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Group Analysis of Equal-Width Equation

Joseph Owuor Owino
Faculty of Applied Sciences and Technology, School of Mathematics and Actuarial Science,
Department of Pure and Applied Mathematics, The Technical University of Kenya, Kenya

Correspondence: josephowuorowino@gmail.com

Abstract. We study a third-order nonlinear equal width equation, which has been used for simulationof a one-dimensional wave propagation in a non-linear medium with dispersion process, by symmetryanalysis. First, Lie point symmetries are obtained and used to reduce reduce the equal width equationthereby constructing exact solutions. Traveling waves are constructed using of a linear combination ofspace and time translation symmetries. We have used the multiplier technique to compute conservationlaws.

1. Introduction
The Equal width Equation [1] is given by,

∆ ≡ ut + αuux + βutxx = 0, (1.1)
where t and x represents time and spatial independent variables ; α and β are the nonlinearityand the dispersion parameters respectively. Equation (1.1) was first studied by Morrison [2] anddescribes nonlinear dispersive waves, particularly those generated in a shallow water channel.Several techniques have been employed to compute solutions of Equation (1.1). A case in point,is in [3], where a Petrov-Galerkin approach applied quadratic B-spline finite element. In [4], theresearchers applied least-squares approach in the construction of numerical solutions. We presenta group analysis approach in this paper by first giving the preliminaries.

2. Preliminaries
This section is a prelude to the sequel.
Received: 3 Nov 2022.
Key words and phrases. equal width equation; Lie group analysis; group-invariant solutions; stationary solutions;symmetry reductions; solitons; traveling waves. 1

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https://doi.org/10.28924/ada/ma.3.13
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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 2
Local Lie groups. [5] We will consider the transformations

T� : x̄
i = ϕi (x i,uα,�), ūα = ψα(x i,uα,�), (2.1)

in the Euclidean space Rn of x = x i independent variables and Rm of u = uα dependent variables.The continuous parameter � ranges from a neighbourhood N ′ ⊂ N ⊂ R of � = 0 for ϕi and ψαdifferentiable and analytic in the parameter �.
Definition 2.1. Let G be a set of transformations in (2.1) . Then G is a local Lie group if:

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

Remark 2.2. The condition (i ) is sufficient for associativity of G.
Prolongations. Consider the system,

∆α
(
x i,uα,u(1), . . . ,u(π)

)
= ∆α = 0, (2.2)

where uα are dependent variables with partial derivatives u(1) = {uαi },
u(2) = {uαij}, . . . ,u(π) = {u

α
i1...iπ
}, of the first, second, . . . , up to the πth-orders. We shall denoteby
Di =

∂

∂x i
+ uαi

∂

∂uα
+ uαij

∂

∂uα
j

+ . . . , (2.3)
the total differentiation operator with respect to the variables x i and δj

i
, the Kronecker delta. Then

Di (x
j) = δ

j
i
, ′, uαi = Di (u

α), uαij = Dj(Di (u
α)), . . . , (2.4)

where uαi defined in (2.4) are differential variables [6].(1) Prolonged groups Let G given by
x̄ i = ϕi (x i,uα,�), ϕi

∣∣∣
�=0

= x i, ūα = ψα(x i,uα,�), ψα
∣∣∣
�=0

= uα, (2.5)
where ∣∣∣

�=0
means evaluated on � = 0.

Definition 2.3. The construction of G in (2.5) is equivalent to the computation of infinitesimaltransformations
x̄ i ≈ x i + ξi (x i,uα)�, ϕi

∣∣∣
�=0

= x i,

ūα ≈ uα + ηα(x i,uα)�, ψα
∣∣∣
�=0

= uα,
(2.6)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 3
obtained from (2.1) by a Taylor series expansion of ϕi (x i,uα,�) and ψi (x i,uα,�) in � about
� = 0 and 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
. (2.7)

Remark 2.4. By using the symbol of infinitesimal transformations, X, (2.6) becomes
x̄ i ≈ (1 + X)x i, ūα ≈ (1 + X)uα, (2.8)

where
X = ξi (x i,uα)

∂

∂x i
+ ηα(x i,uα)

∂

∂uα
, (2.9)

is the generator G in (2.5).
Remark 2.5. The change of variables formula

Di = Di (ϕ
j)D̄j, (2.10)

is employed to construct transformed derivatives from (2.1). The D̄j is total differentiation
x̄ i . As a result

ūαi = D̄i (ū
α), ūαij = D̄j(ū

α
i ) = D̄i (ū

α
j ). (2.11)

If we apply the change of variable formula given in (2.10) on G given by (2.5), we get
Di (ψ

α) = Di (ϕ
j), D̄j(ū

α) = ūαj Di (ϕ
j). (2.12)

If we expand (2.12), we obtain(
∂ϕj

∂x i
+ u

β
i

∂ϕj

∂uβ

)
ū
β
j

=
∂ψα

∂x i
+ u

β
i

∂ψα

∂uβ
. (2.13)

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

α(x i,uα,u(1),�), Φ
α
∣∣∣
�=0

= uαi . (2.14)
Definition 2.6. The transformations in (2.5) and (2.14) give 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),�). (2.15)

Remark 2.8. In terms of infinitesimal transformations, G[1] is given by (2.6) and (2.15).
(2) Prolonged generators

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 4
Definition 2.9. By the relation (2.12) on G[1] from 2.6, we obtain [7]

Di (x
j + ξj�)(uαj + ζ

α
j �) = Di (u

α + ηα�), which gives (2.16)
uαi + ζ

α
j � + u

α
j �Diξ

j = uαi + Diη
α�, (2.17)

and thus
ζαi =Di (η

α) −uαj Di (ξ
j), (2.18)

is the first prolongation formula.
Remark 2.10. Analogously, one constructs higher order prolongations [7],
ζαij = Dj(ζ

α
i ) −u

α
iκDj(ξ

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

) −uαi1,i2,...,iκ−1jDiκ(ξ
j). (2.19)

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

∂

∂uα
i

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

∂ζα
i1,...,iκ

, κ ≥ 1, (2.20)
for the group generator X in (2.9).

Group invariants.

Definition 2.12. A function Γ(x i,uα) is said to be an invariant of G of in (2.1) if
Γ(x̄ i, ūα) = Γ(x i,uα). (2.21)

Theorem 2.13. A function Γ(x i,uα) is an invariant of the group G given by (2.1) if and only if it
solves the following first-order linear PDE: [8]

XΓ = ξi (x i,uα)
∂Γ

∂x i
+ ηα(x i,uα)

∂Γ

∂uα
= 0. (2.22)

From Theorem (2.13), we have the following result.
Theorem 2.14. The Lie group G in (2.1) [9] has precisely n−1 functionally independent invariants
and 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, (2.23)
of the characteristic equations for (2.22):

dx i

ξi (x i,uα)
=

duα

ηα(x i,uα)
. (2.24)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 5
Symmetry groups.

Definition 2.15. We define the vector field X (2.9) as a Lie point symmetry of (2.2) if the determiningequations
X[π]∆α

∣∣∣
∆α=0

= 0, α = 1, . . . ,m, π ≥ 1, (2.25)
are satisfied for the π-th prolongation of X, namely X[π].
Definition 2.16. The Lie group G is a symmetry group of (2.2) if (2.2) is form-invariant, that is

∆α
(
x̄ i, ūα, ū(1), . . . , ū(π)

)
= 0. (2.26)

Theorem 2.17. The Lie group G (2.1) can be constructed from the infinitesimal transformations in
(2.5) 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. (2.27)
Lie algebras.

Definition 2.18. A vector space Vr of operators [8] X (2.9) is a Lie algebra if for any Xi,Xj ∈Vr,
[Xi,Xj] = XiXj −XjXi, (2.28)

is in Vr for all i, j = 1, . . . , r .
Remark 2.19. The commutator is bilinear, skew symmetric and admits to the Jacobi identity [5].
Theorem 2.20. The set of solutions of (2.25) forms a Lie algebra [10].

Exact solutions. The methods of (G’/G)-expansion method [7], Extended Jacobi elliptic functionexpansion [9] and Kudryashov [11] are usually applied after symmetry reductions.
Conservation laws. [11]
Fundamental operators.

Definition 2.21. The Euler-Lagrange operator δ
δuα

is
δ

δuα
=

∂

∂uα
+
∑
κ≥1

(−1)κDi1, . . . ,Diκ
∂

∂uα
i1i2...iκ

, (2.29)
and the Lie- Bäcklund operator in abbreviated form [11] is

X = ξi
∂

∂x i
+ ηα

∂

∂uα
+ . . . . (2.30)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 6
Remark 2.22. The Lie- Bäcklund operator (2.30) in its prolonged form is

X = ξi
∂

∂x i
+ ηα

∂

∂uα
+
∑
κ≥1

ζi1...iκ
∂

∂uα
i1i2...iκ

, (2.31)
for

ζαi = Di (W
α) + ξjuαij , . . . ,ζ

α
i1...iκ

= Di1...iκ(W
α) + ξjuαji1...iκ, j = 1, . . . ,n. (2.32)and the Lie characteristic function

Wα = ηα −ξjuαj . (2.33)
Remark 2.23. The characteristic form of Lie- Bäcklund operator (2.31) is

X = ξiDi + W
α ∂

∂uα
+ Di1...iκ(W

α)
∂

∂uα
i1i2...iκ

. (2.34)
The method of multipliers.

Definition 2.24. A function Λα (x i,uα,u(1), . . .) = Λα, is a multiplier of (2.2) if [7]
Λα∆α = DiT

i, (2.35)
where DiT i is a divergence expression.
Definition 2.25. To find the multipliers Λα, one solves the determining equations (2.36) [10],

δ

δuα
(Λα∆α) = 0. (2.36)

Ibragimov’s conservation theorem . The technique [5] enables one to construct conserved vectorsassociated with each Lie point symmetry of (2.2).
Definition 2.26. The adjoint equations of (2.2) are

∆∗α
(
x i,uα,vα, . . . ,u(π),v(π)

)
≡

δ

δuα
(vβ∆β) = 0, (2.37)

for a new dependent variable vα.
Definition 2.27. The Formal Lagrangian L of (2.2) and its adjoint equations (2.37) is [8]

L = vα∆α(x i,uα,u(1), . . . ,u(π)). (2.38)
Theorem 2.28. Every infinitesimal symmetry Xof (2.2) leads to conservation laws [6]

DiT
i
∣∣∣
∆α=0

= 0, (2.39)
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

− . . .

]
.

(2.40)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 7
3. Main results

3.1. Lie point symmetries of equal width equation(1.1). We start first by computing Lie pointsymmetries of the equal width Equation (1.1), which admits the one-parameter Lie group of trans-formations with infinitesimal generator
X = τ(t,x,u)

∂

∂t
+ ξ(t,x,u)

∂

∂x
+ η(t,x,u)

∂

∂u
(3.1)

if and only if
X[3]∆

∣∣∣∣
∆=0

= 0. (3.2)
where

X[3] = X + ζ1 ∂
∂ut

+ ζ2
∂

∂ux
+ ζ122

∂

∂utxx
, (3.3)

is the third prolongation of the Lie point symmetry X as defined in (2.20) and
ζ1 = Dt(η) −utDt(τ) −uxDt(ξ), (3.4)
ζ12 = Dx (ζ1) −uttDx (τ) −utxDx (ξ), (3.5)
ζ2 = Dx (η) −utDx (τ) −uxDx (ξ), (3.6)

ζ122 = Dx (ζ12) −uttxDx (τ) −utxxDx (ξ), (3.7)
as defined in (2.19), and

Dt =
∂

∂t
+ ut

∂

∂u
+ utx

∂

∂ux
+ utt

∂

∂ut
+ · · · , (3.8)

Dx =
∂

∂x
+ ux

∂

∂u
+ uxx

∂

∂ux
+ utx

∂

∂ut
+ . . . . (3.9)

Applying the definitions of Dt and Dx given in (3.8) and (3.9), we obtain the expanded form of the
ζs as

ζ1 = ηt + ut(ηu −τt) + ux (−ξt) + utux (−ξu) + u2t (−τu),

ζ12 = ηtx + ux (ηtu −ξtx ) + utx (ηu −τt −ξx ) + ut(ηxu −τtx ) + utux (ηuu −ξxu −τtu)

+ ututx (−2τu) + u2t (−τxu) + u
2
t ux (−τuu) + uxx (−ξt) + u

2
x (−ξtu) + uxutx (−2ξu)

+ utu
2
x (−ξuu) + utuxx (−ξu) + utt(−τx ) + uxutt(−τu)

ζ2 = ηx + ux (ηu −ξx ) + ut(−τx ) + utux (−τu) + u2x (−ξu),

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 8
ζ122 = ηtxx + ux (2ηtxu −ξtxx ) + uxx (ηtu − 2ξtx ) + u2x (ηtuu − 2ξtxu) + utxx (ηu −τt − 2ξx )

+ utux (2ηxuu −ξxxu − 2τtxu) + uxutx (2ηuu − 4ξxu −τtu) + utx (2ηxu − 2τtx −ξxx )

+ ut(ηxxu −τtxx ) + utuxx (ηuu − 2ξxu −τtu) + utu2x (ηuuu − 2ξxuu −τtuu) + u
2
tx (−2τu)

ututxx (−2τu) + ututx (−4τxu) + u2t (−τxxu) + uxu
2
t (−2τxuu) + utuxutx (−4τuu),

+ u2xu
2
t (−τuuu) + uxxx (−ξt) + u

2
t uxx (−τuu) + uxuxx (−4ξtu) + u

3
x (−ξtuu) + uxxutx (−3ξu)

uxutxx (−2ξu) + u2xutx (−3ξuu) + uxutuxx (−3ξuu) + utu
3
x (−ξuuu) + utuxxx (−ξu)

+ uttx (−2τx ) + utt(−τxx ) + uxutt(−2τxu) + u2xutt(−τuu) + uxxutt(−τu) + uxuttx (−2τu −ξu)

(3.10)
Now from Equation (3.2), we have

ζ1 + αηux + αζ2u + βζ122
∣∣
utxx =−

ut
β
−α
β
uux

= 0, (3.11)
If we substitute for ζ1, ζ2 and ζ122 in the determining Equation (3.11), we obtain the following;

ηt + ut(ηu −τt) + ux (−ξt) + utux (−ξu) + u2t (−τu) + αηux

+ αu{ηx + ux (ηu −ξx ) + ut(−τx ) + utux (−τu) + u2x (−ξu)}

+ β

{
ηtxx + ux (2ηtxu −ξtxx ) + uxx (ηtu − 2ξtx ) + u2x (ηtuu − 2ξtxu) + utxx (ηu −τt − 2ξx )

+ utux (2ηxuu −ξxxu − 2τtxu) + uxutx (2ηuu − 4ξxu −τtu) + utx (2ηxu − 2τtx −ξxx )

+ ut(ηxxu −τtxx ) + utuxx (ηuu − 2ξxu −τtu) + utu2x (ηuuu − 2ξxuu −τtuu) + u
2
tx (−2τu)

ututxx (−2τu) + ututx (−4τxu) + u2t (−τxxu) + uxu
2
t (−2τxuu) + utuxutx (−4τuu),

+ u2xu
2
t (−τuuu) + uxxx (−ξt) + u

2
t uxx (−τuu) + uxuxx (−4ξtu) + u

3
x (−ξtuu) + uxxutx (−3ξu)

uxutxx (−2ξu) + u2xutx (−3ξuu) + uxutuxx (−3ξuu) + utu
3
x (−ξuuu) + utuxxx (−ξu)

+ uttx (−2τx ) + utt(−τxx ) + uxutt(−2τxu) + u2xutt(−τuu) + uxxutt(−τu) + uxuttx (−2τu −ξu)

}
∣∣∣∣∣
u
txx=−ut

β
−α
β
uux

= 0

(3.12)
Now replacing utxx by −utβ − αβuux in Equation (3.12), we have

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 9

ηt + ut(ηu −τt) + ux (−ξt) + utux (−ξu) + u2t (−τu) + αηux

+ αu{ηx + ux (ηu −ξx ) + ut(−τx ) + utux (−τu) + u2x (−ξu)}

+ β

{
ηtxx + ux (2ηtxu −ξtxx ) + uxx (ηtu − 2ξtx ) + u2x (ηtuu − 2ξtxu)+[

−
ut
β
−
α

β
uux

]
(ηu −τt − 2ξx )

+ utux (2ηxuu −ξxxu − 2τtxu) + uxutx (2ηuu − 4ξxu −τtu) + utx (2ηxu − 2τtx −ξxx )

+ ut(ηxxu −τtxx ) + utuxx (ηuu − 2ξxu −τtu) + utu2x (ηuuu − 2ξxuu −τtuu) + u
2
tx (−2τu)

+ ut

[
−
ut
β
−
α

β
uux

]
(−2τu) + ututx (−4τxu) + u2t (−τxxu) + uxu

2
t (−2τxuu) + utuxutx (−4τuu),

+ u2xu
2
t (−τuuu) + uxxx (−ξt) + u

2
t uxx (−τuu) + uxuxx (−4ξtu) + u

3
x (−ξtuu) + uxxutx (−3ξu)

ux

[
−
ut
β
−
α

β
uux

]
(−2ξu) + u2xutx (−3ξuu) + uxutuxx (−3ξuu) + utu

3
x (−ξuuu) + utuxxx (−ξu)

+ uttx (−2τx ) + utt(−τxx ) + uxutt(−2τxu)

+ u2xutt(−τuu) + uxxutt(−τu) + uxuttx (−2τu −ξu)

}
= 0

(3.13)
which can be written as

ηt + αuηx + βηtxx + ut(βηxxu −βτtxx + 2ξx −αuτx )

+ ux
(

2βηtxu −βξtxx −ξt + αuξx + αuτt + αη
)

+ utux (2ξu + 2βηxuu −βξxxu − 2βτtxu + αuτu) + u2t (τu −βτxxu)+

u2x (2αuξu + βηtuu − 2βξtxu) + β

{
uxx (ηtu − 2ξtx )

+ uxutx (2ηuu − 4ξxu −τtu) + utx (2ηxu − 2τtx −ξxx )

+ utuxx (ηuu − 2ξxu −τtu) + utu2x (ηuuu − 2ξxuu −τtuu) + u
2
tx (−2τu)

+ ututx (−4τxu) + uxu2t (−2τxuu) + utuxutx (−4τuu),

+ u2xu
2
t (−τuuu) + uxxx (−ξt) + u

2
t uxx (−τuu) + uxuxx (−3ξtu) + u

3
x (−ξtuu) + uxxutx (−3ξu)

+ u2xutx (−3ξuu) + uxutuxx (−3ξuu) + utu
3
x (−ξuuu) + utuxxx (−ξu)

+ uttx (−2τx ) + utt(−τxx ) + uxutt(−2τxu) + u2xutt(−τuu) + uxxutt(−τu) + uxuttx (−2τu)

}
= 0

(3.14)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 10
Since the functions τ,ξ and η depend only on t,x and u and are independent of the derivativesof u, we can then split the above equation on the derivatives of u and obtain

τx = τu = ξu = ξt = ξx = ηuu = ηtu =0, (3.15)
η + uτt =0, (3.16)

ηt + αuηx + βηtxx =0 (3.17)
From Equation (3.15), we find that

τ =τ(t), (3.18)
ξ =C1, (3.19)
η =A(x)u + B(t,x). (3.20)

Now substituting η into Equation (3.17) yields
Bt(t,x) + αu

[
A(x)xu + Bx (t,x)

]
+ βBtxx (t,x) = 0. (3.21)

Separation of (3.21) on powers of u gives the following equations
u2 :A(x)x = 0, (3.22)
u :Bx (t,x) = 0, (3.23)
u0 :Bt(t,x) + βBtxx (t,x) = 0. (3.24)

Integration of Equations (3.22) and (3.23) with respect to x gives that
A(x) = C2 (3.25)

B(t,x) = B(t). (3.26)
Now use Equation (3.26) in Equation (3.24) to obtain Btxx (t,x) = 0 and as a result

Bt(t,x) = 0. (3.27)
Integrating Equation (3.27) with respect to t gives

B(t,x) = C3. (3.28)
If we substitute η = C2u + C3 into Equation (3.16), we have

C2u + C3 + τtu = 0. (3.29)
From Equation (3.29), if we obtain

τ(t) = −C2t −C3
t

u
+ C4. (3.30)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 11
and finally;

τ = −C2t −C3
t

u
+ C4, (3.31)

ξ =C1 (3.32)
η =C2u + C3. (3.33)

We have obtained a four-dimensional Lie algebra of symmetries spanned by
X1 =

∂

∂x
, (3.34)

X2 =u
∂

∂u
− t

∂

∂t
, (3.35)

X3 =
∂

∂u
−
t

u

∂

∂t
, (3.36)

X4 =
∂

∂t
. (3.37)

3.2. Commutator Table for Symmetries. We evaluate the commutation relations for the symmetrygenerators. By definition of Lie bracket [9], for example, we have that
[X1,X4] = X1X4 −X4X1 =

(
∂

∂x

∂

∂t

)
−
(
∂

∂t

∂

∂x

)
= 0. (3.38)

Remark 3.1. The remaining commutation relations are obtained analogously. We present all
commutation relations in table (1) below.

[Xi,Xj] X1 X2 X3 X4

X1 0 0 0 0
X2 0 0 -X3 X4
X3 0 −X3 0 1uX4
X4 0 -X4 - 1uX4 0

Table 1. A commutator table for Lie algebra of equal width equation.

3.3. Group Transformations. The corresponding one-parameter group of transformations can bedetermined by solving the Lie equations [6]. Let T�i be the group of transformations for each
Xi, i = 1, 2, 3, 4. We display how to obtain T�i from Xi by finding one-parameter group for theinfinitesimal generator X1, namely,

X1 =
∂

∂x
. (3.39)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 12
In particular, we have the Lie equations

dt̄

d�
=0, t̄

∣∣∣
�=0

= t,

dx̄

d�
=1, x̄

∣∣∣
�=0

= x,

dū

d�
=0, ū

∣∣∣
�=0

= u.

(3.40)
Solving the system (3.40) one obtains,

t̄ = t, x̄ = x + �, ū = u, (3.41)
and hence the one-parameter group T�4 corresponding to the operator X1 is

T�1 : (t̄, x̄, ū) = (t,x + �1,u). (3.42)
All the five one-parameter groups are presented below :

T�1 : (t̄, x̄, ū) = (t,x + �1,u)

T�2 : (t̄, x̄, ū) = (te
−�2,x,ue�2 )

T�3 : (t̄, x̄, ū) = (te
−�3

u ,x,u + �3)

T�4 : (t̄, x̄, ū) = (t + �4,x,u).

(3.43)

3.4. Symmetry transformations. We now show how the symmetries we have obtained can be usedto transform special exact solutions of the equal width equation into new solutions. The Lie groupanalysis vouches for fundamental ways of e constructing exact solutions of PDEs, that is, grouptransformations of known solutions and construction of group-invariant solutions. We will illustratethese methods with examples. If ū = g(t̄, x̄) is a solution of equation (1.1)
φ(t,x,u,�) = g(f1(t,x,u,�), f2(t,x,u,�)), (3.44)

is also a solution. The one parameter groups dictate to the following generated solutions:
T�1 : u =g(t,x + �1)

T�2 : u =g(te
−�2,x)e−�2,

T�3 : u =g(te
−�3

u ,x) − �3,

T�4 : u =g(t + �4,x).

(3.45)

3.5. Construction of Group-Invariant Solutions. Now we compute the group invariant solutions ofBurger’s equation.
(i) X1 = ∂∂xThe associated Lagrangian equations

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 13
dt

0
=

dx

1
=

du

0
, (3.46)

yield two invariants, J1 = t and J2 = u. Thus using J2 = Φ(J1), we have
u(t,x) = Φ(t). (3.47)

The derivatives are given by :
ut =Φ

′(t),

ux =0,

utxx =0.

If we substitute these derivatives into Equation (1.1) , we obtain the first order ordinarydifferential equation
Φ′(t) = 0,

whose space invariant solution is
Φ(t) = C1, (3.48)

and the group-invariant solution associated to the X1 is
u(t,x) = C1.

(ii) X2 = u ∂∂u − t ∂∂t The Lagrangian equations associated to this symmetry are
dt

−t
=

dx

0
=

du

u
. (3.49)

This gives the constants J1 = x and J2 = tu , giving the solution
u =

f (x)

t
. (3.50)

We obtain the derivatives as follows:
ut = −

f (x)

t2
, (3.51)

ux =
f ′(x)

t
(3.52)

utxx = −
f ′′(x)

t2
(3.53)

If we substitute the above derivatives in Equation (1.1), we obtain the second order ordinarydifferential equation
f (x) −αf (x)f ′(x) + βf ′′(x) = 0. (3.54)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 14
Hence the group invariant solution to Equation to (1.1) will be given by

u(t,x) =
f (x)

t
, (3.55)

where f satisfies Equation (3.54).(iii) X3 = ∂∂u − tu ∂∂tThe Lagrangian system associated with the operator X3 is
dt

− t
u

=
dx

0
=

du

1
, (3.56)

whose invariants are J1 = x and J2 = tu. So, u = g(x)t is the group-invariant solution.(iv) X4 = ∂∂tCharacteristic equations associated to the operator X4 are
dt

1
=

dx

0
=

du

0
, (3.57)

yieldsJ1 = x and J2 = u. As a result, the group-invariant solution of (1.1) for this case is
J2 = φ(J1), for some φ an arbitrary function. That is,

u(t,x) = φ(x). (3.58)
The derivatives of given function are

ut = 0, (3.59)
ux = φ

′(x), (3.60)
utxx = 0. (3.61)

Substitution of the value of φ(x) into Equation (1.1) yields a first order nonlinear ordinarydifferential equation
φ(x)φ′(x) = 0. (3.62)

From Equation (3.62), either φ(x) = 0 or φ′(x) = 0. The case φ(x) = 0 =⇒ φ′(x) = 0,and the equation is satisfied. The case φ(x) 6= 0 implies that φ′(x) = 0 and by integration,
φ(x) = C1, hence the group invariant solution is given by

u(t,x) = C2. (3.63)
3.6. Soliton. We obtain a traveling wave solution of the equal width Equation(1.1) by consideringa linear combination of the symmetries X1 and X4, namely, [7]

X = cX1 + X4 = c
∂

∂x
+
∂

∂t
, for some constant c. (3.64)

The characteristic equations are
dt

1
=

dx

c
=

du

0
(3.65)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 15
We get two invariants, J1 = x −ct and J2 = u. So the group-invariant solution is

u(t,x) = Q(x −ct), (3.66)
for some arbitrary function ϕ and c the velocity of the wave.Substitution of u into (1.1) yields a second order ordinary differential equation

cQ′ −αQQ′ + βcQ′′′ = 0, (3.67)
which can be integrated with respect to Q to give

cQ−α
Q2

2
+ βcQ′ = 0, (3.68)

where we have used 0 as a constant of integration. Equation (3.68) can be rearranged and variablesseparated to have
dξ

2βc
=

dQ

αQ2 − 2cQ
, ξ = x −ct. (3.69)

The right hand side can be resolved into partial fractions to obtain
ξ

2βc
=

1

2c

∫ [
α

αQ− 2c
−

1

Q

]
dQ =

1

2c
ln

∣∣∣∣∣αQ− 2cQ
∣∣∣∣∣ + ln |C3|, (3.70)

where C3 is a constant of integration. After rewriting, we have
Q(x −ct) =

2cC3

αC3 −e
x−ct
β

. (3.71)
Finally, the soliton solutions are given by

u(t,x) =
2cC3

αC3 −e
x−ct
β

. (3.72)
4. Conservation laws of equation (1.1)

We will employ multipliers in the construction of conservation laws.
4.1. The multipliers. We make use of the Euler-Lagrange operator defined as defined in [6] to lookfor a zeroth order multiplier Λ = Λ(t,x,u). The resulting determining equation for computing Λ is

δ

δu
[Λ{ut + αuux + βutxx}] = 0. (4.1)

where
δ

δu
=

∂

∂u
−Dt

∂

∂ut
−Dx

∂

∂ux
−DtD2x

∂

∂utxx
+ . . . (4.2)

Expansion of Equation (4.1) yields
Λu(ut + αuux + βutxx ) + αux Λ −Dt(Λ) −αDx (uΛ) −βDtD2x (Λ) = 0. (4.3)

Invoking the total derivatives

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 16

Dt =
∂

∂t
+ ut

∂

∂u
+ utx

∂

∂ux
+ utt

∂

∂ut
+ · · · , (4.4)

Dx =
∂

∂x
+ ux

∂

∂u
+ uxx

∂

∂ux
+ utx

∂

∂ut
+ . . . . (4.5)

on Equation (4.3) produces
Λt + αuΛx + βΛtxx + 2β(Λtxu)ux + β(Λtu)uxx + β(Λtuu)u

2
x + 2β(Λxu)utx

+ 2β(Λuu)uxutx + β(Λxxu)ut + 2β(Λxuu)uxux + β(Λuu)utuxx + β(Λuuu)utu
2
x = 0

(4.6)
Splitting Equation (4.6) on derivatives of u produces an overdetermined system of four partialdifferentialequations, namely,

Λuu =0, (4.7)
Λxu =0, (4.8)
Λtu =0 (4.9)

Λt + αuΛx + βΛtxx =0 (4.10)
By Equation (4.7), we have

Λ = A(t,x)u + B(t,x), (4.11)
which if used in Equations (4.8-4.9), implies that

Λ = C1u + B(t,x). (4.12)
If we substitute (4.12) into Equation (4.10), we obtain

Bt(t,x) + αuBx (t,x) + βBtxx (t,x) = 0. (4.13)
Separation of Equation (4.13) into powers of u gives us

u :Bx (t,x) = 0, (4.14)
u0 :Bt(t,x) + βBtxx (t,x) = 0. (4.15)

Equation (4.14) insists that
Btxx (t,x) = 0 =⇒ Bt(t,x) = 0 = Bx (t,x), (4.16)

and thus
B(t,x) = C2. (4.17)

As a result
Λ(t,x,u) = C1u + C2. (4.18)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 17
Essentially, we extract the two multiplies

Λ1 =1 (4.19)
Λ2 =u. (4.20)

Remark 4.1. Recall that a multiplier Λ for Equation(1.1) has the property that for the density
Tt = Tt(t,x,u,ux ) and flux Tx = Tx (t,x,u,ux,utx ),

Λ (ut + αuux + βutxx ) = DtT
t + DxT

x. (4.21)
We derive a conservation law corresponding to each of the multipliers.

(i). Conservation law for the multiplier Λ1 = 1Expansion of equation (4.21) gives
1{ut + αuux + βutxx} = Ttt + utT

t
u + utxT

t
ux

+ Txx + uxT
x
u + uxxT

x
ux

+ utxxT
x
utx
. (4.22)

Splitting Equation (4.22) on the third derivative of u yields
utxx : T

x
utx

= β, (4.23)
Rest : ut + αuux = Ttt + utTtu + utxTtux + Txx + uxTxu + uxxTxux. (4.24)

The integration of Equation (4.23) with respect to utx gives
Tx = βutx + A(t,x,u,ux ). (4.25)

Substituting the expression of Tx from (4.25) into Equation (4.22) we get
{ut + αuux} =Ttt + utT

t
u + utxT

t
ux

+ Ax + uxAu + uxxAux (4.26)
which splits on second derivatives of u, to give

uxx : Aux = 0, (4.27)
utx : T

t
ux

= 0, (4.28)
Rest : {ut + αuux} = Ttt + utTtu + Ax + uxAu. (4.29)

Integrating equations (4.27) and (4.28) with respect to ux manifests that Tt = Tt(t,x,u) and
A = A(t,x,u). Using values of A and Tt in Equation (4.29), we have

{ut + αuux} = Ttt + utT
t
u + Ax + uxAu, (4.30)

which separates on first derivatives to give us
ut : T

t
u = 1, (4.31)

ux : Au = αu, (4.32)
Rest : Ttt + Ax = 0. (4.33)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 18
Equations (4.31-4.32), can be integrated with respect u to obtain

Ttu = u + B(t,x), (4.34)
A = α

u2

2
+ C(t,x), (4.35)

If we use the obtained values in (4.33), we have
Bt(t,x) + Cx (t,x) = 0. (4.36)

Since B(t,x) and C(t,x) contribute to the trivial part of the conservation law, we take
B(t,x) = C(t,x) = 0 and obtain the conserved quantities

Tt =u, (4.37)
Tx =α

u2

2
+ βutx (4.38)

from which the conservation law corresponding to the multiplier Λ1 = 1 is given by
Dt(u) + Dx

(
α
u2

2
+ βutx

)
= 0. (4.39)

(ii). Conservation law for the multiplier Λ2 = u
u{ut + αuux + βutxx} = Ttt + utT

t
u + utxT

t
ux

+ Txx + uxT
x
u + uxxT

x
ux

+ utxxT
x
utx
. (4.40)

Splitting Equation (4.40) on the third derivative of u yields
utxx : T

x
utx

= βu, (4.41)
Rest : ut + αuux = Ttt + utTtu + utxTtux + Txx + uxTxu + uxxTxux. (4.42)

The integration of Equation (4.41) with respect to utx gives
Tx = βuutx + A(t,x,u,ux ). (4.43)

Substituting the expression of Tx from (4.43) into Equation (4.40) we get
u{ut + αuux} =Ttt + utT

t
u + utxT

t
ux

+ Ax + uxAu + uxβutx + uxxAux. (4.44)
which splits on second derivatives of u, to give

uxx : Aux = 0, (4.45)
utx : T

t
ux

= −βux, (4.46)
Rest : u{ut + αuux} = Ttt + utTtu + Ax + uxAu. (4.47)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 19
Integrating equations (4.45) and (4.46) with respect to ux manifests that
Tt = −βu

2
x

2
+ B(t,x,u) and A = A(t,x,u). Using values of A and Tt in Equation(4.47), we have

u{ut + αuux} = Ttt + utT
t
u + Ax + uxAu, (4.48)

which separates on first derivatives to give us
ut : B(t,x,u)u = u, (4.49)
ux : Au = αu

2, (4.50)
Rest : Bt + Ax = 0. (4.51)

Equations (4.49-4.50), can be integrated with respect u to obtain
B =

u2

2
+ C(t,x), (4.52)

A = α
u3

3
+ D(t,x), (4.53)

If we use the obtained values in (4.51), we have
Ct(t,x) + Dx (t,x) = 0. (4.54)

Since C(t,x) and D(t,x) contribute to the trivial part of the conservation law, we take
C(t,x) = D(t,x) = 0 and obtain the conserved quantities

Tt = −β
u2x
2

+
u2

2
, (4.55)

Tx =βuutx + α
u3

3
(4.56)

from which the conservation law corresponding to the multiplier Λ2 = u is given by
Dt

(
−β

u2x
2

+
u2

2

)
+ Dx

(
βuutx + α

u3

3

)
= 0. (4.57)

Remark 4.2. It can be shown that the two sets of conserved quantities are conservation laws. Giventhat Λ1 = 1 , the verification reaffirms that the equal width equation is itself a conversation law.
5. Conclusion

In this manuscript, an infinite dimensional Lie algebra of Lie point symmetries has been appliedto study a third-order equal width equation. A commutator table has been constructed for theobtained Lie algebra. We have also used symmetry reductions to compute exact group-invariantsolutions, including a soliton. Conservation laws have also been derived for the model with the useof zeroth order multipliers.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.13 20
Acknowledgement

The author thanks referees and the editor for their careful reading and comments.
Author’s contribution

The author wrote the article as a scholarly duty and passion to disseminate mathematical re-search and hereby declares that there is no conflict of interest.
References

[1] J.R. Cannon, The one-dimensional heat equation, Cambridge University Press, Cambridge, 1984.[2] P.J. Morrison, J.D. Meiss, J.R. Cary, Scattering of regularized-long-wave solitary waves, Physica D: NonlinearPhenomena. 11 (1984) 324–336. https://doi.org/10.1016/0167-2789(84)90014-9.[3] L.R.t. Gardner, G.A. Gardner, F.A. Ayoub, N.K. Amein, Simulations of the EW undular bore, Commun. Numer. Meth.Engng. 13 (1997) 583–592. https://doi.org/10.1002/(sici)1099-0887(199707)13:7<583::aid-cnm90>3.
0.co;2-e.[4] L.R.T. Gardner, G.A. Gardner, Solitary waves of the equal width wave equation, J. Comput. Phys. 101 (1992) 218–223.
https://doi.org/10.1016/0021-9991(92)90054-3.[5] J.O. Owino, Group analysis on one-dimensional heat equation, Int. J. Adv. Multidisc. Res. Stud. 2 (2022) 525-540.[6] J. Owuor, Exact symmetry reduction solutions of a nonlinear coupled system of Korteweg-De Vries Equations, Int.J. Adv. Multidisc. Res. Stud. 2 (2022) 76-87.[7] J. Owuor Owino, B. Okelo, Lie group analysis of a nonlinear coupled system of Korteweg-De Vries Equations, Eur.J. Math. Anal. 1 (2021) 133–150. https://doi.org/10.28924/ada/ma.1.133.[8] J.O. Owino, A group approach to exact solutions and conservation laws of Burger’s Equation, Int. J. Math. Comp.Res. 10 (2022) 2894-2909. https://doi.org/10.47191/ijmcr/v10i9.03.[9] J. Owuor, Conserved Quantities of a Nonlinear Coupled System of Korteweg-De Vries Equations, Int. J. Math. Comp.Res. 10 (2022) 2673-2681. https://doi.org/10.47191/ijmcr/v10i5.02.[10] J. O. Owino, An application of lie point symmetries in the study of potential Burger’s equation, Int. J. Adv. Multidisc.Res. Stud. 2 (2022) 191-207.[11] J. O. Owino, Group invariant solutions and conserved vectors for a special kdv type equation, Int. J. Adv. Multidisc.Res. Stud. 2 (2022), 9-26.

https://doi.org/10.28924/ada/ma.3.13
https://doi.org/10.1016/0167-2789(84)90014-9
https://doi.org/10.1002/(sici)1099-0887(199707)13:7<583::aid-cnm90>3.0.co;2-e
https://doi.org/10.1002/(sici)1099-0887(199707)13:7<583::aid-cnm90>3.0.co;2-e
https://doi.org/10.1016/0021-9991(92)90054-3
https://doi.org/10.28924/ada/ma.1.133
https://doi.org/10.47191/ijmcr/v10i9.03
https://doi.org/10.47191/ijmcr/v10i5.02

	1. Introduction
	2. Preliminaries
	Local Lie groups
	Prolongations
	Lie algebras
	Conservation laws
	The method of multipliers
	Ibragimov's conservation theorem 

	3. Main results
	3.1. Lie point symmetries of equal width equation(1.1)
	3.2. Commutator Table for Symmetries
	3.3. Group Transformations
	3.4. Symmetry transformations
	3.5.  Construction of Group-Invariant Solutions
	3.6. Soliton

	4.  Conservation laws of equation (1.1)
	4.1. The multipliers

	5. Conclusion
	Acknowledgement
	Author's contribution
	References

