Conservation Laws for a Generalized Coupled Korteweg-de Vries System

We construct conservation laws for a generalized coupledKdV system,which is a third-order systemof nonlinear partial differential equations. We employ Noether’s approach to derive the conservation laws. Since the system does not have a Lagrangian, we make use of the transformation u = U x , V = V x and convert the system to a fourth-order system inU,V.This new systemhas a Lagrangian, and so the Noether approach can now be used to obtain conservation laws. Finally, the conservation laws are expressed in the u, V variables, and they constitute the conservation laws for the third-order generalized coupled KdV system. Some local and infinitely many nonlocal conserved quantities are found.

In this study, we consider a special case of the generalized coupled KdV system given by   +   +   + VV  = 0, and construct conservation laws for (2).Recently, the conservation laws of system (2) for special values of the constants  =  = −1 and  =  = −6 were derived in [12] using the multiplier approach.Many nonlinear partial differential equations (PDEs) of mathematical physics and engineering are continuity equations, which express conservation of mass, momentum, energy, or electric charge.It is well known that conservation laws play a crucial role in the solution and reduction of PDEs.For variational problems the conservation laws can be constructed by means of the Noether theorem [13].The application of the Noether theorem depends upon the existence of a Lagrangian.However, there are nonlinear differential equations that do not have a Lagrangian.In such instances, researchers have developed several methods to derive conserved quantities for such equations.See, for example, [14][15][16][17][18][19][20].
The organization of this paper is as follows.In Section 2 we briefly recall some notations and fundamental relations concerning the Noether symmetries approach, which we utilize in the same section to obtain the Noether symmetries and the corresponding conserved vectors.The concluding remarks are summarized in Section 3.

Conservation Laws of Coupled KdV Equations
In this section we derive the conservation laws for the generalized coupled KdV system (2).This system does not have a Lagrangian.In order to apply the Noether theorem we transform our system (2) to a fourth-order system, using the transformations  =   and V =   .Then system (2) becomes It can readily be verified that the second-order Lagrangian for system (3) is given by because where / and / are the standard Euler operators defined by Consider the vector field which has the second-order prolongation defined by Here The Lie-Bäcklund operator  defined in ( 7) is a Noether operator corresponding to the Lagrangian (4) if it satisfies  [2] ( where  1 (, , , ),  2 (, , , ) are the gauge terms.Expansion of (10) yields The splitting of (11) with respect to different combinations of derivatives of  and  results in an overdetermined system of PDEs for  1 ,  2 ,  1 ,  2 ,  1 , and  2 .Solving this system of PDEs we arrive at the following two cases for which Noether symmetries exist.
Case 1.  ̸ = .In this case we obtain the following Noether symmetries and gauge terms: The above results will now be used to find the components of the conserved vectors for the second-order Lagrangian.Here we can choose  = 0,  = 0 as they contribute to the trivial part of the conserved vector.We recall that the conserved vectors for the second-order Lagrangian are given by [13,21] Here  1 and  2 are the Lie characteristic functions, given by Using (13) together with (12) and  =   , V =   we obtain the following independent conserved vectors for system (2): and for the arbitrary functions () and (), Conserved vector ( 14) is a nonlocal conserved vector, and ( 15) is a local conserved vector for system (2).We now derive two particular cases from conserved vector ( 16) by letting () = 1 and () = 0, which gives a nonlocal conserved vector and by choosing () = 0 and () = 1, we get the nonlocal conserved vector Case 2.  = .
The second case gives the following Noether symmetries and gauge terms: Again we can set  = 0 and  = 0 as they contribute to the trivial part of the conserved vector.The independent conserved vectors for system (2), in this case, are and for the arbitrary functions () and (), we obtain Conserved vectors (20) are nonlocal, whereas ( 21) is a local conserved vector for system (2).Conserved vector (22) for () = 1 and () = 0 gives a nonlocal conserved vector and for () = 0 and () = 1 it gives a nonlocal conserved vector We note that for arbitrary values of () and () infinitely many nonlocal conservation laws exist for system (2).

Conclusion
In this paper we studied the third-order generalized coupled Korteweg-de Vries system (2).This system did not have a Lagrangian.In order to apply Noether theorem the transformations  =   and V =   were utilized, and the system was transformed to fourth-order system (3) in  and  variables.This system admitted the Lagrangian (4).Noether theorem was then used to derive the conservation laws in  and  variables.Finally, by reverting back to our original variables  and V we obtained the conservation laws for the thirdorder generalized coupled KdV system (2).The conservation laws obtained consisted of some local and infinite number of nonlocal conserved vectors.