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Conservation of the Cylindrical and Elliptic
Cylindrical K-P Equations
Joseph Boon Zik Hong
Univeriti Teknologi Malaysia

Kamran Fakhar
Kwantlen Polytechnic University

S. Ahmad
Universiti Teknologi Malaysia

A. H. Kara
University of the Witwatersrand

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following peer review. The definitive publisher-authenticated version: Hong, J. B.Z., Fakhar, K., Ahmad, S. and A.H. Kara (2015).
Conservation of the cylindrical and elliptic cylindrical K-P equations. Applied Mathematics & Information Sciences, 9 (2), 631-635, is
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Appl. Math. Inf. Sci. .., No. ., 1-5 (2014)

1

Applied Mathematics & Information Sciences
An International Journal
c 2014 NSP
⃝
Natural Sciences Publishing Cor.

Conservation of the cylindrical and elliptic cylindrical
K-P equations
B Z H Joseph1 , K Fakhar2 , S Ahmad1 and A H Kara3,4
1 Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81300 Skudai, Johor, Malaysia
2 Mathematics Department, University of British Columbia, Vancouver, Canada
3 School of Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa
4 Dept of Maths and Stats, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

Abstract: We study the invariance properties and exact solutions of the Kadomtsev-Petviashvili equation and construct its conservation
laws and that of its transformed elliptic and elliptic-cylindrical versions. Then, it is shown how the conservation laws and related
quantities of the transformed versions may be attained by applying the transformation variables as opposed to independent calculations
which are often cumbersome for high order partial differential equations of ‘many’ variables.
Keywords: cylindrical and elliptic cylindrical Kadomtsev-Petviashvili equation ; Lie symmetries; reduction; conservation laws

with h = h(T, ζ , ν ) lead to the elliptic cylindrical KP
equation (ecKP)

1 Introduction
The Kadomtsev-Petviashvili (KP) equations originates in
the study of the surface wave problems for an
incompressible fluid described by the full set of Euler
equations with free surface and rigid horizontal bottom
boundary conditions. It has been extensively studied in a
number of papers ([1–5], inter alia, and references
therein). After some well known adjustments, the
equation takes the form
(ut + uux + uxxx )x + 3s2 uyy = 0.

(hT + 6hhζ + hζ ζ ζ + 2(T 2T−a2 ) h − 12s2a(Tν2 −a2 ) hζ )ζ
2 2

2

+ (T 23s−a2 ) hνν = 0.

For a detailed account on the nature, application and
reasons for the respective transformations, we refer the
reader to [6]. The inverse transformations in the
respective cases above are

(1)

t = τ,

The transformation

τ = t,

y2
χ = x+
,
12s2t

(2)

w
3s2
)χ + 2 wvv = 0
2τ
τ

(3)

and the transformation
T = t,

ζ = x+

ty2
12s2 (t 2 − a2 )

,

x=χ−

τ v2
,
12s2

y = τv

(6)

√
T 2 − a2 ν .

(7)

and
y
v=
t

leads to the cylindrical KP (cKP) equation ([6,7]), with
w = w(τ , χ , v),
(wτ + 6wwχ + wχ χ χ +

(5)

y
ν=√
t 2 − a2
(4)

t = T,

x=ζ−

T ν2
,
12s2

y=

In this paper, we, firstly study the invariance
properties of equation (1) and show how this lead to exact
solutions. That is, we determine the one parameter Lie
groups of transformations (Lie point symmetry
generators) to successively reduce the equation. We then
construct the conserved vectors of the equation using the
method of multipliers and the homotopy operator. Next,
we list, independently, the conservation laws of the
transformed equations cKP and ecKP. As a final emphasis
of the study, we show that the conserved vectors of the

∗ Corresponding author e-mail: Abdul.Kara@wits.ac.za
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BZH Joseph et al. : Conservation laws of the cylindrical and elliptic cylindrical K-P equation...

cKP and ecKP are, in fact, obtainable from the
transformed variables. Since, there is a one to one
correspondence between the multipliers and conserved
vectors, we need only perform the transformation on the
multipliers. Thus, all the properties of the KP obtained via
a study of the conservation laws are, equivalently,
obtainable for the cKP and ecKP. These include
integrability, convergence, conserved quantities and so on
([8]).

2 Symmetries, reductions and conservation
laws - KP
The Lie symmetry approach on differential equations is
well known; for details see e.g., [9,10].
We present some of the definitions and notations
below. Intrinsic to a Lie algebraic treatment of differential
equations is the universal space A (see [10]).The space
A is the vector space of all differential functions of all
finite orders and forms an algebra. Consider an rth-order
system of partial differential equations of n independent
variables x = (x1 , x2 , . . . , xn ) and m dependent variables
u = (u1 , u2 , . . . , um )
Gµ (x, u, u(1) , . . . , u(r) ) = 0, µ = 1, . . . , m̃,

(8)

where u(1) , u(2) , . . . , u(r) denote the collections of all first,
second, . . ., rth-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
i
α
∂x
∂u
∂uj

(9)

where the summation convention is used whenever
appropriate. A current T = (T 1 , . . . , T n ) is conserved if
it satisfies
Di T i = 0
(10)
along the solutions of (8). It can be shown that every
admitted conservation law arises from multipliers
Qµ (x, u, u(1) , . . .) such that
Q µ G µ = Di T i

(11)

holds identically (that is, off the solution space) for some
current T . The conserved vector may then be obtained
by the homotopy operator (see [8,11]). Other works on
symmetries and conservation laws can be found in [12,13].
Definition A Lie-Bäcklund operator is given by
X = ξi

∂
∂
∂
,
+ ηα
+ ∑ ηiα1 ...is
i
α
∂x
∂u
∂ uαi1 ...is
s≥1

(12)

where ξ i , η α ∈ A and the additional coefficients are
determined uniquely by the prolongation formulae

ηiα

= Di (W α ) + ξ j uαi j ,

ηiα1 ...is = Di1 . . . Dis (W α ) + ξ j uαji1 ...is ,
c 2014 NSP
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(13)
s > 1.

In (13), W α is the Lie characteristic function given by
W α = η α − ξ j uαj .

(14)

A Lie symmetry generator of (8) is a one parameter
Lie group transformation that leaves the given differential
equation invariant under the transformation of all
independent variables and dependent variables. In this
paper, we will assume that X is a Lie point operator, i.e.,
ξ and η are functions of x and u and are independent of
derivatives of u. A Lie-Bäcklund operator of the form
X̃ = η α ∂ /∂ uα + · · · is called a canonical or evolutionary
representation of X.

2.1 Symmetries and reductions
A one parameter Lie group of transformations that leave
invariant (1) will be written as a vector field
X = τ (t, x, y, u)∂t + ξ (t, x, y, u)∂x + η (t, x, y, u)∂u
+ϕ (t, x, y, u)∂u .

(15)

This would be a generator of point symmetry of the
system. The tedious calculations reveal the following
point symmetry generators
X1 = 6F1 (t)∂x + F1′ ∂u ,
X2 = −36F2 (t)∂y + s62 yF2′ ∂x + s12 yF2′′ ∂u ,
X3 = 108F3 (t)∂t + (36xF3′ − s62 y2 F3′′ )∂x + 72yF3′ ∂y
+(6xF3′′ − 72uF3′ − s12 y2 F3′′′ )∂u
(16)
where the Fi s are arbitrary functions of t. In X3 , for e.g.,
F3 = t leads to the scaling transformation
3t ∂t + x∂x + 2y∂y − 2u∂u . Also, we can easily show that a
linear combination of the appropriate choice of Fi s lead
to the translations
X1 = ∂t ,

X2 = ∂x ,

X3 = ∂y .

The first two, in the first instance yield the transformation
α = x − ct, y = y and u = u, so that equation (1) become
−cuαα + 6uα2 + 6uuαα + uαααα + 3s2 uyy = 0,

(17)

which admits a Lie point symmetry generator X1 = ∂α and
X2 = ∂y . These symmetries yield the transformation γ =
y − kα and u = u. The equation (17) under new invariant
transformation becomes
(−ck2 + 3s2 + 6k2 u)uγγ + 6k2 uγ2 + k4 uγγγγ = 0.

(18)

By integrating equation (18) with respect to γ twice, we
have
(−ck2 + 3s2 )u + 3k2 u2 + k4 u′′ = bγ + d,

(19)

where b and d are integrating constant. For b ̸= 0, equation
(19) have no symmetry generator, which implies no further

Appl. Math. Inf. Sci. .., No. ., 1-5 (2014) / www.naturalspublishing.com/Journals.asp

reduction is possible. To this end, we consider b = 0 with
d ̸= 0. Suppose H = u, K = u′ and hence

implies
√ √
1
du
2
= 2 du − (−ck2 + 3s2 )u2 − k2 u3 ,
dγ
k
2
and consequently the solution of equation (1) is given as
1

√

du− 12 (−ck2 +3s2 )u2 −k2 u3
√
= k22 [y − k(x − ct)]

du
(20)

2.2 Conservation Laws
The conserved vectors
(T t , T x , T y ), i.e.,

of

(1)

will

be

written

Dt T t + Dx T x + Dy T y = 0

3 Conservation laws - cKP and ecKP
Here, firstly, we use a combination of the multiplier and
homotopy approach, as in the previous section, to obtain
the multipliers and their corresponding conservation laws
for cKP and ecKP, respectively. As would be expected,
they admit four independent conservation laws given
below.
The CKP admits the following four conserved vectors
(i)

dK
u′′
d − (−ck2 + 3s2 )H − 3k2 H 2
= ′ =
,
dH
u
k4 K

∫

3

(21)

′

1
3 2 1
T1τ = 72s
2 [−2v τ G wχ
1
2
+G (−36s vw + v(v2 τ + 36s2 χ )wχ )],
′′
′
χ
T1 = 72s12 τ v(τ (w(2v2 τ 2 G1 + G1 (v2 τ − 36s2 χ
′
−24v2 τ 2 wχ )) − 2v2 τ 2 G1 (wτ + 2wχ χ χ ))
+G1 (−216s2 τ w2 + 12w(3s2 χ + τ (v2 τ + 36s2 χ )wχ )
+τ ((v2 τ + 36s2 χ )wτ
−72s2 wχ χ + 2(v2 τ + 36s2 χ )wχ χ χ ))),
′
v
T1 = 121τ 2 [−2v2 τ 2 G1 (−3w + vwv )
+G1 (−3(v2 τ + 12s2 χ )w + v(v2 τ + 36s2 χ )wv )].

(ii)
′

along the solutions of the differential equation. By
omitting the calculation details, we directly write below
the set of multipliers Q and their corresponding conserved
vectors for the system (1).
1
2 1
2
1
(i) Q1 = − 18s
2 y(y Ft − 18s xF (t))
′

1
2 1
2 1
T1t = − 36s
2 y(y F ux + 18s F (u − xux )),
′′
1
2 1 + 6F 1 ′ (3s2 x + 2y2 u ))
T1x = − 36s
x
2 y(u(−y F
′
+y2 F 1 (ut + 2uxxx )
+18s2 F 1 (6u2 − xut − 12xuux + 2uxx − 2xuxxx )),
′
y
T1 = −3s2 xF 1 (u − yuy ) − 16 y2 F 1 (−3u + yuy )

1
2 2 2
T2τ = 24s
2 [−2v τ G w χ
2
2
+G (−12s w + (v2 τ + 12s2 χ )wχ )],
′′
χ
T2 = 24s12 τ (τ (w(2v2 τ 2 G2
′
+G2 (v2 τ − 12s2 χ − 24v2 τ 2 wχ ))
′
−2v2 τ 2 G2 (wτ + 2wχ χ χ ))
2
+G (−72s2 τ w2 + 12w(s2 χ + τ (v2 τ + 12s2 χ )wχ )
+τ ((v2 τ + 12s2 χ )wτ
−24s2 wχ χ + 2(v2 τ + 12s2 χ )wχ χ χ ))),
′
v
T2 = 41τ 2 [−2vτ 2 G2 (−2w + vwv )
2
+G (−2vτ w + (v2 τ + 12s2 χ )wv )].

(iii)

(ii) Q2 = − 6s12 (y2 Ft2 − 6s2 xF 2 (t))

T3τ = 12 vG3 wχ ,
′
χ
T3 = 21τ [v(−τ wG3 + G3 (w(1 + 12τ wχ )
+τ (wτ + 2wχ χ χ )))],
T3v = − τ12 (3s2 G3 (w − vwv )).

′
y2 F 2 u

1
T2t = 12
(− s2 x − 6F 2 (u − xux )),
1
x
2 2 ′′ + 6F 2 ′ (s2 x + 2y2 u ))
T2 = − 12s
x
2 [u(−y F
′
+y2 F 2 (ut + 2uxxx )
+6s2 F 2 (6u2 − xut − 12xuux + 2uxx − 2xuxxx )],
′
′
y
T2 = yuF 2 + 12 (6s2 xF 2 − y2 F 2 )uy

(iii) Q3 = yF 3 (t)
T3t = 12 yF 3 ux ,
′
T3x = 12 y(−u(F 3 − 12F 3 ux ) + F 3 (ut + 2uxxx )),
y
2
3
T3 = −3s F (u − yuy )
(iv) Q4 = F 4 (t)
T4t = 12 F 4 ux ,
4′

T4x = u(− F2 + 6F 4 ux ) + 12 F 4 (ut + 2uxxx )3s2 F 4 uy ,
T4y = 3s2 F 4 uy

(iv)

T4τ = 12 G4 wχ ,
′
χ
T4 = 21τ [−τ wG4 + G4 (w(1 + 12τ wχ )
+τ (wτ + 2wχ χ χ ))],
T4v = τ12 (3s2 G4 wv ).
The corresponding multipliers are
1
2
2 2 1
2
1
1
Q̄1 = 36s
2 v[36G (τ ) χ s − 2v τ Gτ + v τ G ],
1
2
2
2 2 2
2
2
Q̄2 = 12s
2 [12G (τ ) χ s − 2v τ Gτ + v τ G ]
.
3
Q̄3 = G (τ )v,
Q̄4 = G4 (τ )

Similarly, the corresponding multipliers and conserved
vectors for ecKP are

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BZH Joseph et al. : Conservation laws of the cylindrical and elliptic cylindrical K-P equation...

(i)
Q1
=
1
2 H 1 ν 2 + 2a2 H 1 ν 2 + 36H 1 (T )ζ s2 + T H 1 ν 2 ]
ν
[−2T
2
T
T
36s
1
1
2
2
1
2
T1T = 72s
2 [−36s ν hH + ν ((36s ζ + T ν )H
′
2
2
2
1
+2(a − T )ν H )hζ ],
ζ
T1 = 432s4 (a12 −T 2 ) ν (−1296a2 s4 h2 H 1
+1296s4 T 2 h2 H 1 − 6s2 h(H 1 (36s2 T ζ
+7a2 ν 2 − 12(a2 − T 2 )(36s2 ζ + T ν 2 )hζ )
′′
−(a2 − T 2 )(2(−a2 + T 2 )ν 2 H 1
′
+H 1 (−36s2 ζ + T ν 2 + 24(a2 − T 2 )ν 2 hζ )))
′
+2(a2 − T 2 )ν 2 H 1 (6s2 (a2 − T 2 )hT + a2 ν 2 hζ
+12s2 (a2 − T 2 )hζ ζ ζ )
+H 1 (6s2 (a2 − T 2 )(36s2 ζ
+T ν 2 )hT + a2 ν 2 (36s2 ζ
+T ν 2 )hζ + 12s2 (a2 − T 2 )(−36s2 hζ ζ
+(36s2 ζ + T ν 2 )hζ ζ ζ ))),
ν
T1 = 12(a21−T 2 (3h((12s2 ζ + T ν 2 )H 1
′

+2(a2 − T 2 )ν 2 H 1 ) − ν ((36s2 ζ
′
+T ν 2 )H 1 + 2(a2 − T 2 )ν 2 H 1 )hν )

(iv) Q4 = H 4 (T )
T4T = 12 H 4 hζ ,
′
ζ
T4 = 12s2 (a12 −T 2 ) [−6s2 h((a2 − T 2 )H 4
+H 4 (T − 12(a2 − T 2 )hζ )) + H 4 (6s2 (a2 − T 2 )hT
+a2 ν 2 hζ + 12s2 (a2 − T 2 )hζ ζ ζ )],
1
ν
2 4
T4 = − a2 −T
2 (3s H hν )

3.1 Conservation laws via transformations
The transformation (2) and transformation (4) transform
the equation (1) into cKP and ecKP, respectively. By
exploiting this fact, we show that one can avoid the
lengthly procedure and can directly obtain the
conservation laws by using these transformations. It is
sufficient to calculate the multipliers via the
transformations and as there is a one to one corresponding
between the conserve vector and the multiplier, the
conserved vectors can be constructed directly from the
homotopy integral. For illustration, we transform some
multipliers.
(a) Transformation of multipliers from KP to cKP
(i) The transformation of Q1 is

(ii)
1
2
2 2 2
2 2 2
2
Q2 = 12s
2 [−2T HT ν + 2a HT ν + 12H (T )ζ s
2
2
−T H ν ]

1
2 1
2
1
Q1 = − 18s
2 y(y Ft − 18s xF (t))

1
τv
2
1
2 2 1
= − 18s
2 (τ v)[τ v Fτ − 18s ( χ + 12s2 )F (τ )]
1
2
2
1
2
1
= − 36s2 (τ v)(2τ v Fτ − 36s χ F − 3τ v2 F 1 ).
2

If we let τ F 1 (τ ) = G1 (τ ), we get
1
2
2
2
2
2
T2T = 24s
2 [−12s hH + ((12s ζ + T ν )H
′
+2(a2 − T 2 )ν 2 H 2 )hζ ],
ζ
T2 = 144s4 (a12 −T 2 ) (−432a2 s4 h2 H 2 + 432s4 T 2 h2 H 2
−6s2 h(3H 2 (4s2 T ζ + a2 ν 2 − 4(a2 − T 2 )(12s2 ζ
′′
+T ν 2 )hζ ) − (a2 − T 2 )(2(−a2 + T 2 )ν 2 H 2
′
+H 2 (−12s2 ζ + T ν 2 + 24(a2 − T 2 )ν 2 hζ )))
′
+2(a2 − T 2 )ν 2 H 2 (6s2 (a2 − T 2 )hT + a2 ν 2 hζ
2
2
2
+12s (a − T )hζ ζ ζ )
+H 2 (6s2 (a2 − T 2 )(12s2 ζ + T ν 2 )hT + a2 ν 2 (12s2 ζ
+T ν 2 )hζ + 12s2 (a2 − T 2 )(−12s2 hζ ζ
+(12s2 ζ + T ν 2 )hζ ζ ζ ))),
′
T2ν = 4(a21−T 2 [2ν h(T H 2 + 2(a2 − T 2 )H 2 )
′

−((12s2 ζ + T ν 2 )H 2 + 2(a2 − T 2 )ν 2 H 2 )hν ]

(iii) Q3 = ν H 3 (T )
T3T = 21 ν H 3 hζ ,
′
ζ
T3 = 12s2 (a12 −T 2 ) [ν (−6s2 h((a2 − T 2 )H 3
+H 3 (T − 12(a2 − T 2 )hζ )) + H 3 (6s2 (a2 − T 2 )hT
+a2 ν 2 hζ + 12s2 (a2 − T 2 )hζ ζ ζ ))],
1
ν
2 3
T3 = a2 −T
2 (3s H (h − ν hν ))
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1
1 1
2 2 1 1
Q1 = − 36s
2 (τ v)(2τ v ( τ Gτ − τ 2 G )
2

1
2 1
− 36s
τ χ G − 3v G )
1
1
= 36s2 v[36G χ s2 − 2v2 τ 2 G1τ + v2 τ G1 ],

which matches Q̄1 of cKP.
(ii) The transformation of Q2 is
Q2 = − 6s12 (y2 Ft2 − 6s2 xF 2 (t))

τv
2
= − 6s12 ((τ v)2 Fτ2 − 6s2 (χ + 12s
2 )F (τ ))
2

τv
1
2 2 2
2
2
= 12s
2 [−2τ v Fτ + 12s ( χ + 12s2 )F ]
2

where, if we let F 2 (τ ) = G2 (τ ), we obtain
Q2 =

1
[12G2 (τ )χ s2 − 2v2 τ 2 G2τ + v2 τ G2 ],
12s2

which is the same as Q̄2 of cKP.
(b) Transformation of multipliers from KP to ecKP
The multiplier Q2 of KP leads to
Q2 = − 6s12 (y2 Ft2 − 6s2 xF 2 (t))
√
T ν2
2
= − 6s12 [( T 2 − a2 ν )2 FT2 − 6s2 (ζ − 12s
2 )F (T )]
1
2
2
2
2
2
2
= 12s2 (−2T ν FT + 2a ν FT
+12s2 ζ F 2 − T ν 2 F 2 )

Appl. Math. Inf. Sci. .., No. ., 1-5 (2014) / www.naturalspublishing.com/Journals.asp

Again, if we let F 2 (T ) = H 2 (T ), we obtain
1
2 2 2
2 2 2
Q2 = 12s
2 [−2T HT ν + 2a HT ν
2
2
2
2
+12H (T )ζ s − T H ν ],

which is Q2 of ecKP.
It is clear, therefore, that one can obtain the other
multipliers and hence the conservation laws for cKP and
ecKP from that of the KP.

4 Conclusion
In this study, via a knowledge of the Lie symmetry
generators, we have successively reduced the fourth order
KP to a first order ODE and further obtained an exact
solution. The conservation laws for KP, cKP, and ecKP
have been obtained independently. We have demonstrated
that the transformations used to transform KP to cKP and
ecKP can, in fact, be used to obtain the conservation laws
for cKP and ecKP. This fact is significant in the sense that
the interesting features, well known or otherwise, of the
KP, like the exact solutions, multipliers and conservation
laws can easily be achieved for the cKP and ecKP.
Acknowledgement AHK thanks the African Institute
of Mathematics (AIMS) for providing facilities to carry
out the research.

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(2002), 60-72.

5

Kamran
Fakhar
obtained
his
PhD in 2005. He is a fellow
of Interdisciplinary Center for
Theoretical Studies at China.
He is a recipient numerous
awards, scholarships and
merit certificates. Currently
he is working as a research
associate
at
Department
of Mathematics, University of
British Columbia, Canada. Dr. Kamran’s research
interests are Symmetries and differential equations,
Conservation laws, Fluid Mechanics, Engineering
Mathematics; he has published numerous articles in these
areas. He has supervised a number of MSc/PhD students.
Abdul Hamid Kara
is a professor in the School of
Mathematics at the University
of the Witwatersrand in
Johannesburg, South Africa
from where he obtained
his PhD. His field of expertise
involves
the
analysis
of differential equations
using invariance properties
and conservation laws. He has
published, collaboratively, in the areas of mathematical
physics, relativity and cosmology, applications of
differential geometry, classification of Lagrangians, inter
alia. Prof Kara has supervised a number of MSc/PhD
students and is an NRF rated researcher in South Africa.

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