Symmetries and conservation laws of evolution equations via

NTMSCI 5, No. 1, 128-136 (2017)

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New Trends in Mathematical Sciences http://dx.doi.org/10.20852/ntmsci.2017.130

Symmetries and conservation laws of evolution equations via multiplier and nonlocal conservation methods Emrullah Yasar and Yakup Yildirim Department of Mathematics, Faculty of Arts and Sciences, Uludag University, Bursa, Turkey Received: 20 May 2016, Accepted: 1 September 2016 Published online: 11 February 2017.

Abstract: In this work, we have applied a new technique which is a union of multiplier and Ibragimov’s nonlocal conservation method for constructing the local conservation laws of nonlinear evolution equations. One can conclude that the higher order solutions of adjoint equation can be obtained by the multiplier functions. The Lax equation and generalized Hirota-Satsuma coupled KdV system are chosen to illustrate the effectiveness of the method. Thus, we have obtained a plenty of local (some of them are the higher order) conservation laws. The combined method presents a wider applicability for handling the conservation laws of nonlinear wave equations. Keywords: Lax equation, generalized Hirota-Satsuma coupled KdV system, symmetry, conservation laws.

1 Introduction As well known, the method of Lie symmetry is one of the most powerful method for investigating the differential equations. There exist extensive literature for this topic [1]-[3]. Lie symmetry method has a close relationship with the concept of conservation laws. Conservation laws are very important tools in the study of differential equations from mathematical as well as physical point of view.If the under study system has conservation laws then its integrability is quite possible. They are also used for existence, uniqueness and Lyapunov stability analysis and construction of numerical schemes. Moreover, conservation laws are used in obtaining the new nonlocal symmetries, nonlocal conservation laws and linerization [4]. The first study in the literature for obtaining the conservation laws using the Lie symmetries is given by E. Noether [5].In this seminal work,it is showed that for Euler-Lagrange differential equations, to each Noether symmetry associated with the Lagrangian there corresponds a conservation law which can be determined explicitly by a formula. The application of Noether’s theorem depends upon the knowledge of a suitable Lagrangian ([5]-[6]). There are some new approaches in the literature about construction of conservation laws such as direct method, partial Lagrangian method, the characteristic method, the variational approach (multiplier approach) and nonlocal conservation theorem method ([7], [8]-[12]). It is proved in [11], if a differential equation (ordinary or partial) has a symmetry then the under study equation always admits a conservation law. In order to obtain the local conservation laws, later Ibragimov [13] proposed the concept of nonlinearly self adjoint differential equations. Very recently, Gangwei et. al [14] presented a new way for obtaining the conservation laws of partial differential equations (PDEs) with the help of combination of multiplier and nonlocal conservation method. In this new method, the c 2017 BISKA Bilisim Technology

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E. Yasar and Y. Yildirim: Symmetries and conservation laws of evolution equations via multiplier...

most important point is that the solution of the adjoint equation is equivalent the multiplier function. Therefore, one can obtain local conservation laws of the system using the both method together. In this work, we considered two famous evolution equations from the soliton theory. The first one is the Lax equation. Lax equation ut + 10uuxxx + 20uxuxx + 30u2ux + uxxxxx = 0, (1) first appeared in the work of [15] (see also [16]) when studying nonlinear equations of evolutions with linear operators. It was found that the Lax equation belongs to the completely integrable hierarchy of higher order KdV equations. This equation have infinite sets of conservation laws [17]. On the other hand, by introducing a 4 x 4 matrix spectral problem with three potentials, Wu et al. derived a new hierarchy of nonlinear evolution equations [18]. One of typical equations in the hierarchy is the new and generalized Hirota-Satsuma coupled KdV equation ut −

uxxx + 3uux − 3vx w − 3vwx = 0, 2 vt + vxxx − 3uvx = 0, wt + wxxx − 3uwx = 0.

(2)

Soliton solutions and Miura transformationa for this system were derived in [18]-[19], respectively. The plan of the paper is organized as follows : In Section 2, we give briefly the description of the nonlocal conservation and multiplier methods. In addition, the relationships of both methods were emphasized. Section 3 is devoted to the conservation laws of Eq.(1) and Eq.(2) with the help of this combined method. In Section 4, some concluding remarks are given.

2 Necessary preliminaries In this section, we present the notations and some of the definitions below. For the details see e.g., [4],[7], [11], [13] and [14]. Consider a sth-order PDESs of n independent variables x = (x1 , x2 , ..., xn ) and m dependent variables u = (u1 , u2 , ..., um ) E α (x, u, u(1) , ..., u(s) ) = 0,

α = 1, ..., m,

(3)

where u(1) , u(2) ..., u(s) denote the collections of all first, second,...,sth-order partial derivatives, that is, uαi = Di (uα ), uαi j = D j Di (uα ),... respectively, with the total differentiation operator with respect to xi given by Di =

∂ ∂ ∂ + uαi j α + ..., i = 1, ..., n, + uαi ∂ xi ∂ uα ∂uj

(4)

where the summation convention is used whenever appropriate. As usual A is the vector space of differential functions of finite orders. The basic operators defined in A are stated below. The n−tuple vector T = T 1 , T 2 , ..., T n , T j ε A, j = 1, ..., n is a conserved vector of Eq. (3) if T i satisfies Di T i (3) = 0.

(5) c 2017 BISKA Bilisim Technology

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Equation (5) defines a local conservation law of system (3).Every admitted conservation laws of Eq. (3) arises from multipliers Qα x, u, u(1) , ... such that (6) Qα E α = Di T i , holds identically. In the multiplier approach for conservation laws, one takes the variational derivative of (6) that is,

δ (Qα Eα ) = 0, δ uβ

(7)

holds for arbitrary functions of u x1 , x2 , ..., xn where

δ ∂ ∂ = α + ∑ (−1)s Di1 ...Dis α , α = 1, ..., m, δ uα ∂u ∂ u i1 ...is s>1

(8)

is the Euler(-Lagrange) operator. In [14], the authors modified the Ibragimov’s nonlocal conservation method in terms of the multiplier function. We now present the nonlocal conservation method using the multiplier function which we described in the above. It is interesting to note that the higher order solutions of Ibragimov’s adjoint equation which is consequences of the concept of nonlinearly self-adjointness can be obtained by the multiplier functions. According to [14], the solutions of the adjoint equation are constructed through the multipliers satisfying

δ (Qµ Eα ) = 0, α = 1, ..., m. δ uα

(9)

For each known Qµ , we define a formal Lagrangian function L = Q µ Eα

(10)

and whose Euler-Lagrange equations inherit, all Lie symmetry generators (point, Lie-Backlund and nonlocal) of Eq. (3). Using the Noether theorem, we obtain the following conservation laws of Eq. (3) Ci = N i L.

(11)

where N i is the Noether operator, Ni = ξ i + W α

δ ∂ + ∑ Di1 ...Dis (W α ) α , i = 1, ..., m, δ uαi ∂ u i1 ...is s>1

and X = ξi

∂ ∂ + ηα α ∂ xi ∂u

(12)

(13)

is a symmetry generator of (3) and W α = η α − ξ j uαj is the characteristic function. In the following, for Eq.(1), Qµ = Q, and for system (2), we will write (Q1 , Q2 , Q3 ) as (Q, P, K). For two independent variables, (x1 , x2 ) will be written as (x,t) so that (C1 ,C2 ) will be written as (Cx ,Ct ).

3 Conservation laws of Eq. (1) and (2). c 2017 BISKA Bilisim Technology

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3.1 Lax equation We first consider Lax equation (1) ut + 10uuxxx + 20uxuxx + 30u2ux + uxxxxx = 0 whose Lie point symmetry generators are X1 =

∂ ∂ ∂ 5t ∂ x ∂ , X2 = , X3 = − − +u . ∂t ∂x 2 ∂t 2 ∂x ∂u

(14)

By (7), the multipliers up to fourth-order Qi that lead to nontrivial conservation laws are Q1 = −30tu2 − 10tuxx + x, u2 utx 35u4 − 20u2uxx − 5uu2x − 2uuxxxx + ux uxxx − xx + , 2 2 2 Q3 = 10u3 + 10uuxx + 5u2x + uxxxx , Q2 = −

Q4 = 3u2 + uxx , Q5 = u, Q6 = 1.

(15)

It is readily seen that each Qi (i = 1, ..., 6) satisfies the adjoint equation of Eq.(1) −wt − 10wx uxx − 30u2wx − 10uxwxx − 10uwxxx − wxxxxx = 0.

(16)

For instance, for the multiplier Q5 = u, the left-hand side of (16) becomes −ut − 10uuxxx − 20ux uxx − 30u2ux − uxxxxx ,

(17)

which vanishes on Eq.(1). On the other hand the situation is the same for higher order multipliers. For example, for the higher-order multiplier Q4 , the left-hand side of (16) becomes, after some simplification, − 6u(ut + 10uuxxx + 20uxuxx + 30u2ux + uxxxxx ) − (ut + 10uuxxx + 20uxuxx + 30u2ux + uxxxxx )xx

(18)

which also vanishes on Eq. (1). We now present the conserved vectors of Eq.(1) using the combined method namely multiplier function and Ibragimov’s method omitting the calculations. In our scheme, we use the multipliers Q1 , Q3 , Q4 , Q5 , Q6 ; Q2 is cumbersome, but the procedure would be the same. Based on the stated multiplier and symmetry, the conserved vectors are given below. Case 1. Q6 , X1 , Ct = 10uuxxx + 20uxuxx + 30u2ux + uxxxxx , Cx = −(30u2 + 10uxx )ut − 10uxutx − 10uutxx − utxxxx c 2017 BISKA Bilisim Technology


Case 2. Q6 , X2 , Ct = −ux , Cx = ut + 10uuxxx + 20ux uxx + 30u2ux + uxxxxx − (30u2 + 10uxx )ux − 10uxuxx − 10uuxxx − uxxxxx Case 3. Q6 , X3 , 5tut xux 5t (ut + 10uuxxx + 20uxuxx + 30u2ux + uxxxxx ) + u + + , 2 2 2 x 5tut xux Cx = − (ut + 10uuxxx + 20ux uxx + 30u2ux + uxxxxx ) + (u + + )(30u2 + 10uxx ) 2 2 2 5tutxx xuxxx 5tutxxxx xuxxxxx 3ux 5tutx xuxx + + ) + 10u(2uxx + + ) + 3uxxxx + + + 10ux ( 2 2 2 2 2 2 2 Ct = −

Case 4. Q5 , X1 , Ct = u(ut + 10uuxxx + 20uxuxx + 30u2ux + uxxxxx ) − uut , Cx = −ut u 20uxx + 30u2 + uxxxx + utx uxxx − utxx (10u2 + uxx ) + utxxx ux − uutxxxx Case 5. Q5 , X2 , Ct = −uux Cx = u(ut + 10uuxxx + 20ux uxx + 30u2ux + uxxxxx ) − ux u 20uxx + 30u2 + uxxxx + uxx uxxx − uxxx (10u2 + uxx ) + uxxxx ux − uuxxxxx Case 6. Q5 , X3 5tu 5tut xux (ut + 10uuxxx + 20ux uxx + 30u2ux + uxxxxx ) + u(u + + ), 2 2 2 xu 5tut xux Cx = − (ut + 10uuxxx + 20uxuxx + 30u2ux + uxxxxx ) + (u + + ) u 20uxx + 30u2 + uxxxx 2 2 2 5tutxx xuxxx 5uxxx 5tutxxx xuxxxx 3ux 5tutx xuxx 2 + + )uxxx + (2uxx + + )(10u + uxx ) − ( + + )ux −( 2 2 2 2 2 2 2 2 5tutxxxx xuxxxxx + )u + (3uxxxx + 2 2 Ct = −

Case 7. Q4 , X1 Ct = (3u2 + uxx )(ut + 10uuxxx + 20uxuxx + 30u2ux + uxxxxx ) − ut (3u2 + uxx ), Cx = −ut ((3u2 + uxx )(30u2 + 20uxx) − utx − 6uxuxxx − 4uuxxxx − 2u2xx − 60uu2x − 30u2uxx − 10uxx(3u2 + uxx ) + 10u(6uuxx + 6u2x + uxxxx )) − utx (ut + 4uuxxx + 2uxuxx + 30u2ux + 10ux(3u2 + uxx ) − 10u(6uux + uxxx )) − utxx (10u(3u2 + uxx ) + 6u2x + 6uuxx + uxxxx ) + utxxx (6uux + uxxx ) − utxxxx (3u2 + uxx ) Case 8. Q4 , X2 Ct = −ux (3u2 + uxx ), Cx = (3u2 + uxx )(ut + 10uuxxx + 20uxuxx + 30u2ux + uxxxxx ) − ux ((3u2 + uxx )(30u2 + 20uxx ) − utx − 6uxuxxx − 4uuxxxx − 2u2xx − 60uu2x − 30u2uxx − 10uxx(3u2 + uxx ) + 10u(6uuxx + 6u2x + uxxxx )) − uxx (ut + 4uuxxx + 2uxuxx + 30u2ux + 10ux (3u2 + uxx ) − 10u(6uux + uxxx )) − uxxx (10u(3u2 + uxx ) + 6u2x + 6uuxx + uxxxx ) + uxxxx (6uux + uxxx ) − uxxxxx (3u2 + uxx )

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Case 9. Q4 , X3 , 5t 5tut xux (3u2 + uxx )(ut + 10uuxxx + 20ux uxx + 30u2ux + uxxxxx ) + (u + + )(3u2 + uxx ), 2 2 2 5tut xux x + )((3u2 + uxx )(30u2 + 20uxx ) Cx = − (3u2 + uxx )(ut + 10uuxxx + 20uxuxx + 30u2ux + uxxxxx ) + (u + 2 2 2 3ux 5tutx + − utx − 6uxuxxx − 4uuxxxx − 2u2xx − 60uu2x − 30u2uxx − 10uxx(3u2 + uxx ) + 10u(6uuxx + 6u2x + uxxxx )) + ( 2 2 xuxx 5tutxx xuxxx 2 2 + )(ut + 4uuxxx + 2uxuxx + 30u ux + 10ux(3u + uxx ) − 10u(6uux + uxxx )) + (2uxx + + )(10u(3u2 + uxx ) 2 2 2 5tutxxxx xuxxxxx 5uxxx 5tutxxx xuxxxx + + )(6uux + uxxx ) + (3uxxxx + + )(3u2 + uxx ) + 6u2x + 6uuxx + uxxxx ) − ( 2 2 2 2 2 Ct = −

Case 10. Q3 , X1 , Ct = (10u3 + 10uuxx + 5u2x + uxxxx )(ut + 10uuxxx + 20uxuxx + 30u2ux + uxxxxx ) − ut (10u3 + 10uuxx + 5u2x + uxxxx ), Cx = −ut ((10u3 + 10uuxx + 5u2x + uxxxx )(30u2 + 20uxx) − 10uutx − 10uxx (10u3 + 10uuxx + 5u2x + uxxxx ) − utxxx ) − utx (10uut + 10ux(10u3 + 10uuxx + 5u2x + uxxxx ) + utxx ) − utxx (10u(10u3 + 10uuxx + 5u2x + uxxxx ) − utx ) − utxxx ut − utxxxx (10u3 + 10uuxx + 5u2x + uxxxx ) Case 11. Q3 , X2 Ct = −ux (10u3 + 10uuxx + 5u2x + uxxxx ), Cx = (10u3 + 10uuxx + 5u2x + uxxxx )(ut + 10uuxxx + 20uxuxx + 30u2ux + uxxxxx ) − ux ((10u3 + 10uuxx + 5u2x + uxxxx )(30u2 + 20uxx ) − 10uutx − 10uxx(10u3 + 10uuxx + 5u2x + uxxxx ) − utxxx ) − uxx (10uut + 10ux(10u3 + 10uuxx + 5u2x + uxxxx ) + utxx ) − uxxx (10u(10u3 + 10uuxx + 5u2x + uxxxx ) − utx ) − uxxxx ut − uxxxxx (10u3 + 10uuxx + 5u2x + uxxxx ) Case 12. Q1 , X1 , Ct = (−30tu2 − 10tuxx + x)(ut + 10uuxxx + 20uxuxx + 30u2ux + uxxxxx ) − ut (−30tu2 − 10tuxx + x), Cx = −ut ((−30tu2 − 10tuxx + x)(30u2 + 20uxx ) + 10t(30u2uxx + 60uu2x + 10uuxxxx + 30uxuxxx + 20u2xx + utx + u6x ) − 10uxx(−30tu2 − 10tuxx + x) + 10u(−60tuuxx − 60tu2x − 10tuxxxx ) − 180tu2xx − 240tuxuxx − 60tuuxxxx − 10tu6x) − utx (−10t(ut + 10uuxxx + 20ux uxx + 30u2ux + uxxxxx ) + 10ux(−30tu2 − 10tuxx + x) − 10u(−60tuux − 10tuxxx + 1) + 180tuxuxx + 60tuuxxx + 10tu5x) − utxx (10u(−30tu2 − 10tuxx + x) − 60tu2x − 60tuuxx − 10tuxxxx ) − utxxx (60tuux + 10tuxxx − 1) − utxxxx (−30tu2 − 10tuxx + x) Case 13. Q1 , X2 Ct = −ux (−30tu2 − 10tuxx + x), Cx = (−30tu2 − 10tuxx + x)(ut + 10uuxxx + 20uxuxx + 30u2ux + uxxxxx ) − ux ((−30tu2 − 10tuxx + x)(30u2 + 20uxx ) + 10t(30u2uxx + 60uu2x + 10uuxxxx + 30uxuxxx + 20u2xx + utx + u6x ) − 10uxx(−30tu2 − 10tuxx + x) + 10u(−60tuuxx − 60tu2x − 10tuxxxx ) − 180tu2xx − 240tuxuxx − 60tuuxxxx − 10tu6x) − uxx (−10t(ut + 10uuxxx + 20uxuxx + 30u2ux + uxxxxx ) + 10ux(−30tu2 − 10tuxx + x) − 10u(−60tuux − 10tuxxx + 1) + 180tuxuxx + 60tuuxxx + 10tu5x) − uxxx (10u(−30tu2 − 10tuxx + x) − 60tu2x − 60tuuxx − 10tuxxxx ) − uxxxx (60tuux + 10tuxxx − 1) − uxxxxx (−30tu2 − 10tuxx + x).



134

3.2 Generalized Hirota-Satsuma coupled KdV equation We now consider generalized Hirota-Satsuma coupled KdV equation. The symmetries and multipliers up to second-order (Qi , Pi , Ki ) are ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ , X3 = −v + w , X4 = 3t + x − 2u − 4v (19) X1 = , X2 = ∂t ∂x ∂v ∂w ∂t ∂x ∂u ∂v and x 3u2 uxx Q1 = −tu + , P1 = tw, K1 = tv, Q2 = 0, P2 = wx , K2 = −vx , Q3 = − vw − , 3 2 2 P3 = −uw + wxx , K3 = −uv + vxx, Q4 = −u, P4 = w, K4 = v Q5 = 1, P5 = 0, K5 = 0

(20)

respectively. Based on the symmetry X4 and the obtained multipliers, the components of conserved vectors given as follows. Case 1. (Q5 , P5 , K5 ) uxxx + 3uux − 3vx w − 3vwx ) − 2u − xux − 3tut , 2 1 uxxx + 3uux − 3vx w − 3vwx ) − 3u(2u + xux + 3tut ) + (4uxx + xuxxx + 3tutxx ) + 3w(4v + 3tvt + xvx ) Cx = x(ut − 2 2 Ct = 3t(ut −

Case 2. (Q4 , P4 , K4 ) uxxx + 3uux − 3vx w − 3vwx ) + w(vt + vxxx − 3uvx ) + v(wt + wxxx − 3uwx )) 2 + u(2u + 3tut + xux ) − w(4v + 3tvt + xvx ) − v(3twt + xwx ), uxxx + 3uux − 3vx w − 3vwx ) + w(vt + vxxx − 3uvx ) + v(wt + wxxx − 3uwx )) Cx = x(−u(ut − 2 uxx ux u − (2u + 3tut + xux )(−3u2 + ) + (3ux + 3tutx + xuxx ) − (4uxx + 3tutxx + xuxxx ) 2 2 2 − wxx (4v + 3tvt + xvx ) + wx (5vx + 3tvtx + xvxx ) − w(6vxx + 3tvtxx + xvxxx ) Ct = 3t(−u(ut −

− vxx (3twt + xwx ) + vx (3twtx + wx + xwxx ) − v(2wxx + 3twtxx + xwxxx ) Case 3. (Q1 , P1 , K1 ) x uxxx Ct = 3t((−tu + )(ut − + 3uux − 3vx w − 3vwx ) + tw(vt + vxxx − 3uvx) + tv(wt + wxxx − 3uwx )) 3 2 x − (2u + 3tut + xux )(−tu + ) − tw(4v + 3tvt + xvx ) − tv(3twt + xwx ), 3 uxxx x x + 3uux − 3vx w − 3vwx ) + tw(vt + vxxx − 3uvx) + tv(wt + wxxx − 3uwx )) C = x((−tu + )(ut − 3 2 x tuxx 1 1 1 x − (2u + 3tut + xux )(3u(−tu + ) + ) − (−tux + )(3ux + 3tutx + xuxx ) + (−tu + )(4uxx + 3tutxx + xuxxx ) 3 2 2 3 2 3 x − (4v + 3tvt + xvx )(−3w(−tu + ) − 3tuw + twxx) + twx (5vx + 3tvtx + xvxx ) − tw(6vxx + 3tvtxx + xvxxx ) 3 x − (3twt + xwx )(−3v(−tu + ) − 3tuv + tvxx) + tvx(3twtx + wx + xwxx ) − tv(2wxx + 3twtxx + xwxxx ) 3 Case 4. (Q2 , P2 , K2 ) Ct = 3t(wx (vt + vxxx − 3uvx) − vx (wt + wxxx − 3uwx )) − wx (4v + 3tvt + xvx ) + vx (3twt + xwx ), Cx = x(wx (vt + vxxx − 3uvx) − vx (wt + wxxx − 3uwx )) + wt (4v + 3tvt + xvx ) + wxx (5vx + 3tvtx + xvxx ) − wx (6vxx + 3tvtxx + xvxxx ) − vt (3twt + xwx ) − vxx (3twtx + wx + xwxx ) + vx (2wxx + 3twtxx + xwxxx ) c 2017 BISKA Bilisim Technology

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Case 5. (Q3 , P3 , K3 ), uxx uxxx 3u2 − vw − )(ut − + 3uux − 3vx w − 3vwx ) + (−uw + wxx )(vt + vxxx − 3uvx ) 2 2 2 uxx 3u2 − vw − ) − (4v + 3tvt + xvx )(−uw + wxx ) + (−uv + vxx)(wt + wxxx − 3uwx )) − (2u + 3tut + xux )( 2 2 − (3twt + xwx )(−uv + vxx ),

Ct = 3t((

3u2 uxx uxxx − vw − )(ut − + 3uux − 3vx w − 3vwx ) + (−uw + wxx)(vt + vxxx − 3uvx ) 2 2 2 uxx utx 3u2 − vw − )+ − wvxx − 2vx wx − vwxx ) + (−uv + vxx)(wt + wxxx − 3uwx )) − (2u + 3tut + xux )(3u( 2 2 2 ut 1 3u2 uxx − (3ux + 3tutx + xuxx )(− + wvx + vwx ) + (4uxx + 3tutxx + xuxxx )( − vw − ) 2 2 2 2 uxx 3u2 − vw − ) − 3u(−uw + wxx) − wtx + ux wx + 2uwxx − wuxx ) − (4v + 3tvt + xvx )(−3w( 2 2 3u2 uxx − (5vx + 3tvtx + xvxx )(−2uwx + wux + wt ) − (6vxx + 3tvtxx + xvxxx )(−uw + wxx ) − (3twt + xwx )(−3v( − vw − ) 2 2 − 3u(−uv + vxx) − vtx + ux vx + 2uvxx − vuxx )

Cx = x((

− (3twtx + wx + xwxx )(−2uvx + vux + vt ) − (2wxx + 3twtxx + xwxxx )(−uv + vxx). It is readily seen that the situations are the same for the other symmetry generators.

4 Concluding remarks In this work we have constructed conservation laws of the Lax equation and generalized Hirota-Satsuma coupled KdV system which is not derivable from a variational principle. Namely, no recourse to a Lagrangian formulation is made. Using the combined approach which is a union of multiplier and Ibragimov’s nonlocal conservation method, a plenty of local conservation laws were obtained. We conclude that the solutions of adjoint equation can be obtained by the multiplier functions. The conserved vectors obtained here can be used in double reductions and solutions of the underlying equations [20].

Competing interests The authors declare that they have no competing interests.

Authors’ contributions All authors have contributed to all parts of the article. All authors read and approved the final manuscript.

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