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Lie Group Analysis of a Nonlinear Coupled System of Korteweg-de Vries Equations

Joseph Owuor Owino∗, Benard Okelo
Department of Pure and Applied Mathematics, Jaramogi Oginga Odinga University of Science and

Technology, Box 210-40601, Bondo, Kenya
bnyaare@yahoo.com, josephowuorowino@gmail.com
∗Correspondence: josephowuorowino@gmail.com

Abstract. In this paper, we consider coupled Korteweg-de Vries equations that model the propagationof shallow water waves, ion-acoustic waves in plasmas, solitons, and nonlinear perturbations alonginternal surfaces between layers of different densities in stratified fluids, for example propagation ofsolitons of long internal waves in oceans. The method of Lie group analysis is used to on the systemto obtain symmetry reductions. Soliton solutions are constructed by use of a linear combination oftime and space translation symmetries. Furthermore, we compute conservation laws in two waysthat is by multiplier method and by an application of new conservation theorem developed by NailIbragimov.

1. Introduction
The dynamics of shallow-water waves, ion-acoustic waves in plasmas, and long internal waves inoceans can be described by coupled KdV equations. The equations are derived from the classicalkdV equation. This section extends the previous study of kdV equations to that of a couplednonlinear system. From the Kortweg-de Vries equation

qt + αqqx + βqxxx = 0, (1)
for α and β as constants, we let

q(t,x) = u(t,x) + iv(t,x), (2)
where i2 = −1. Then substituting (2) into (1) and separating the real and imaginary parts, weobtain

∆1 ≡ ut + αuux −αvvx + βuxxx = 0, ∆2 ≡ vt + αuvx + αvux + βvxxx = 0, (3)
Received: 3 Sep 2021.
Key words and phrases. coupled KdV equations; Lie group analysis; group-invariant solutions; stationary solutions;symmetry reductions; soliton; multipliers; conservation laws.133

https://adac.ee
https://doi.org/10.28924/ada/ma.1.133


Eur. J. Math. Anal. 1 (2021) 134
which is a nonlinear system of coupled KdV equations. We perform Lie symmetry analysis on (3),that is , we obtain Lie point symmetries, invariant solutions and conservation laws of (3).This paperuses symmetry analysis method to construct exact solutions and conservation laws for a nonlinearcoupled kdV system (3).

2. Preliminaries
In this section, we outline preliminary concepts which are useful in the sequel. In Euclideanspaces Rn of x = x i independent variables and Rm of u = uα dependent variables, we considerthe transformations

T� : x̄
i = ϕi (x i,uα,�), ūα = ψα(x i,uα,�), (4)

involving the continuous parameter � which ranges from a neighbourhood N ′ ⊂ N ⊂ R of � = 0where the functions ϕi and ψα differentiable and analytic in the parameter �.
Definition 2.1. The set G of transformations given by (4) is a local Lie group if it holds true that

(1) (i). (Closure) Given T�1,T�2 ∈ G, for �1,�2 ∈ N ′ ⊂ N , then T�1T�2 = T�3 ∈ G, �3 =
φ(�1,�2) ∈N .(2) (ii). (Identity) There exists a unique T0 ∈G if and only if � = 0 such that T�T0 = T0T� = T�.(3) (iii). (Inverse) There exists a unique T�−1 ∈G for every transformation T� ∈G,where � ∈N ′ ⊂N and �−1 ∈N such that T�T�−1 = T�−1T� = T0.

Remark 2.2. Associativity of the group G in (4) follows from (1).
In the system,

∆α
(
x i,uα,u(1), . . . ,u(π)

)
= ∆α = 0, (5)the variables uα are dependent. The partial derivatives u(1) = {uαi },u(2) = {uαij}, . . . ,u(π) =

{uαi1...iπ}, are of the first, second, . . . , up to the πth-orders.Denoting
Di =

∂

∂x i
+ uαi

∂

∂uα
+ uαij

∂

∂uα
j

+ . . . , (6)
the total differentiation operator with respect to the variables x i and δj

i
, the Kronecker delta, wehave

Di (x
j) = δ

j
i
, ′, uαi = Di (u

α), uαij = Dj(Di (u
α)), . . . , (7)

where uαi defined in (7) are differential variables [7].Consider the local Lie group G given by the transformations
x̄ i = ϕi (x i,uα,�), ϕi

∣∣∣
�=0

= x i, ūα = ψα(x i,uα,�), ψα
∣∣∣
�=0

= uα, (8)
where the symbol ∣∣∣

�=0
means evaluated on � = 0.



Eur. J. Math. Anal. 1 (2021) 135
Definition 2.3. The construction of the group G given by (8) is an equivalence of the computationof infinitesimal transformations

x̄ i ≈ x i + ξi (x i,uα)�, ϕi
∣∣∣
�=0

= x i, ūα ≈ uα + ηα(x i,uα)�, ψα
∣∣∣
�=0

= uα, (9)
obtained from (4) by a Taylor series expansion of ϕi (x i,uα,�) and ψi (x i,uα,�) in � about � = 0and keeping only the terms linear in �, where

ξi (x i,uα) =
∂ϕi (x i,uα,�)

∂�

∣∣∣
�=0
, ηα(x i,uα) =

∂ψα(x i,uα,�)

∂�

∣∣∣
�=0
. (10)

Remark 2.4. The symbol of infinitesimal transformations, X, is used to write (9) as
x̄ i ≈ (1 + X)x i, ūα ≈ (1 + X)uα, (11)

where
X = ξi (x i,uα)

∂

∂x i
+ ηα(x i,uα)

∂

∂uα
, (12)

is the generator of the group G given by (8).
Remark 2.5. To obtain transformed derivatives from (4), we use a change of variable formulae

Di = Di (ϕ
j)D̄j, (13)

where D̄j is the total differentiation in the variables x̄ i . This means that
ūαi = D̄i (ū

α), ūαij = D̄j(ū
α
i ) = D̄i (ū

α
j ). (14)

If we apply the change of variable formula given in (13) on G given by (8), we get
Di (ψ

α) = Di (ϕ
j), D̄j(ū

α) = ūαj Di (ϕ
j). (15)

Expansion of (15) yields (
∂ϕj

∂x i
+ u

β
i

∂ϕj

∂uβ

)
ū
β
j

=
∂ψα

∂x i
+ u

β
i

∂ψα

∂uβ
. (16)

The variables ūαi can be written as functions of x i,uα,u(1), that is
ūαi = Φ

α(x i,uα,u(1),�), Φ
α
∣∣∣
�=0

= uαi . (17)
Definition 2.6. The transformations in the space of the variables x i,uα,u(1) given in (8) and (17)form the first prolongation group G[1].
Definition 2.7. Infinitesimal transformation of the first derivatives is

ūαi ≈ u
α
i + ζ

α
i �, where ζαi = ζαi (x i,uα,u(1),�). (18)

Remark 2.8. In terms of infinitesimal transformations, the first prolongation group G[1] is given by(9) and (18).



Eur. J. Math. Anal. 1 (2021) 136
Definition 2.9. By using the relation given in (15) on the first prolongation group G[1] given byDefinition 2.6, we obtain [5]
Di (x

j + ξj�)(uαj + ζ
α
j �) = Di (u

α + ηα�), which gives uαi + ζαj � + uαj �Diξj = uαi + Diηα�,(19)
and thus

ζαi =Di (η
α) −uαj Di (ξ

j), (20)
is the first prolongation formula.
Remark 2.10. Similarly, we get higher order prolongations [8],

ζαij = Dj(ζ
α
i ) −u

α
iκDj(ξ

κ), . . . , ζαi1,...,iκ = Diκ(ζ
α
i1,...,iκ−1

) −uαi1,i2,...,iκ−1jDiκ(ξ
j). (21)

Remark 2.11. The prolonged generators of the prolongations G[1], . . . ,G[κ] of the group G are
X[1] = X + ζαi

∂

∂uα
i

, . . . ,X[κ] = X[κ−1] + ζαi1,...,iκ
∂

∂ζα
i1,...,iκ

, κ ≥ 1, (22)
where X is the group generator given by (12).
Definition 2.12. A function Γ(x i,uα) is called an invariant of the group G of transformations givenby (4) if

Γ(x̄ i, ūα) = Γ(x i,uα). (23)
Theorem 2.13. A function Γ(x i,uα) is an invariant of the group G given by (4) if and only if it
solves the following first-order linear PDE: [5]

XΓ = ξi (x i,uα)
∂Γ

∂x i
+ ηα(x i,uα)

∂Γ

∂uα
= 0. (24)

From Theorem (2.13), we have the following result.
Theorem 2.14. The local Lie group G of transformations in Rn given by (4) [7] has precisely n− 1
functionally independent invariants. One can take, as the basic invariants, the left-hand sides of
the first integrals

ψ1(x
i,uα) = c1, . . . ,ψn−1(x

i,uα) = cn−1, (25)
of the characteristic equations for (24):

dx i

ξi (x i,uα)
=

duα

ηα(x i,uα)
. (26)



Eur. J. Math. Anal. 1 (2021) 137
Definition 2.15. The vector field X (12) is a Lie point symmetry of the PDE system (5) if thedetermining equations

X[π]∆α

∣∣∣
∆α=0

= 0, α = 1, . . . ,m, π ≥ 1, (27)
are satisfied, where ∣∣∣

∆α=0
means evaluated on ∆α = 0 and X[π] is the π-th prolongation of X.

Definition 2.16. The Lie group G is a symmetry group of the PDE system given in (5) if the PDEsystem (5) is form-invariant, that is
∆α
(
x̄ i, ūα, ū(1), . . . , ū(π)

)
= 0. (28)

Theorem 2.17. Given the infinitesimal transformations in (8), the Lie group G in (4) is found by
integrating the Lie equations

dx̄ i

d�
= ξi (x̄ i, ūα), x̄ i

∣∣∣
�=0

= x i,
dūα

d�
= ηα(x̄ i, ūα), ūα

∣∣∣
�=0

= uα. (29)
Definition 2.18. A vector space Vr of operators [5] X (12) is a Lie algebra if for any two operators,
Xi,Xj ∈Vr , their commutator

[Xi,Xj] = XiXj −XjXi, (30)
is in Vr for all i, j = 1, . . . , r .
Remark 2.19. The commutator satisfies the properties of bilinearity, skew symmetry and the Jacobiidentity [5].
Theorem 2.20. The set of solutions of the determining equation given by (27) forms a Lie algebra [5].

The methods of (G’/G)-expansion method [20], Extended Jacobi elliptic function expansion [21]and Kudryashov [22] are usually applied after symmetry reductions. Let a system of πth-orderPDEs be given by (5).
Definition 2.21. The Euler-Lagrange operator δ/δuα is

δ

δuα
=

∂

∂uα
+
∑
κ≥1

(−1)κDi1, . . . ,Diκ
∂

∂uα
i1i2...iκ

, (31)
and the Lie- Bäcklund operator in abbreviated form [5] is

X = ξi
∂

∂x i
+ ηα

∂

∂uα
+ . . . . (32)

Remark 2.22. The Lie- Bäcklund operator (32) in its prolonged form is
X = ξi

∂

∂x i
+ ηα

∂

∂uα
+
∑
κ≥1

ζi1...iκ
∂

∂uα
i1i2...iκ

, (33)



Eur. J. Math. Anal. 1 (2021) 138
where

ζαi = Di (W
α) + ξjuαij , . . . ,ζ

α
i1...iκ

= Di1...iκ(W
α) + ξjuαji1...iκ, j = 1, . . . ,n. (34)

and the Lie characteristic function is
Wα = ηα −ξjuαj . (35)

Remark 2.23. The characteristic form of Lie- Bäcklund operator (33) is
X = ξiDi + W

α ∂

∂uα
+ Di1...iκ(W

α)
∂

∂uα
i1i2...iκ

. (36)
Remark 2.24. Noether’s Theorem is applicable to systems from variational problems
Definition 2.25. A function Λα (x i,uα,u(1), . . .) = Λα, is a multiplier of the PDE system given by(5) if it satisfies the condition that [16]

Λα∆α = DiT
i, (37)

where DiT i is a divergence expression.
Definition 2.26. To find the multipliers Λα, one solves the determining equations (38) [3],

δ

δuα
(Λα∆α) = 0. (38)

The technique [9] enables one to construct conserved vectors associated with each Lie pointsymmetry of the PDE system given by (5).
Definition 2.27. The adjoint equations of the system given by (5) are

∆∗α
(
x i,uα,vα, . . . ,u(π),v(π)

)
≡

δ

δuα
(vβ∆β) = 0, (39)

where vα is the new dependent variable.
Definition 2.28. Formal Lagrangian L of the system (5) and its adjoint equations (39) is [9]

L = vα∆α(x i,uα,u(1), . . . ,u(π)). (40)
Theorem 2.29. Every infinitesimal symmetry Xof the system given by (5) leads to conservation
laws [9]

DiT
i
∣∣∣
∆α=0

= 0, (41)
where the conserved vector

T i = ξiL + Wα
[
∂L
∂uα
i

−Dj

(
∂L
∂uα
ij

)
+ DjDk

(
∂L
∂uα
ijk

)
− . . .

]
+

Dj(W
α)

[
∂L
∂uα
ij

−Dk

(
∂L
∂uα
ijk

)
+ . . .

]
+ DjDk(W

α)

[
∂L
∂uα
ijk

− . . .

]
.

(42)



Eur. J. Math. Anal. 1 (2021) 139
3. Main results

We now present our results in this section. An illustrative example with a simple kdV equationcan be found in [6]. The infinitesimal transformations of the Lie group with parameter � are
t̄ = t + ξt(t,x,u,v)�, x̄ = x + ξx (t,x,u,v)�, ū = u + ηu(t,x,u,v)�, v̄ = v + ηv (t,x,u,v)�.(43)

The vector field
X = ξt(t,x,u,v)

∂

∂t
+ ξx (t,x,u,v)

∂

∂x
+ ηu(t,x,u,v)

∂

∂u
+ ηv (t,x,u,v)

∂

∂v
, (44)

is a Lie point symmetry of (3) if
X[3]∆1∣∣∣

∆1=0, ∆2=0
= 0, X[3]∆2∣∣∣

∆1=0, ∆2=0
= 0. (45)

Expanding (45) and and splitting on derivatives of v and u, we have an overdetermined system often PDEs, namely,
ξtu = 0, ξ

t
v = 0, ξ

t
x = 0, ξ

x
u = 0, ξ

x
v = 0, ξ

t
tt = 0, ξ

x
tt = 0, 3ξ

x
x −ξ

t
t = 0,

3ηv + 2ξttv = 0, 3αη
u + 2αξttu − 3ξ

x
t = 0.

(46)
Solving the system (46) yields

ξt = A1 + 3A2t, ξ
x = A2x + αA3t + A4, η

u = −2A2u + A3, ηv = −2A2v, (47)
for arbitrary constants A1,A2,A3,A4. Hence from (47), the infinitesimal symmetries of the coupledKdV Equations (3) is a Lie algebra generated by the vector fields

X1 =
∂

∂t
, X2 =

∂

∂x
, X3 = αt

∂

∂x
+
∂

∂u
, X4 = 3t

∂

∂t
+ x

∂

∂x
− 2u

∂

∂u
− 2v

∂

∂v
. (48)

The set of all infinitesimal symmetries of coupled KdV equations forms a Lie algebra and yield thefollowing commutation relations in Table 1.
[Xi,Xj] X1 X2 X3 X4

X1 0 0 αX2 3X1
X2 0 0 0 X2
X3 -αX2 0 0 -2X3
X4 -3X1 -X2 2X3 0Table 1: A commutator table for the Lie algebra generated by the symmetries of coupled KdVequation.



Eur. J. Math. Anal. 1 (2021) 140
The following Lie groups, for i = 1, 2, 3, 4, are obtained

T�1 : t̄ = t + �1, x̄ = x, ū = u, v̄ = v, (49)
T�2 : t̄ = t, x̄ = x + �2, ū = u, v̄ = v, (50)
T�3 : t̄ = t, x̄ = x + α�3t, ū = u + �3, v̄ = v, (51)
T�4 : t̄ = te

3�4, x̄ = xe�4, ū = ue−2�4, v̄ = ve−2�4. (52)
The symmetries obtained yield the following symmetry reductions.

X1 =
∂

∂t
. (53)

Solving the characteristic equations
dt

1
=

dx

c
=

du

0
=

dv

0
, (54)

associated to the operator X1 gives the invariants
J1 = x, J2 = u, J3 = v. (55)

Hence, we have
u = ϕ(x), v = ψ(x), (56)

for arbitrary functions ϕ and ψ. Substituting the expressions for u and v given by (56) into thesystem (3), we get a system of third order ordinary DEs namely,
α
[
ϕ(x)ϕ′(x) −ψ(x)ψ′(x)

]
+ βϕ′′′(x) = 0, α (ϕ(x)ψ(x))

′
+ βψ′′′(x) = 0. (57)

Integration of the system (57) yields;
α

2

[
ϕ(x)2 −ψ(x)2

]
+ βϕ′′(x) = C1, (58)

α [ϕ(x)ψ(x)] + βψ′′(x) = C2, (59)
for arbitrary constants C1 and C2. If we take

C1 = C2 = 0, (60)
the system (58)-(59) becomes

α

2

[
ϕ(x)2 −ψ(x)2

]
+ βϕ′′(x) = 0, (61)

α [ϕ(x)ψ(x)] + βψ′′(x) = 0. (62)



Eur. J. Math. Anal. 1 (2021) 141
To find more solutions of the system (61)-(62), we determine its Lie point symmetries. Using theLie’s algorithm for computing point symmetries, we see that the Lie point symmetries of (61)-(62)are

X∗1 =
∂

∂x
, X∗2 = x

∂

∂x
− 2ϕ

∂

∂ϕ
− 2ψ

∂

∂ψ
. (63)

Proceeding as above, we see that the symmetry X∗1 yields the trivial solution
u = 0, v = 0. (64)

The second symmetry X∗2 has the characteristic equations
dx

x
=

dϕ

−2ϕ
=

dψ

−2ψ
, (65)

which provides the invariants
J1 = x

2ϕ, J2 = x
2ψ. (66)

Letting
ϕ =

λ

x2
, ψ =

µ

x2
, (67)

substituting the values of ϕ and ψ into (61)-(62) and solving the resulting equations yield:
Case one. Taking

µ = 0 (68)
gives

λ = 0 (69)
or

λ = −
12β

α
. (70)

When
λ = 0, and µ = 0, (71)

we also get the trivial solution (64). One can easily see that if
λ = −

12β

α
, and µ = 0, (72)

then
ϕ = −

12β

αx2
, ψ = 0, (73)

which is a solution of the system (61)-(62). Hence
u1(t,x) = −

12β

αx2
, v1(t,x) = 0, (74)



Eur. J. Math. Anal. 1 (2021) 142
is a solution of the coupled KdV system (3).

Case two. Taking
λ = −

6β

α
(75)

gives
µ = ±

6βi

α
, (76)

with i2 = −1. Consequently,
u2(t,x) = −

6β

αx2
, v2(t,x) =

6iβ

αx2
, (77)

and
u3(t,x) = −

6β

αx2
, v3(t,x) = −

6iβ

αx2
, (78)

are solutions of the coupled KdV system . Hence Lie group analysis has given us three steady-statesolutions for the coupled KdV system under the time translation symmetry X1 = ∂∂t .
X2 =

∂

∂x
. (79)

Solving the characteristic equations
dt

0
=

dx

1
=

du

0
=

dv

0
, (80)

associated to X2 gives the invariants
J1 = t, , J2 = u J3 = v. (81)

Therefore, the group-invariant solution is
u = φ(t), v = h(t), (82)

for arbitrary functions h and φ. Substitution of the solutions from (82) into (3), we get a system offirst order ordinary DEs, namely,
φ′(t) = 0, h′(t) = 0, (83)

which is integrated once with respect to t to yield,
φ(t) = C1, h(t) = C2, (84)

for arbitrary constants C1 and C2. Consequently, the space translation group-invariant solution ofthe system (3) is
u(t,x) = C1, v(t,x) = C2. (85)

X3 = αt
∂

∂x
+
∂

∂u
. (86)



Eur. J. Math. Anal. 1 (2021) 143
Solving the characteristic equations

dt

0
=

dx

αt
=

du

1
=

dv

0
, (87)

associated to Galilean boost gives the invariants
J1 = t, J2 = v, J3 = −u +

x

αt
, t 6= 0. (88)

Thus the invariant solution of (3) is
u =

x

αt
−g(t), v = f (t), t 6= 0, (89)

for arbitrary functions f and g. Substitution of the values of u and v from (89) into the System (3),we get a nonlinear system of coupled first order ordinary DEs, namely,
tg′(t) + g(t) = 0, tf ′(t) + f (t) = 0, (90)

whose solutions are
g(t) =

C1
t
f (t) =

C2
t
, (91)

for arbitrary constants C1 and C2. Hence the Galilean boost group-invariant solution of the system(3) is
u(t,x) =

x + A

αt
, v(t,x) =

C2
t

(92)
where A = −αC1 and t 6= 0.

The scaling

X4 = 3t
∂

∂t
+ x

∂

∂x
− 2u

∂

∂u
− 2v

∂

∂v
(93)

. By solving of the characteristic equations
dt

3t
=

dx

x
= −

du

2u
= −

dv

2v
, (94)

associated to this symmetry, we obtain the invariants
J1 =

x3

t
, J2 = ux

2, J3 = vx
2. (95)

Generally, the group-invariant solution pair is
u(t,x) =

f (λ)

x2
, v(t,x) =

g(λ)

x2
, where λ = x3

t
, (96)

and the functions f and g satisfy the system of third order nonlinear coupled ordinary DEs
2α(g2 − f 2) −λ2f ′ + 3αλ(f f ′ −gg′) + β(−24f + 24λf ′ + 27λ3f ′′′) =0, (97)

−4αf g −λ2g′ + 3αλ(f g)′ + β(−24g + 24λg′ + 27λ3g′′′) =0. (98)
X = X1 + cX2. (99)



Eur. J. Math. Anal. 1 (2021) 144
We consider a symmetry X, which is a linear combination of the time and space translationssymmetries, that is,

X =
∂

∂t
+ c

∂

∂x
, (100)

for a constant c. The invariants associated to this symmetry X are
J1 = x −ct, J2 = u, J3 = v. (101)

Hence, the invariant solution for the symmetry X is
u = f (x −ct), v = g(x −ct), (102)

for arbitrary functions f and g. Substitution of u and v from (102) into the system (3) yields asystem of nonlinear third order ordinary DEs, namely
−cf ′(ξ) + α

{
f (ξ)f ′(ξ) −g(ξ)g′(ξ)

}
+ βf ′′′(ξ) = 0, −cg′(ξ) + α(f (ξ)g(ξ))′ + βg′′′(ξ) = 0,(103)

which on integrating once with respect to ξ yields
−cf +

1

2
α(f 2 −g2) + βf ′′ + C1 = 0, −cg + αf g + βg′′ + C2 = 0, (104)

for arbitrary constants C1 and C2.
Remark 3.1. If we take the constants C1 = C2 = 0, then when the wave velocity c = 0, we canrecover the stationary solutions given in (3).
Remark 3.2. Traveling wave solutions of the system (3) must satisfy the system (104).

Computation of conservation laws for the coupled KdV Equations (3) is done using two meth-ods; the method of multipliers and a theorem due to Ibragimov. We seek local conservation lawmultipliers for the system (3), whose determining equations are
δ

δu

[
Λ1∆1 + Λ

2∆2
]

= 0,
δ

δv

[
Λ1∆1 + Λ

2∆2
]

= 0, (105)
where

δ

δu
=

∂

∂u
−Dt

∂

∂ut
−Dx

∂

∂ux
+ D2x

∂

∂uxx
−D3x

∂

∂uxxx
+ . . . , (106)

δ

δv
=

∂

∂v
−Dt

∂

∂vt
−Dx

∂

∂vx
+ D2x

∂

∂vxx
−D3x

∂

∂vxxx
+ · · · , (107)

are the Euler-Lagrange operators and
Dt =

∂

∂t
+ ut

∂

∂u
+ vt

∂

∂v
+ utx

∂

∂ux
+ vtx

∂

∂vx
+ utt

∂

∂ut
+ vtt

∂

∂vt
+ · · · , (108)

Dx =
∂

∂x
+ ux

∂

∂u
+ vx

∂

∂v
+ uxx

∂

∂ux
+ vxx

∂

∂vx
+ utx

∂

∂ut
+ vtx

∂

∂vt
+ · · · , (109)



Eur. J. Math. Anal. 1 (2021) 145
are total derivatives operators. We look for second order multipliers, that is,

Λn = Λn(t,x,u,ux,uxx,v,vx,vxx ), n = 1, 2. (110)
The determining Equations (105) become

δ

δu

[
Λ1{ut + αuux −αvvx + βuxxx} + Λ2{vt + αuvx + αvux + βvxxx}

]
= 0, (111)

δ

δv

[
Λ1{ut + αuux −αvvx + βuxxx} + Λ2{vt + αuvx + αvux + βvxxx}

]
= 0. (112)

Expanding (111)-(112) and splitting on derivatives of u and v yields an overdetermined system of22 PDEs, namely
Λ1xx = 0, Λ

2
xx = 0 Λ

1
vx = 0, Λ

2
vx = 0, Λ

1
xvxx

= 0, Λ2xvxx = 0, βΛ
1
vv −αΛ

2
vxx

= 0,

βΛ2vv + αΛ
1
vvxx

= 0, Λ1vvxx = 0, Λ
2
vvxx

= 0, Λ1vxxvxx = 0, Λ
2
vxxvxx

= 0, Λ1u + Λ
2
v = 0,

Λ1t + α
(

Λ2xv + Λ
1
xu
)

= 0, Λ2t + α
(

Λ2xu − Λ
1
xv
)

= 0, Λ2u − Λ
1
v = 0, Λ

1
ux

= 0, Λ2ux = 0,

Λ1uxx + Λ
2
vxx

= 0, Λ2uxx − Λ
1
vxx

= 0, Λ2vx = 0 Λ
1
vx

= 0.(113)
Calculations reveal the solution of the system (113) as

Λ1 =
α

2β

(
c3{u2 −v2} + 2c4uv

)
+ (c2t + c5)u + (c1t + c6)v + c3uxx + c4vxx + c7 −

1

α
c2x,

Λ2 =
α

2β

(
c4{u2 −v2}− 2c3uv+

)
+ (c1t + c6)u − (c2t + c5)v + c4uxx −c3vxx + c8 −

1

α
c1x,(114)

for arbitrary constants c1, . . . ,c8.
Remark 3.3. Essentially, the nonlinear coupled system of KdV Equations (3) has eight sets of localconservation law multipliers.

Solving (105)„ we obtain conserved vectors corresponding to each set of multipliers as shownbelow.
(i) The multiplier (

Λ11, Λ
2
1

)
=
(
tv,tu −

x

α

)
, (115)

has the conserved vectors
Tt1 = tuv −

xv

α
, Tx1 = β

[
t{vuxx + uvxx −vxux} +

1

α
{vx −xvxx}

]
+ α

[
t

(
u2v −

v3

3

)] (116)
−xuv. (117)

(ii) The multiplier (
Λ12, Λ

2
2

)
=
(
tu −

x

α
,−tv

)
, (118)



Eur. J. Math. Anal. 1 (2021) 146
has the conserved vectors

Tt2 =
t

2
{u2 −v2}−

xu

α
, Tx2 = β

[
t

(
uuxx −vvxx +

1

2
{v2x −u

2
x}
)

+
1

α
{ux −xuxx}

]
+

αt

[
u3

3
−uv2

]
+
x

2
{v2 −u2}.

(119)
(iii) The multiplier (

Λ13, Λ
2
3

)
=

(
α

2β
{u2 −v2} + uxx,−{

αuv

β
+ vxx}

)
, (120)

has the conserved vectors
Tt3 =

α

2β

(
u3

3
−uv2

)
, Tx3 =

α

2

[
(u2 −v2)uxx −v2vxx

]
−αuvvxx + (121)

β

2

[
u2xx −v

2
xx

]
+ utux −vtvx +

α2

4β

[
1

2
{u4 + v4}− 3u2v2

]
. (122)

(iv) The multiplier (
Λ14, Λ

2
4

)
=

(
{
αuv

β
+ vxx},

α[u2 −v2]
2β

+ uxx

)
, (123)

has the conserved vectors
Tt4 =

α

2β

(
u2v −

v3

3

)
, (124)

Tx4 =
α2

2β

[
(u3v −uv3)

]
+ vtux + utvx +

α

2
(u2 −v2)vxx + {αuv + βvxx}uxx. (125)

(v) The multiplier (
Λ15, Λ

2
5

)
= (u,−v) , (126)

has the conserved vectors
Tt5 =

1

2
{u2 −v2}, Tx5 = β

(
uuxx −vvxx +

v2x −u2x
2

)
+ α

(
u3

3
−uv2

)
. (127)

(vi) The multiplier (
Λ16, Λ

2
6

)
= (v,u) , (128)

has the conserved vectors
Tt6 = uv, T

x
6 = β (vuxx + uvxx −uxvx ) + α

(
u2v −

v3

3

)
. (129)

(vii) The multiplier (
Λ17, Λ

2
7

)
= (1, 0) , (130)

has the conserved vectors
Tt7 = u, T

x
7 =

α

2
{u2 −v2} + βuxx. (131)



Eur. J. Math. Anal. 1 (2021) 147
(viii) The multiplier has (

Λ18, Λ
2
8

)
= (0, 1) , (132)

the conserved vectors
Tt8 = v, T

x
8 = αuv + βvxx. (133)

Remark 3.4. It can be verified that
DtT

t
i + DxT

x
i

∣∣∣
∆1=0, ∆2=0

= 0, (134)
for i = 1, . . . , 8.
Remark 3.5. The expressions in (134) are eight conservation laws for the coupled KdV system (3).
Remark 3.6. The presence of multipliers(

Λ17, Λ
2
7

)
= (1, 0) ,

(
Λ18, Λ

2
8

)
= (0, 1) (135)

manifest that the coupled KdV equations are themselves conservation laws.
At this point, we derive conserved vectors for coupled KdV equations (3) by a new theorem due toIbragimov. The adjoint equations for the nonlinear system coupled KdV Equations (3) are

∆∗1 ≡ ft + α ufx + αvgx + βfxxx = 0, ∆
∗
2gt −αvfx + αugx + βgxxx = 0. (136)

The formal Lagrangian L for the nonlinear coupled system of the KdV Equations (3) and its adjointEquations (136) is given by
L = f{ut + αuux −αvvx + βuxxx} + g{vt + αuvx + αvux + βvxxx}, (137)

where f and g are new variables. We shall use the Lie point symmetries of the system (3) ,namely
X1 = ∂t, X2 = ∂x, X3 = αt∂x + ∂u, X4 = 3t∂t + x∂x − 2u∂u − 2v∂v, (138)

to derive conserved vectors corresponding to each symmetry below.
Case (i) The symmetry X1 = ∂∂t , yields Lie characteristic functions given by

W 11 = −ut, W
2
1 = −vt. (139)

Hence by Ibragimov’s theorem [9], the associated conserved vector is given by
Tt1 =α [f{uux −vvx} + g{vux + uvx}] + β{f uxxx + gvxxx},

Tx1 =α [f{−uut + vvt}−g{vut + uvt}]

+ β{fxutx + gxvtx −utfxx −vtgxx − f utxx −gvtxx}.

(140)



Eur. J. Math. Anal. 1 (2021) 148
Case (ii) The symmetry X2 = ∂∂x , yields Lie characteristic functions

W 12 = −ux, W
2
2 = −vx. (141)

Therefore by Ibragimov’s theorem [9], the associated conserved vector is
Tt2 = −uxf −vxg, T

x
2 = f ut + gvt + β{−uxfxx −vxgxx + fxuxx + gxvxx}. (142)

Case (iii) The symmetry
X3 = αt

∂

∂x
+
∂

∂u
(143)

yields Lie characteristic functions given by
W 13 = 1 −αtux, W

2
3 = −αtvx. (144)

Hence by Ibragimov’s theorem [9], the associated conserved vector is given by
Tt3 = f −αt{uxf + vxg} ,

Tx3 = α

[
f u + gv + t{utf + vtg} + βt{

fxx
αt
−uxfxx −vxgxx + fxuxx + gxvxx}

]
.

(145)
Case (iv) The symmetry

X4 = 3t
∂

∂t
+ x

∂

∂x
− 2u

∂

∂u
− 2v

∂

∂v
(146)

yields the Lie characteristic functions
W 14 = −2u − 3tut −xux, W

2
4 = −2v − 3tvt −xvx. (147)

Consequently by Ibragimov’s theorem [9], the corresponding conserved vector is given by
Tt4 = α [3t{f uux − f vvx + guvx + gvux}] + β [3t{f uxxx + gvxxx}]

− 2{f u + gv}−x{f ux + gvx},

Tx4 = x{f ut + gvt} + β
[

3
(
fxux + gxvx + t{fxutx + gxvtx}

)]
−α

[
2
(
f{u2 −v2} + 2guv

)
+ 3t

(
f{uut −vvt} + g{vut + uvt}

)]
−β [x{uxfxx + vxgxx − fxuxx −gxvxx} + 2{ufxx + vgxx}]

−β [3t{fxxut + gxxvt + f utxx + gvtxx} + 4{f uxx + gvxx}] .

(148)

Remark 3.7. The appearance of arbitrary functions f (t,x) and g(t,x) in the conserved vectorsproves the existence of infinite conservation laws for coupled KdV system obtained by Ibagimov’smethod.



Eur. J. Math. Anal. 1 (2021) 149
4. Conclusion

In this paper, Lie group analysis was employed in studying a nonlinear coupled kdV system.A four-dimensional Lie algebra of symmetries was found for the nonlinear coupled system KdVequations. This was spanned by space and time translations, Galilean boost and scaling symmetrieswhere the scaling symmetry acts on four variables. Associated to each symmetry, we obtainedsymmetry reductions that gave six nontrivial solutions for the coupled system. All the group-invariant solutions describe the various states of the system. The obtained solutions can be usedas a benchmark against numerical simulations. Lastly, we constructed infinite conservation laws ofa nonlinear coupled KdV system by using multipliers and a theorem proposed by Nail Ibragimov.
Acknowledgement

The first author acknowledges the financial support of AIMS-South Africa and MasterCard Foun-dation. The authors are also grateful to the referees for their careful reading of the manuscript andvaluable comments.
References

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https://doi.org/10.1186/1687-1847-2014-135

	1. Introduction
	2. Preliminaries
	3. Main results
	4. Conclusion
	Acknowledgement
	References

