EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5371
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Conservation Laws and Symmetry Multi-Reductions of
Two (2 + 1)-Dimensional Equations: The

Zakharov-Kuznetsov (ZK) Equation and a Nonlinear
Wave Equation

Molahlehi Charles Kakuli1,2,∗, Winter Sinkala1, Phetogo Masemola2

1 Department of Mathematical Sciences and Computing, Faculty of Natural Sciences,
Walter Sisulu University, Private Bag X1, Mthatha 5117, Republic of South Africa
2 School of Mathematics, University of the Witwatersrand, Braamfontein 2000, South Africa

Abstract. The construction of invariant solutions is a key application of Lie symmetry analysis in
studying partial differential equations. The generalised double reduction method, which uses both
symmetries and conservation laws of a PDE or system of PDEs, provides a powerful framework for
constructing such solutions. This paper contributes to the application of the generalised double
reduction method by analysing two (2 + 1)-dimensional equations: the Zakharov-Kuznetsov (ZK)
equation and a nonlinear wave equation. We extend the work of Bokhari et al. [6, 7] on the nonlin-
ear wave equation by performing a second symmetry reduction using previously unused inherited
symmetries. For the ZK equation, we identify its Lie point symmetries, construct four conservation
laws using the multiplier method, and determine their associated Lie point symmetries. This al-
lows for symmetry reductions using each conservation law. This paper provides a detailed account
of the generalised double reduction method, including the exploitation of inherited symmetries at
each reduction step.

2020 Mathematics Subject Classifications: 22E70, 35C05, 35K15, 35Q80, 68-04

Key Words and Phrases: Double reduction, Zakharov-Kuznetsov equation, Lie symmetry anal-
ysis, conservation law, invariant solution

1. Introduction

Partial differential equations (PDEs) are widely used as models of real-world physical
phenomena. Analytical solutions to PDEs are highly desirable whenever possible. Lie
symmetry analysis [4, 5, 9, 32, 33] provides powerful routines for seeking analytical solu-
tions of PDEs known as group-invariant solutions. This approach has been successfully
applied to find exact solutions of many PDEs, including those in physics, engineering, and
other fields [1, 16, 23–27, 34, 36, 42, 43].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5371

Email addresses: ckakuli@wsu.ac.za (M. C. Kakuli), phetogo.masemola@wits.ac.za (P. Masemola)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 2 of 23

Based on pioneering work by Kara et al. [18–20] (see also [40, 41]), Sjöberg [38, 39]
showed that the association of conservation laws with symmetries provides a new avenue
for obtaining invariant solutions of PDEs. This association results in double reduction of a
PDE. For a PDE of order q with two independent variables andm dependent variables that
admits a nontrivial conserved law, with at least one associated symmetry, Sjöberg [38, 39]
developed a double reduction method that reduces the PDE to an ODE of order (q − 1).
There are many articles on the application of the double reduction method involving two
independent variables and one dependent variables [8, 14, 15, 17, 37].

Recently, Bokhari et al. [7], and also Anco and Gandarias [2] generalised the double
reduction theory to the case involving several independent variables. According to Bokhari
et al. [7], a nonlinear system of qth-order PDEs with n independent and m dependent
variables can be reduced to a nonlinear system of (q − 1) th-order ODEs. The reduction
is possible only if in every reduction, there is at least one symmetry associated with
a nontrivial conservation law. Naz et al. [31] utilised the double reduction theory to
find some exact solutions of a class of nonlinear regularised long wave equations. Other
applications of the generalised the double reduction theory include Bokhari et al [6], Sait
et al [35] and Muatjetjeja et al [28].

The first part of this paper is essentially the extension of the seminal paper by Bokhari
et al [6] on the generalisation of the double reduction theory. In [6] the theory was applied
on the nonlinear (2 + 1)-dimensional wave equation

utt − (f(u)ux)x − (g(u)uy)y = 0 (1.1)

involving two arbitrary functions f(u) and g(u). We have presented an extended account
of the application and included a second multi-reduction of the conservation law of the
equation by finding and using inherited symmetries that were not determined. We have
also included for illustrative purposes solutions of the wave equation for particular speci-
fications of the arbitrary functions.

In the second part of the paper we consider another equation, the (2 + 1)-dimensional
Zakharov-Kuznetsov (ZK) equation [3, 12, 13, 29]

ut + µux + νuux + αuxxx + βuxyy = 0, (1.2)

where µ, ν, α, and β are arbitrary constants, are found by employing the generalised double
reduction theory. The ZK equation originated from the study of weakly nonlinear ion-
acoustic waves in a strongly magnetised plasma consisting of cold ions and hot isothermal
electrons. It was first introduced by Vladimir Zakharov and Boris Kuznetsov in 1974 [45]
and serves as a two-dimensional generalisation of the well-known Korteweg-de Vries (KdV)
equation, alongside the Kadomtsev-Petviashvili (KP) equation [10, 44].

Over time, researchers have investigated various aspects of the ZK equation, including
its local, global, and scattering properties, as well as seeking novel exact solutions through

techniques such as the
(
G′

G

)2
-expand method, Lie symmetry analysis, and the homotopy

perturbation method [10, 11, 16, 22, 44]. Research to find analytical solutions of the ZK
equation and its variants has continued aimed at contributing to the understanding of



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 3 of 23

complex physical phenomena modelled by the ZK equation that arise in diverse fields
[10, 16, 22, 44].

The main goal of our present work is to obtain reductions of the ZK equation by
exploiting the generalised double reduction theory [7]. We obtain four non-trivial con-
servation laws of the ZK equation (when µ = 0 and ν = 1) by the multiplier method.
The generalised double reduction theorem is then applied leading to second-order ODEs
in each of the cases where multireduction is possible. Our application of the generalised
double reduction method is instructive in that we demonstrate the use of “nonassociated”
symmetries in the reduction routine. We show that an associated symmetry with a re-
duced conserved form may be inherited from a nonassociated symmetry with the original
conserved form.

2. Fundamentals of the Double Reduction Theorem

In this section, we present the double reduction routine for a qth-order (q ≥ 1) partial
differential equation with n independent variables x = (x1, x2, . . . , xn) and one dependent
variable u = u(x), namely

F (x, u, u(1), u(2), . . . , u(q)) = 0, (2.1)

where u(q) denotes the collection {uq} of qth-order partial derivatives. In this connection,
we first present the following well-known definitions and results (see, e.g., [7, 18, 21, 30]).

Definition 2.1. The total derivative operator with respect to xi is

Di =
∂

∂xi
+ ui

∂

∂u
+ uij

∂

∂uj
+ · · · , i = 1, 2, . . . , n, (2.2)

where ui denotes the derivative of u with respect to xi. Similarly, uij denotes the derivative
of u with respect to xi and xj.

Definition 2.2. The Lie-Bäcklund operator is

X = ξi
∂

∂xi
+ η

∂

∂u
ξi, η ∈ A, (2.3)

where A is the space of differential functions. The operator (2.3) is an abbreviated form
of the infinite formal sum

X = ξi
∂

∂xi
+ η

∂

∂u
+
∑
s≥1

ζi1i2...is
∂

∂ui1i2...is
, (2.4)

where the additional coefficients are determined uniquely by the prolongation formulae,

ζi = Di (W ) + ξjuij

ζi1...is = Di1...Dis
(W ) + ξjuji1...is , s > 1,

(2.5)

in which W is the Lie characteristic function,

W = η − ξjuj . (2.6)



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 4 of 23

Definition 2.3. An n-tuple T =
(
T 1, T 2, . . . , Tn

)
, i = 1, 2, . . . , n, such that

DiT
i = 0 (2.7)

holds for all solutions of (2.1) is known as a conservation law of (2.1).

Definition 2.4. A multiplier Λ for Equation (2.1) is a non-singular function on the
solution space of (2.1) with the property

DiT
i = ΛE (2.8)

for arbitrary function u
(
x1, x2, . . . , xn

)
.

Definition 2.5. The determining equations for multipliers are obtained by taking the
variational derivative

δ

δu
(ΛE) = 0, (2.9)

where the Euler operator δ/δu is defined by

δ

δu
=

∂

∂u
−Di

∂

∂ui
+Dij

∂

∂uij
−Dijk

∂

∂uijk
+ · · · . (2.10)

Definition 2.6. A Lie-Bäcklund symmetry generator X of the form (2.3) is associated
with a conserved vector T of the system (2.1) if X and T satisfy the relations

X
(
T i
)
+ T iDjξ

j − T jDjξ
i,= 0, i = 1, . . . , n. (2.11)

Theorem 2.1. Suppose DiT
i = 0 is a conservation law of the PDE system (2.1). Then

under a similarity transformation of a symmetry X of the form (2.3) for the PDE, there
exist functions T̃ i such that X is still symmetry for the PDE D̃iT̃

i = 0, where T̃ i is given
by 

T̃ 1

T̃ 2

·
·
·
T̃n


= J

(
A−1

)T


T 1

T 2

·
·
·
Tn

 , (2.12)

where

A =


D̃1x1 D̃1x2 . . . D̃1xn
D̃2x1 D̃2x2 . . . D̃2xn
...

...
...

...

D̃nx1 D̃nx2 . . . D̃nxn

 , A−1 =


D1x̃1 D1x̃2 . . . D1x̃n
D2x̃1 D2x̃2 . . . D2x̃n
...

...
...

...
Dnx̃1 Dnx̃2 . . . Dnx̃n

 ,

and J = det(A).



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 5 of 23

Colollary 2.1. (The necessary and sufficient condition for reduced conserved form [7]).
The conserved form DiT

i = 0 of the PDE system (2.1) can be reduced under a similarity
transformation of a symmetry X to a reduced conserved form D̃iT̃

i = 0 if and only if X
is associated with the conservation law T .

Colollary 2.2. (see [7]). A nonlinear system of qth-order PDEs with n independent and
m dependent variables which admits a nontrivial conserved form that has at least one
associated symmetry in every reduction from the n reductions (the first step of double
reduction) can be reduced to a (q − 1) th-order nonlinear system of ODEs.

Colollary 2.3. (The Inherited Symmetries [7]). Any symmetry Y for the conserved form
DiT

i = 0 of PDE system (2.1) can be transformed under the similarity transformation of
a symmetry X for the PDE to the symmetry Ỹ for the PDE D̃iT̃

i = 0.

3. Application of the generalised double reduction theory to the
nonlinear wave equation (1.1)

It was established in [6, 7] that for arbitrary functions f(u) and g(u), the nonlinear
(2 + 1) wave equation (1.1) has the conserved vector(

T t, T x, T y
)
= (−ut, f(u)ux, g(u)uy) , (3.1)

and admits the Lie point symmetries

X1 =
∂

∂t
, X2 =

∂

∂x

X3 =
∂

∂y
, X4 = t

∂

∂t
+ x

∂

∂x
+ y

∂

∂y
.

(3.2)

Furthermore, the symmetries X1, X2 and X3 are associated with the conserved vector
(3.1), i.e., if X = κ1X1 + κ2X2 + κ3X3, where κi’s are arbitrary constants, then

X

 T t

T x

T y

−

 Dtξ
t Dxξ

t Dyξ
t

Dtξ
x Dxξ

x Dyξ
x

Dtξ
y Dxξ

y Dyξ
y

 T t

T x

T y


+
(
Dtξ

t +Dxξ
x +Dyξ

y
) T t

T x

T y

 = 0.

(3.3)

So we can get a reduced conserved form by the combination of them X = ∂
∂t+κ2

∂
∂x+κ3

∂
∂y ,

where the generator X has a canonical form X = ∂
∂q . From the characteristic equations

dt

1
=

dx

κ2
=

dy

κ3
=

du

0
=

dr

0
=

ds

0
=

dq

1
=

dw

0
, (3.4)

we obtain canonical coordinates



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 6 of 23

r = y − κ3t, s = x− κ2t, q = t, w(r, s) = u. (3.5)

The inverse canonical coordinates are

t = q, x = κ2q + s, y = κ3q + r, u = w. (3.6)

It follows therefore that the partial derivatives ut, ux and uy expressed in terms of the
canonical variables (3.5) are

ut = −κ2ws − κ3wr, ux = ws, uy = wr. (3.7)

According to Theorem 2.1, we obtain the reduced conserved form T r

T s

T q

 = J
(
A−1

)T  T t

T x

T y

 , (3.8)

where

A =

 Drt Drx Dry
Dst Dsx Dsy
Dqt Dqx Dqy

 =

 0 0 1
0 1 0
1 κ2 κ3

 , (3.9)

A−1 =

 Dtr Dts Dtq
Dxr Dxs Dxq
Dyr Dys Dyq

 =

 −κ3 −κ2 1
0 1 0
1 0 0

 , (3.10)

with J = det(A) = −1. Substituting for the partial derivatives in (3.8) using (3.7), we
obtain

T r = κ23wr + κ2κ3ws − g(w)wr,

T s = κ2κ3wr + κ22ws − f(w)ws,

T q = −κ3wr − κ2ws.

(3.11)

Therefore, the reduced conserved form is

DrT
r +DsT

s = 0. (3.12)

After writing the symmetries (3.2) in the canonical variables (3.5), we obtain

X̃1 = κ3
∂

∂r
+ κ2

∂

∂s
, X̃2 =

∂

∂s
, X̃3 =

∂

∂r
, X̃4 = r

∂

∂r
+ s

∂

∂s
, (3.13)

and it turns out that they are all inherited by (3.12). Furthermore, all the inherited
symmetries (3.13) are associated with the conservation law (3.12), i.e., if X̃ = δ1X̃1 +
δ2X̃2 + δ3X̃3 + δ4X̃4, then

X̃

(
T r

T s

)
−
(

Drξ
r Dsξ

r

Drξ
s Dsξ

s

)(
T r

T s

)
+ (Drξ

r +Dsξ
s)

(
T r

T s

)
= 0. (3.14)

We consider reduction of the conservation law (3.12) under two cases: under the inherited
symmetry Y1 = r ∂

∂r + s ∂
∂s , and under the inherited symmetry Y2 = ∂

∂r + γ ∂
∂s , where γ is

an arbitrary constant.



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 7 of 23

3.1. Reduction of (T r, T s) under Y1 = r ∂
∂r

+ s ∂
∂s

The generator Y1 = r ∂
∂r + s ∂

∂s has a canonical form Y = ∂
∂m when

dr

r
=

ds

s
=

dw

0
=

dn

0
=

dm

1
=

dv

0
, (3.15)

or
n =

s

r
, m = ln r, v(n) = w. (3.16)

The inverse canonical coordinates are given by

r = em, s = emn, w = v, (3.17)

and so the partial derivatives in the conserved vector (T r, T s) in terms of the canonical
coordinates (3.16) are given by

wr = −e−mnvn, ws = −e−mnvn. (3.18)

By using the formula (2.12) we get the reduced conserved form(
Tn

Tm

)
= J

(
A−1

)T ( T r

T s

)
, (3.19)

where

A =

(
Dnr Dns
Dmr Dms

)
=

(
0 em

em emn

)
,

A−1 =

(
Drn Drm
Dsn Dsm

)
=

(
− s

r2
1
r

1
r 0

)
,

and J = det(A) = −e2m.
From (3.18) and (3.19) we obtain that

Tn = vn
(
2κ2κ3n− κ23n

2 + n2g(v)− κ22 + f(v)
)
,

Tm = vn
(
κ23n− κ2κ3 − ng(v)

)
.

(3.20)

Therefore, the reduced conservation law is

DnT
n = 0, (3.21)

or
vn
(
2κ2κ3n− κ23n

2 + n2g(v)− κ22 + f(v)
)
= k, (3.22)

where k is a constant. The solution of the nonlinear wave equation (1.1) follows from the
solution of (3.22) via (3.16) and (3.5).

For illustrative purposes, let us consider particular choices of the arbitrary functions
in equation (1.1). If we let f(u) = 1, g(u) = u and set κ2 = 1 and κ3 = 0, then the ODE
(3.22) reduces to

n2v(n)v′(n) = k, (3.23)



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 8 of 23

the solution of which is
2k

n
+ v2 = λ, (3.24)

where λ is an arbitrary constant. In light of the change of variables (3.16) and (3.5), with
κ2 = 1, κ3 = 0, n = x−t

y and v = u, the solution (3.24) translates into the following
solution of the nonlinear wave equation (1.1):

2ky

x− t
+ u2 = λ. (3.25)

3.2. Reduction of (T r, T s) under Y2 = ∂
∂r

+ γ ∂
∂s

From the generator Y2 the canonical coordinates are

n = s− γr, m = r, v(n) = w, (3.26)

and the inverse canonical coordinates are given by

r = m, s = γm+ n, w = v. (3.27)

Therefore, the partial derivatives in (T r, T s) in terms of the canonical coordinates (3.26)
are given by

wr = −γvn, ws = vn. (3.28)

By using the formula (2.12) we get the reduced conserved form(
Tn

Tm

)
= J

(
A−1

)T ( T r

T s

)
, (3.29)

where

A =

(
Dnr Dns
Dmr Dms

)
=

(
0 1
1 γ

)
,

A−1 =

(
Drn Drm
Dsn Dsm

)
=

(
−γ 1
1 0

)
,

and J = det(A) = −1.
From (3.28) and (3.29) we obtain that

Tn = vn
(
f(v) + γ2g(v)− (κ2 − γκ3)

2
)
,

Tm = vn(κ3(γκ3 − κ2)− γg(v)).
(3.30)

Therefore, the reduced conservation law is

DnT
n = 0, (3.31)

or
vn
(
f(v) + γ2g(v)− (κ2 − γκ3)

2
)
= k, (3.32)



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 9 of 23

where k is a constant. The solution of the nonlinear wave equation (1.1) follows from the
solution of (3.32) via (3.26) and (3.5).

If for illustrative purposes we let f(u) = g(u) = u in the nonlinear wave equation (1.1),
and set γ = κ3

κ2
, the ODE (3.32) reduces to

(Mv(n) + v(n)− L)v′(n) = k, (3.33)

where

M =
κ23
κ22

, L =

(
κ2 −

κ23
κ2

)2

.

The solution of (3.33) is

λ− 2kn− 2Lw + (M + 1)w2 = 0, (3.34)

where λ is an arbitrary constant. In light of (3.5) and (3.26), and if we set γ = κ3
κ2
, we

obtain

n =
κ23t

κ2
− κ3y

κ2
− κ2t+ x and w = u. (3.35)

Therefore (3.34) becomes

k

(√
L t+

κ3
κ2

y − x

)
+

λ

2
− Lu+

1

2
(M + 1)u2 = 0, (3.36)

which is the solution in implicit form of the non-linear wave equation (1.1).

4. Symmetries and Conservation Laws of the ZK Equation

First, we will derive the Lie point symmetries of (1.2), in the case µ = 0, ν = 1, αβ ̸= 0,
i.e.,

ut + uux + αuxxx + βuxyy = 0. (4.1)

If the operator

X =ξ1(t, x, y, u)
∂

∂t
+ ξ2(t, x, y, u)

∂

∂x
+ ξ3(t, x, y, u)

∂

∂y
+ η(t, x, y, u)

∂

∂u

is a generator of a Lie point symmetry of (4.1), then it must satisfy the invariance condition

X [3] [ut + uux + αuxxx + βuxyy]
∣∣∣
(4.1)

= 0, (4.2)

where X(3) is the third prolongation of X and can be computed according to (2.5). Equa-
tion (4.2), after expansion and separation, yields an overdetermined system of linear first-
order partial differential equations for the unknown coefficients ξ1, ξ2, ξ3, and η. The
solution of the system leads to the following Lie point symmetries of (4.1):

X1 =
∂

∂t
, X2 =

∂

∂x
, X3 =

∂

∂y
, X4 = t

∂

∂x
+

∂

∂u

X5 = 3t
∂

∂t
+ x

∂

∂x
+ y

∂

∂y
− 2u

∂

∂u
.

(4.3)



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 10 of 23

The conservation laws for (4.1) are constructed by the multiplier approach. We consider
multipliers of the form Λ(x, t, u, ux, uy, ut, uxx, uxy, utx, uyy, utt, uty) for (4.1). The deter-
mining equation for the multipliers is

δ

δu
[Λ (ut + uux + αuxxx + βuxyy)] = 0, (4.4)

where δ
δu is the standard Euler operator (2.10). Expanding and then separating (4.4)

with respect to different combinations of partial derivatives of u results in the following
overdetermined system for the multipliers:

Λtt = 0, Λtu − Λt

u
= 0, Λtuyy = 0,

Λuu −
Λuyy

β
= 0, Λu,uyy = 0, Λuyyuyy = 0,

Λx +
Λt

u
= 0, Λux = 0, Λuy = 0,

Λut = 0, Λuxx −
αΛuyy

β
= 0, Λuxy = 0,

Λutx = 0, Λutt = 0, Λuty = 0,

(4.5)

provided that αβ ̸= 0. The solution of system (4.5) is

Λ = c1tu− c1x+ c2uyy +
αc2
β

uxx +
c2
2β

u2 + c3u+ c4, (4.6)

where c1, c2, c3 and c4 are arbitrary constants. From (4.6), we obtain single parameter
multipliers

Λ1 = tu− x, Λ2 = uyy +
1

2β
u2 +

α

β
uxx, Λ3 = u, Λ4 = 1. (4.7)

According to (2.8), the multipliers in (4.7) satisfy

Λ (ut + uux + αuxxx + βuxyy) = DtT
t +DxT

x +DyT
y (4.8)

for arbitrary functions u(t, x, y). We obtain four nontrivial conserved vectors

T1 =
(
T t
1, T

x
1 , T

y
1

)
, T2 =

(
T t
2, T

x
2 , T

y
2

)
, T3 =

(
T t
3, T

x
3 , T

y
3

)
, T4 =

(
T t
4, T

x
4 , T

y
4

)
, (4.9)

where T t
i , T

x
i , T

y
i , i = 1, . . . , 4, are given by

T1 :



T t
1 =

tu2

2
− ux,

T x
1 =

tu3

3
− u2x

2
+ αux −

α

2
tu2x + αtuuxx − αxuxx +

β

2
tuuyy − βxuyy,

T y
1 =

βtuuxy
2

+ βuy −
βtuxuy

2
,

(4.10)



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 11 of 23

T2 :



T t
2 =

uuyy
2

+
u3

6β
+

αuuxx
2β

,

T x
2 = αuxxuyy +

u2uyy
2

+
u4

8β
+

αu2uxx
2β

− αuutx
2β

+
αuxut
2β

+
α2u2xx
2β

+
βu2yy
2

,

T y
2 =

utuy
2

− uuty
2

,

(4.11)

T3 :



T t
3 =

u2

2
,

T x
3 =

u3

3
− αu2x

2
+ αuuxx +

βuuyy
2

,

T y
3 =

βuuxy
2

− βuxuy
2

,

(4.12)

T4 :


T t
4 = u,

T x
4 =

u2

2
+ αuxx + βuyy,

T y
4 = 0,

(4.13)

5. Double Reduction of the ZK equation

To determine symmetries associated with the conserved vectors (4.9), we set a lin-
ear combination of the symmetries (4.3), i.e., X =

∑5
i=1 κiXi, where κi’s are arbitrary

constants, and then apply the association condition

X

(
T t

T x

T y

)
−

(
Dtξ

1 Dxξ
1 Dyξ

1

Dtξ
2 Dxξ

2 Dyξ
2

Dtξ
3 Dxξ

3 Dyξ
3

)(
T t

T x

T y

)
+
(
Dtξ

1 +Dxξ
2 +Dyξ

3
)( T t

T x

T y

)
= 0

(5.1)
for each of conserved vector Ti in (4.9). We obtain that T1 is associated with only X3, T2

and T3 are associated with the linear combination κ1X1+κ2X2+κ4X4, and T4 is associated
with the linear combination κ1X1 + κ2X2 + κ3X3 + κ5X5. For the multi-reductions that
follow we use these associations:

X3 −→ T1

X1 + κ2X2 + κ3X3 −→ T2

X1 + κ2X2 + κ3X3 −→ T3

X1 + κ2X2 + κ3X3 −→ T4

X5 −→ T4.

(5.2)

5.1. Multi-Reduction of the ZK equation by T3

We obtain the first reduction of the conserved vector T3 by writing the vector in
canonical variables determined from the associated symmetry

X = X1 + κ2X2 + κ3X3. (5.3)



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 12 of 23

Writing (5.3) in the canonical form X = ∂
∂q , we obtain from the corresponding character-

istic equations
dt

1
=

dx

κ2
=

dy

κ3
=

du

0
=

dr

0
=

ds

0
=

dq

1
=

dw

0
, (5.4)

the canonical coordinates

r = y − κ3t, s = x− κ2t, q = t, w = u, (5.5)

where w = w(r, s). Inverse canonical coordinates are given by

t = q, x = κ2q + s, y = κ3q + r, u = w. (5.6)

From Theorem 2.1, we obtain the reduced conserved form T r
3

T s
3

T q
3

 = J
(
A−1

)T  T t
3

T x
3

T y
3

 , (5.7)

where

A =

 Drt Drx Dry
Dst Dsx Dsy
Dqt Dqx Dqy

 =

 0 0 1
0 1 0
1 κ2 κ3

 , (5.8)

A−1 =

 Dtr Dts Dtq
Dxr Dxs Dxq
Dyr Dys Dyq

 =

 −κ3 −κ2 1
0 1 0
1 0 0

 , (5.9)

and
J = det(A) = −1. (5.10)

Expressing the first and second partial derivatives ut, ux, utt, uxy and uxx in terms of
the canonical coordinates (5.5), we obtain

ut = −κ2ws − κ3wr, ux = ws, uy = wr,

uxx = wss, utx = −κ2wss − κ3wrs, uyy = wrr.
(5.11)

Substituting for the partial derivatives uxx and uyy in (5.7) using (5.11), we obtain

T r
3 =

κ3w
2

2
− βwwrs

2
+

βwrws

2
,

T s
3 =

κ2w
2

2
− w3

3
− βwwrr

2
− αwwss +

αw2
s

2
,

T q
3 = −w2

2
,

(5.12)

leading to the reduced conservation law

DrT
r
3 +DsT

s
3 = 0. (5.13)



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 13 of 23

Equation (5.13) inherits the symmetries X1, X2 and X3 from (4.3), which when written
in terms of the canonical variables (5.5), are:

X̃1 = κ3
∂

∂r
+ κ2

∂

∂s
, X̃2 =

∂

∂s
, X̃3 =

∂

∂r
. (5.14)

It turns out that all the symmetries in (5.14) are associated with the conservation law
(5.13), i.e.

X̃

(
T r
3

T s
3

)
−
(

Drξ
r Dsξ

r

Drξ
s Dsξ

s

)(
T r
3

T s
3

)
+ (Drξ

r +Dsξ
s)

(
T r
3

T s
3

)
= 0, (5.15)

if X̃ = δ1X̃1 + δ2X̃2 + δ3X̃3, where δ′is are arbitrary constants.
So, we can get a further reduction of the conserved vector (T r, T s) by

Y =
∂

∂r
+ γ

∂

∂s
, (5.16)

where γ is an arbitrary constant. The generator Y has a canonical form Y =
∂

∂m
when

dr

r
=

ds

s
=

dw

0
=

dn

0
=

dm

1
=

dv

0
, (5.17)

which results in canonical coordinates

n = s− γr, m = r, v = w, (5.18)

where v = v(n). The inverse canonical coordinates are given by

r = m, s = γm+ n, w = v. (5.19)

Therefore, the partial derivatives in the components (5.12) in terms of the canonical co-
ordinates (5.18) are given by

wr = −γvn, ws = vn, wrr = γ2vnn, wrs = −γvnn, wss = vnn. (5.20)

According to Theorem 2.1, we have that(
Tn
3

Tm
3

)
= J

(
A−1

)T ( T r
3

T s
3

)
, (5.21)

where

A =

(
Dnr Dns
Dmr Dms

)
=

(
0 1
1 γ

)
, (5.22)

A−1 =

(
Drn Drm
Dsn Dsm

)
=

(
−γ 1
1 0

)
, (5.23)



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 14 of 23

and
J = det(A) = −1. (5.24)

Substituting the partial derivatives in (5.20) into (5.21) results in a reduced conserved
vector with the following components:

T r
3 = (α+ βγ2)

(
vvnn − v2n

2

)
+

1

2
v2(γκ3 − κ2) +

v3

3
,

T s
3 =

1

2
βγv2n − κ3v

2

2
− 1

2
βγvvnn.

(5.25)

This leads to the reduced conservation law DrT
r
3 = 0, from which we obtain the second-

order ODE

(α+ βγ2)

(
vvnn − v2n

2

)
+

1

2
v2(γκ3 − κ2) +

v3

3
= k, (5.26)

where k is an arbitrary constant.

5.2. Multi-Reduction of the ZK equation by T4

We see from (5.1) that the symmetries associated with T4 are X1, X2, X3 and X5. We
perform multi-reduction by T4 under two cases, namely reduction from using the linear
combination X = X1 + κ2X2 + κ3X3 and reduction from using X5.

5.2.1. Reduction via X = X1 + κ2X2 + κ3X3

The canonical variables (5.5) obtained earlier under T3 apply here. Therefore, the reduced
conserved form resulting from T4 is T r

4

T s
4

T q
4

 = J
(
A−1

)T  T t
4

T x
4

T y
4

 , (5.27)

where A, A−1, J and the partial derivatives are the same as the ones computed earlier in
(5.8)– (5.11). Therefore, we obtain

T r
4 = κ3w,

T s
4 = κ2w − w2

2
− βwrr − αwss

T q
4 = −w,

(5.28)

and the reduced conservation law

DrT
r
4 +DsT

s
4 = 0. (5.29)

Like in the T3 case, the symmetries X1, X2, and X3, in (4.3) written in terms of the
canonical variables (5.5), become

X̃1 = κ3
∂

∂r
+ κ2

∂

∂s
, X̃2 =

∂

∂s
, X̃3 =

∂

∂r
, (5.30)



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 15 of 23

and are inherited by (5.29). Also, they are all associated with the conservation law (5.29).
Using Y = ∂

∂r + γ ∂
∂s , where γ is an arbitrary constant, as in the T3 case, we find canonical

coordinates
n = s− γr, m = r, v = w, (5.31)

where v = v(n). Taking advantage of the calculations done in the T3 case, in which the
same canonical variables were used, the reduced conserved vector is given by(

Tn
4

Tm
4

)
= J

(
A−1

)T ( T r
4

T s
4

)
, (5.32)

where A, A−1, and J are given by (5.22), (5.23) and (5.24), respectively.
Substituting the partial derivatives in (5.20) into (5.32) results in the following com-

ponents of the reduced conserved vector:

T r
4 = −κ2v + γκ3v +

v2

2
+ vnn

(
α+ βγ2

)
,

T s
4 = −κ3v.

(5.33)

This leads to the reduced conservation law DrT
r
4 = 0, from which it follows that

v(γκ3 − κ2) +
v2

2
+ vnn

(
α+ βγ2

)
= k, (5.34)

where k is an arbitrary constant.

5.2.2. Reduction via X = X5

We obtain the first reduction of the conservation law T4 by writing it in canonical variables
determined from writing the associated symmetry X = X5 in the form X = ∂

∂q . From the
associated characteristic equations

dt

3t
=

dx

x
=

dy

y
=

du

−2u
=

dr

0
=

ds

0
=

dq

1
=

dw

0
, (5.35)

we obtain the canonical coordinates

r =
y
3
√
t
, s =

x
3
√
t
, q =

ln t

3
, w = t2/3u, (5.36)

where w = w(r, s). Inverse canonical coordinates are given by

t = e3q, x = eqs, y = eqr, u = e−2qw, (5.37)

and the partial derivatives in the conserved vector T4, in terms of the canonical coordinates,
are

uxx = e−4qwss, uyy = e−4qwrr. (5.38)



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 16 of 23

The reduced conserved vector is therefore T r
4

T s
4

T q
4

 = J
(
A−1

)T  T t
4

T x
4

T y
4

 , (5.39)

where

A =

 Drt Drx Dry
Dst Dsx Dsy
Dqt Dqx Dqy

 =

 0 0 eq

0 eq 0
3e3q eqs eqr

 (5.40)

A−1 =

 Dtr Dts Dtq
Dxr Dxs Dxq
Dyr Dys Dyq

 =

 − y
3t4/3

− x
3t4/3

1
3t

0 1
3√t

0
1
3√t

0 0

 , (5.41)

and
J = det(A) = −3e5q. (5.42)

Substituting the partial derivatives (5.38) into (5.39), we obtain

T r
4 = rw,

T s
4 = sw − 3w2

2
− 3βwrr − 3αwss

T q
4 = −w.

(5.43)

Then the reduced conservation law is

DrT
r
4 +DsT

s
4 = 0. (5.44)

None of the symmetries from (4.3) are inherited by (5.44), and therefore no further reduc-
tion of (5.44) is performed in this case.

5.3. Multi-Reduction of the ZK equation by T2

We see from (5.1) that the symmetry associated with T2 is X = ∂
∂t + κ2

∂
∂x + κ3

∂
∂y , the

same symmetry used in the multi-reduction by T3. Therefore, the first reduction can be
achieved through the canonical coordinates

r = y − κ3t, s = x− κ2t, q = t, w = u, (5.45)

where w = w(r, s). The resulting reduced conserved form is T r
2

T s
2

T q
2

 = J
(
A−1

)T  T t
2

T x
2

T y
2

 , (5.46)



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 17 of 23

where A, A−1, and J are given by (5.8), (5.9) and (5.10), respectively. Substituting for
the partial derivatives in the conserved vector T2 using (5.11), we obtain

T r
2 = −κ2wwrs

2
+

κ2wrws

2
+

κ3w
3

6β
+

ακ3wwss

2β
+

κ3w
2
r

2
,

T s
2 =

κ2w
3

6β
+

κ2wwrr

2
+

ακ2w
2
s

2β
− ακ3wwrs

2β
− w4

8β
− w2wrr

2

+
ακ3wrws

2β
− αw2wss

2β
− βw2

rr

2
− αwrrwss −

α2w2
ss

2β
,

T q
2 = −w3

6β
− wwrr

2
− αwwss

2β
.

(5.47)

Then the reduced conservation law is

DrT
r
2 +DsT

s
2 = 0. (5.48)

Like in the T3 case, the symmetries X1, X2, and X3, in (4.3) written in terms of the
canonical variables (5.5), become

X̃1 = κ3
∂

∂r
+ κ2

∂

∂s
, X̃2 =

∂

∂s
, X̃3 =

∂

∂r
, (5.49)

and are inherited by (5.48). Also, they are all associated with the conservation law (5.48).
Using Y = ∂

∂r + γ ∂
∂s , where γ is an arbitrary constant, we find canonical coordinates

n = s− γr, m = r, v = w, (5.50)

where v = v(n). Taking advantage of the calculations done in the T3 case, in which the
same canonical variables were used, the reduced conserved vector is given by(

Tn
2

Tm
2

)
= J

(
A−1

)T ( T r
2

T s
2

)
, (5.51)

where A, A−1, and J are given by (5.22), (5.23) and (5.24), respectively.
Substituting the partial derivatives in (5.20) into (5.51) results in the following com-

ponents of the reduced conserved vector:

Tn
2 =

Q2v2nn
2β

−
(κ2 − γκ3)

(
Qv2n + v3/3

)
2β

+
Qv2vnn

2β
+

v4

8β
,

Tm
2 =

1

2
γκ2v

2
n − 1

2
γκ2vvnn − κ3v

3

6β
− ακ3vvnn

2β
− 1

2
γ2κ3v

2
n,

(5.52)

where Q = α+ βγ2. This leads to the reduced conservation law DnT
n
2 = 0, from which it

follows that
Q2v2nn
2β

−
(κ2 − γκ3)

(
Qv2n + v3/3

)
2β

+
Qv2vnn

2β
+

v4

8β
= k, (5.53)

where k is an arbitrary constant.



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 18 of 23

5.4. Multi-Reduction of the ZK equation by T1

The first reduction of the conserved vector T1 is obtained from writing the vector in
canonical variables determined from writing the associated symmetry X = X3 in the form
X = ∂

∂q . From the corresponding characteristic equations

dt

0
=

dx

0
=

dy

1
=

du

0
=

dr

0
=

ds

0
=

dq

1
=

dw

0
, (5.54)

we obtain the canonical coordinates

r = t, s = x, q = y, w = u, (5.55)

where w = w(r, s). Inverse canonical coordinates are given by

t = r, x = s, y = q, u = w. (5.56)

Therefore, the partial derivatives in the conserved vector T1 in terms of the canonical
coordinates (5.55), are

ux = ws, uy = 0, uxx = wss, uxy = 0, uyy = 0. (5.57)

The reduced conserved vector is T r
1

T s
1

T q
1

 = J
(
A−1

)T  T t
1

T x
1

T y
1

 , (5.58)

where

A =

 Drt Drx Dry
Dst Dsx Dsy
Dqt Dqx Dqy

 =

 1 0 0
0 1 0
0 0 1

 , (5.59)

A−1 =

 Dtr Dts Dtq
Dxr Dxs Dxq
Dyr Dys Dyq

 =

 1 0 0
0 1 0
0 0 1

 , (5.60)

and
J = det(A) = 1. (5.61)

Substituting for the partial derivatives in (5.58) using (5.57), we obtain the components

T r
1 =

rw2

2
− sw,

T s
1 =

rw3

3
+ αrwwss −

1

2
αrw2

s −
sw2

2
− αswss + αws,

T q
1 = 0.

(5.62)



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 19 of 23

The corresponding reduced conservation law

DrT
r
1 +DsT

s
1 = 0, (5.63)

inherits the symmetries X1, X2, X4, and X5 from (4.3), or in terms of the canonical
variables (5.55)

X̃1 =
∂

∂r
, X̃2 =

∂

∂s
, X̃4 = r

∂

∂s
+

∂

∂w
, X̃5 = −3

2
r
∂

∂r
− 1

2
s
∂

∂s
+ w

∂

∂w
. (5.64)

It turns out that none of the symmetries (5.64) are associated with the conservation law
(5.63). Therefore, no further reduction of (5.63) is performed.

6. Concluding remarks

The work presented in this study significantly enhances the existing body of literature
concerning the application of the generalised double reduction theory. The theory presents
a powerful tool for obtaining invariant solutions of systems of PDEs from the association
of symmetries of the system with its nontrivial conservation laws. Given a system of qth-
order PDEs with n independent and m dependent variables, the theory provides for the
reduction of the system to a system of (q − 1)th-order ODEs.

We have applied the generalised double reduction theory to two (2+1)-dimensional par-
tial differential equations, a nonlinear wave equation and the Zakharov-Kuznetsov equa-
tion. For the nonlinear wave equation, we extended the application presented in the
seminal paper by Bokhari et al [6] on the generalised double reduction theory. We pre-
sented a more detailed account of the application and also determined and used additional
inherited symmetries. We furthermore presented exact solutions of the equation for spe-
cific prescriptions of the arbitrary functions in the equation. As for the ZK equation, five
Lie point symmetries of the equation were determined along with four nontrivial conser-
vation laws. Upon establishing association of the conservation laws with the symmetries,
multi-reductions were performed for each of the conservation laws.

Overall, our study has provided a comprehensive guide to using the generalised double
reduction method. We have presented clear illustrative examples and included each key
step of the method.

Author Contributions

Conceptualization, M.C.K. and W.S.; methodology, M.C.K., W.S. and P.M.; soft-
ware, M.C.K. and W.S.; validation, W.S. and P.M.; formal analysis, M.C.K., W.S. and
P.M.; writing—original draft preparation, M.C.K. and W.S.; writing—review and editing,
M.C.K., W.S. and P.M. All authors have read and agreed to the published version of the
manuscript.

Funding

This research received no external funding.



M. C. Kakuli, W. Sinkala, P. Masemola / Eur. J. Pure Appl. Math, 18 (1) (2025), 5371 20 of 23

Acknowledgements

The authors would like to thank the Directorate of Research Development and Inno-
vation of Walter Sisulu University for continued financial support

Conflicts of Interest

The authors declare no conflict of interest.

Data Availability Statement

No data were created or analyzed in this study.

References

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[2] S.C. Anco and M.l. Gandarias. Symmetry multi-reduction method for partial dif-
ferential equations with conservation laws. Commun. Nonlinear Sci. Numer. Simul.,
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