
Now, we construct conservation laws for system (2) by using the Lie point symmetry (7).
The vectors C i = (C t i , C x i ), (i = 1, 2, 3) are called conserved vectors for system (2), if they satisfy the conservation equations, For system (2), a formal Lagrangian can be introduced as where i (x, t), i = 1, 2, 3, are new dependent variables. The Euler-Lagrange operators are defined by here (D α t ) is the adjoint operator of D α t . For the RL-fractional operators The adjoint equations to the system (2) are written as Replacing the formal Lagrangian (11) into (12), we have Since in the system (2), there are no fractional derivatives involved w.r.t. x, we have where are the Lie characteristic functions corresponding to the Lie symmetries X 1 and X 2 .
If we have the RL-time-fractional derivative in the system (2) then the operators N t are given by where J is the integral the operators N x are defined by For any generator X of system (2), we have These equalities yield the conservation laws For the case, when α ∈ (0, 1), using N t i and N x i (i = 1, 2, 3), one can get the components of the conserved vectors and where i = 1, 2 and the functions W i are Also, when α ∈ (1, 2), we get the components of the conserved vectors where i = 1, 2 and the functions W i in the form (17); also the conserved vectors C x 1 , C x 2 , C x 3 coincide with (14), (15) and (16).

Conclusions
In this paper, Lie symmetries and conservation laws have been studied for fractional order coupled KdV system (2). First, we obtained the fractional Lie point symmetries to the KdV system (2) with Riemann-Liouville derivative and we have shown that system (2) can be reduced to a nonlinear system of FDEs. Finally, conservation laws are constructed for system (2), the calculated conserved vectors, might be used for creating the particular solutions for the KdV system by the given method in [23,24].