Research Article On the Solutions and Conservation Laws of a Coupled
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Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 741780, 7 pages
http://dx.doi.org/10.1155/2013/741780
Research Article
On the Solutions and Conservation Laws of a Coupled
Kadomtsev-Petviashvili Equation
Chaudry Masood Khalique
Department of Mathematical Sciences, International Institute for Symmetry Analysis and Mathematical Modelling,
North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa
Correspondence should be addressed to Chaudry Masood Khalique; masood.khalique@nwu.ac.za
Received 6 October 2012; Accepted 2 December 2012
Academic Editor: Asghar Qadir
Copyright © 2013 Chaudry Masood Khalique. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
A coupled Kadomtsev-Petviashvili equation, which arises in various problems in many scientific applications, is studied. Exact
solutions are obtained using the simplest equation method. The solutions obtained are travelling wave solutions. In addition, we
also derive the conservation laws for the coupled Kadomtsev-Petviashvili equation.
1. Introduction
In this paper, we study a new coupled KP equation [10]:
The well-known Korteweg-de Vries (KdV) equation [1]
π’π‘ + 6π’π’π₯ + π’π₯π₯π₯ = 0
(1)
governs the dynamics of solitary waves. Firstly, it was derived
to describe shallow water waves of long wavelength and small
amplitude. It is a crucial equation in the theory of integrable
systems because it has infinite number of conservation laws,
gives multiple-soliton solutions, and has many other physical
properties. See, for example, [2] and references therein.
An essential extension of the KdV equation is the
Kadomtsev-Petviashvili (KP) equation given by [3]
(π’π‘ + 6π’π’π₯ + π’π₯π₯π₯ )π₯ + π’π¦π¦ = 0.
(2)
This equation models shallow long waves in the π₯-direction
with some mild dispersion in the π¦-direction. The inverse
scattering transform method can be used to prove the
complete integrability of this equation. This equation gives
multiple-soliton solutions.
Recently, the coupled Korteweg-de Vries equations and
the coupled Kadomtsev-Petviashvili equations, because of
their applications in many scientific fields, have been the
focus of attention for scientists and as a result many studies
have been conducted [4–9].
7
5
(π’π‘ + π’π₯π₯π₯ − π’π’π₯ − π£π£π₯ + (π’π£)π₯ ) + π’π¦π¦ = 0,
4
4
π₯
(3a)
5
7
(π£π‘ + π£π₯π₯π₯ − π’π’π₯ − π£π£π₯ + 2(π’π£)π₯ ) + π£π¦π¦ = 0,
4
4
π₯
(3b)
and find exact solutions of this equation. The method that
is employed to obtain the exact solutions for the coupled
Kadomtsev-Petviashvili equation ((3a) and (3b)) is the simplest equation method [11, 12]. Secondly, we derive conservation laws for the system ((3a) and (3b)) using the multiplier
approach [13, 14].
The simplest equation method was introduced by
Kudryashov [11] and later modified by Vitanov [12]. The
simplest equations that are used in this method are the
Bernoulli and Riccati equations. This method provides a
very effective and powerful mathematical tool for solving
nonlinear equations in mathematical physics.
Conservation laws play a vital role in the solution process
of differential equations (DEs). The existence of a large
number of conservation laws of a system of partial differential
equations (PDEs) is a strong indication of its integrability
[15]. A conserved quantity was utilized to find the unknown
exponent in the similarity solution which could not have been
obtained from the homogeneous boundary conditions [16].
2
Journal of Applied Mathematics
Also recently, conservation laws have been employed to find
solutions of the certain PDEs [17–19].
The outline of the paper is as follows. In Section 2, we
obtain exact solutions of the coupled KP system ((3a) and
(3b)) using the simplest equation method. Conservation laws
for ((3a) and (3b)) using the multiplier method are derived
in Section 3. Finally, in Section 4 concluding remarks are
presented.
Here π»(π§) satisfies the Bernoulli and Riccati equations, π
is a positive integer that can be determined by balancing
procedure, and A0 , . . . , Aπ, B0 , . . . , Bπ are constants to
be determined. The solutions of the Bernoulli and Riccati
equations can be expressed in terms of elementary functions.
We first consider the Bernoulli equation:
π»σΈ (π§) = ππ» (π§) + ππ»2 (π§) ,
2. Exact Solutions of ((3a) and (3b)) Using
Simplest Equation Method
We first transform the system of partial differential equations
((3a) and (3b)) into a system of nonlinear ordinary differential
equations in order to derive its exact solutions.
The transformation
π’ = πΉ (π§) ,
π£ = πΊ (π§) ,
π§ = π‘ − ππ₯ + (π − 1) π¦, (4)
where π is a real constant, transforms ((3a) and (3b)) to the
following nonlinear coupled ordinary differential equations
(ODEs):
where π and π are constants. Its solution can be written as
π» (π§) = π {
cosh [π (π§ + πΆ)] + sinh [π (π§ + πΆ)]
}.
1 − π cosh [π (π§ + πΆ)] − π sinh [π (π§ + πΆ)]
(8)
Secondly, for the Riccati equation:
5
π4 πΉσΈ σΈ σΈ σΈ (π§) + π2 πΊ (π§) πΉσΈ σΈ (π§)
4
π»σΈ (π§) = ππ»2 (π§) + ππ» (π§) + π
7
− π2 πΉ (π§) πΉσΈ σΈ (π§) + π2 πΉσΈ σΈ (π§)
4
5
− 3ππΉσΈ σΈ (π§) + πΉσΈ σΈ (π§) + π2 πΉσΈ (π§) πΊσΈ (π§)
2
(5a)
7
5
− π2 πΉσΈ (π§)2 + π2 πΉ (π§) πΊσΈ σΈ (π§)
4
4
− π2 πΊ (π§) πΊσΈ σΈ (π§) − π2 πΊσΈ (π§)2 = 0,
4
σΈ σΈ σΈ σΈ
ππΊ
2
σΈ σΈ
(π§) + 2π πΊ (π§) πΉ (π§)
5
− π2 πΉ (π§) πΉσΈ σΈ (π§) + 4π2 πΉσΈ (π§) πΊσΈ (π§)
4
5
− π2 πΉσΈ (π§)2 + 2π2 πΉ (π§) πΊσΈ σΈ (π§)
4
7
− π2 πΊ (π§) πΊσΈ σΈ (π§) + π2 πΊσΈ σΈ (π§)
4
7
− 3ππΊσΈ σΈ (π§) + πΊσΈ σΈ (π§) − π2 πΊσΈ (π§)2 = 0.
4
We now use the simplest equation method [11, 12] to solve the
system ((5a) and (5b)) and as a result we will obtain the exact
solutions of our coupled KP system ((3a) and (3b)). We use
the Bernoulli and Riccati equations as the simplest equations.
We briefly recall the simplest equation method here. Let
us consider the solutions of ((5a) and (5b)) in the form
(9)
(π, π, and π are constants), we shall use the solutions
π» (π§) = −
π
π
1
−
tanh [ π (π§ + πΆ)] ,
2π 2π
2
π» (π§) = −
π
1
π
−
tanh ( ππ§)
2π 2π
2
+
(5b)
(7)
(10)
sech (ππ§/2)
,
πΆ cosh (ππ§/2) − (2π/π) sinh (ππ§/2)
where π2 = π2 − 4ππ > 0 and πΆ is a constant of integration.
2.1. Solutions of ((3a) and (3b)) Using the Bernoulli Equation
as the Simplest Equation. In this case the balancing procedure
yields π = 2 so the solutions of ((5a) and (5b)) are of the form
πΉ (π§) = A0 + A1 π» + A2 π»2 ,
πΊ (π§) = B0 + B1 π» + B2 π»2 .
(11)
π
πΉ (π§) = ∑Aπ (π» (π§))π ,
π=0
π
πΊ (π§) = ∑Bπ (π» (π§))π .
π=0
(6)
Substituting (11) into ((5a) and (5b)) and making use of the
Bernoulli equation (7) and then equating all coefficients of
the functions π»π to zero, we obtain an algebraic system of
equations in terms of A0 , A1 , A2 , B0 , B1 , and B2 .
Journal of Applied Mathematics
3
Solving the system of algebraic equations, with the aid of
Mathematica, we obtain
π = 1,
π = 3,
A0 =
π (π4 + π2 − 3π + 1)
π2
A1
340992
B2 =
× {9849A21 π3
− 3245184A0 π3 − 354564A0 A1 π3
,
+ 90144A1 π3 + 3752A21 π2
36A0 π4
A1 = 4
,
π + π2 − 3π + 1
− 1081728A0 π2 − 135072A0 A1 π2
A2 = 3A1 ,
B0 =
+ 30048A1 π2 + 11256A21 π
1
21312
− 50652A0 A1 π + 90144A1 π
− 25326A21 − 16884A0 A1
× {−49248A0 π9 − 8208A0 π8
− 240384A1 + 1665792} ,
+ 67392A20 π7 − 98496A0 π7
(12)
+ 11232A20 π6 + 279072A0 π6
− 75978A30 π5 + 67392A20 π5
where π is any root of 469π3 − 416π2 + 304π − 256 = 0.
Consequently, a solution of ((3a) and (3b)) is given by
− 98496A0 π5 − 12663A30 π4
− 190944A20 π4 + 270864A0 π4
− 67608A20 π3 − 492480A0 π3
+ 2814A0 A1 π3 − 22536A20 π2
+ 205200A0 π2 + 938A0 A1 π2
+ 2814A0 A1 π + 34704A0
− 7504A0 A1 + 41472π11
10
+ 6912π + 124416π
− 352512π + 186624π
+ π΅2 π2 {
− 884736π4 + 1181952π3
− 1645056π2 + 933120π − 172800} ,
A1
B1 =
1022976
×
− 3245184A0 π
3
− 354564A0 A1 π3 + 90144A1 π3
− 135072A0 A1 π + 30048A1 π
cosh [π (π§ + πΆ)] + sinh [π (π§ + πΆ)]
}
1 − π cosh [π (π§ + πΆ)] − π sinh [π (π§ + πΆ)]
2
cosh [π (π§ + πΆ)] + sinh [π (π§ + πΆ)]
},
1 − π cosh [π (π§ + πΆ)] − π sinh [π (π§ + πΆ)]
(13a)
cosh [π (π§ + πΆ)] + sinh [π (π§ + πΆ)]
}
1 − π cosh [π (π§ + πΆ)] − π sinh [π (π§ + πΆ)]
2
cosh [π (π§ + πΆ)] + sinh [π (π§ + πΆ)]
},
1 − π cosh [π (π§ + πΆ)] − π sinh [π (π§ + πΆ)]
(13b)
where π§ = π‘ − ππ₯ + (π − 1)π¦ and πΆ is a constant of integration.
2.2. Solutions of ((3a) and (3b)) Using Riccati Equation as the
Simplest Equation. The balancing procedure gives π = 2 so
the solutions of ((5a) and (5b)) are of the form
πΉ (π§) = A0 + A1 π» + A2 π»2 ,
+ 3752A21 π2 − 1081728A0 π2
2
+ π΄ 2 π2 {
= π΅0 + π΅1 π {
7
− 705024π6 + 1285632π5
{9849A21 π3
= π΄ 0 + π΄ 1π {
π£ (π‘, π₯, π¦)
9
8
π’ (π‘, π₯, π¦)
2
+ 11256A21 π − 50652A0 A1 π
+ 90144A1 π − 25326A21
− 16884A0 A1 − 240384A1 + 1665792} ,
πΊ (π§) = B0 + B1 π» + B2 π»2 .
(14)
Substituting (14) into ((5a) and (5b)) and making use of the
Riccati equation (9), we obtain algebraic system of equations
in terms of A0 , A1 , A2 , B0 , B1 , and B2 by equating all
coefficients of the functions π»π to zero.
4
Journal of Applied Mathematics
π’ (π‘, π₯) =π΄ 0 + π΄ 1 {−
Solving the algebraic equations one obtains
π = −1,
+
2
A0 = π (8ππ + π + 5) ,
A1 =
12πA0 π
,
8ππ + π2 + 5
+ π΄ 2 {−
πA1
A2 =
,
π
+
B0 = 3 (−2048π2 ππ − 256ππ3 + 208πA0 π
B1 =
B2 =
A1 (336ππ − 29A1 )
,
192ππ − 6A1
2
sech (ππ§/2)
},
πΆ cosh (ππ§/2)−(2π/π) sinh (ππ§/2)
(17a)
+
1
17760000
+ π΅2 {−
× {−270144π2 A20 A1 ππ
+
(15)
+ 1080576π2 A20 A2 π2 + 50652πA20 A1 π3
− 84420πA20 A1 π + 5784000πA1 π
sech (ππ§/2)
}
πΆ cosh (ππ§/2) − (2π/π) sinh (ππ§/2)
π
π
1
−
tanh ( ππ§)
2π 2π
2
π£ (π‘, π₯) =π΅0 + π΅1 {−
−1
− 1280ππ + 15A0 A1 ) × (74A1 ) ,
π
π
1
−
tanh ( ππ§)
2π 2π
2
π
π
1
−
tanh ( ππ§)
2π 2π
2
sech (ππ§/2)
}
πΆ cosh (ππ§/2)−(2π/π) sinh (ππ§/2)
π
π
1
−
tanh ( ππ§)
2π 2π
2
2
sech (ππ§/2)
},
πΆ cosh (ππ§/2) − (2π/π) sinh (ππ§/2)
(17b)
where π§ = π‘ − ππ₯ + (π − 1)π¦ and πΆ is a constant of integration.
A profile of the solution ((13a) and (13b)) is given in
Figure 1. The flat peaks appearing in the figure are an artifact
of Mathematica and they describe the singularities of the
solution.
− 751200πA0 A1 π − 3552000ππB1
+ 1350720πA20 A2 π + 6009600πA0 A2 π
+ 70350A0 A21 + 313000A21
− 422100A20 A2 − 3756000A0 A2
3. Conservation Laws of ((3a) and (3b))
+ 28920000A2 − 4221A0 A21 π4
In this section we present conservation laws for the coupled
KP system ((3a) and (3b)) using the multiplier method [13,
14]. First we present some preliminaries which we will need
later in this section.
− 938A31 π3 π − 14070A0 A21 π2
+ 62600A21 π2 − 480256A32 π4
− 4006400A22 π2 − 1800960A0 A22 π2
+ 112560A21 A2 π2 } ,
where π is any root of 469π3 − 416π2 + 304π − 256 and hence
solutions of ((3a) and (3b)) are
π’ (π‘, π₯, π¦) =π΄ 0 + π΄ 1 {−
+ π΄ 2 {−
π
π
1
−
tanh [ π (π§ + πΆ)]}
2π 2π
2
2
π
π
1
−
tanh [ π (π§ + πΆ)]} ,
2π 2π
2
(16a)
π
π
1
tanh [ π (π§ + πΆ)]}
π£ (π‘, π₯, π¦) = π΅0 + π΅1 {− −
2π 2π
2
+ π΅2 {−
2
π
π
1
−
tanh [ π (π§ + πΆ)]} ,
2π 2π
2
(16b)
3.1. Preliminaries. We briefly present the notation and pertinent results which we utilize below. For details the reader is
referred to [20].
Consider a πth-order system of PDEs of π-independent
variables π₯ = (π₯1 , π₯2 , . . . , π₯π ) and π-dependent variables π’ =
(π’1 , π’2 , . . . , π’π ):
πΈπΌ (π₯, π’, π’(1) , . . . , π’(π) ) = 0,
πΌ = 1, . . . , π,
(18)
where π’(1) , π’(2) , . . . , π’(π) denote the collections of all first,
second,. . ., πth-order partial derivatives, that is, π’ππΌ =
π·π (π’πΌ ), π’πππΌ = π·π π·π (π’πΌ ), . . ., respectively, with the total
derivative operator with respect to π₯π given by
π·π =
π
π
π
+ π’ππΌ πΌ + π’πππΌ πΌ + ⋅ ⋅ ⋅ ,
ππ₯π
ππ’
ππ’π
π = 1, . . . , π,
(19)
where the summation convention is used whenever appropriate.
Journal of Applied Mathematics
π₯
5
−2
0
2
25
20
π’ 15
10
6
π£ 4
2
5
−2
−2
−2
π¦
0
0
π¦ 0
2
2
(a)
π₯
2
(b)
Figure 1: Profile of the travelling wave solution ((13a) and (13b)).
The Euler-Lagrange operator, for each πΌ, is given by
π
πΏ
π
=
+ ∑(−1)π π·π1 , . . . , π·ππ πΌ
,
πΏπ’πΌ ππ’πΌ π ≥1
ππ’π1 π2 ,...,ππ
coupled KP system ((3a) and (3b)), we obtain the zerothorder multipliers (with the aid of GeM [21]), Λ 1 (π‘, π₯, π¦, π’, π£),
Λ 2 (π‘, π₯, π¦, π’, π£) that are given by
(20)
Λ 1 = π3 (π‘) + π¦π4 (π‘) − π¦2 π7σΈ (π‘) + 2π₯π7 (π‘)
πΌ = 1, . . . , π.
+ π¦3 (−π8σΈ (π‘)) + 6π₯π¦π8 (π‘) ,
The π-tuple vector π = (π1 , π2 , . . . , ππ ), ππ ∈ A, π =
1, . . . , π, where A is the space of differential functions, is a
conserved vector of (18) if ππ satisfies
σ΅¨
π·π ππ σ΅¨σ΅¨σ΅¨σ΅¨(18) = 0.
(21)
Equation (21) defines a local conservation law of system (18).
A multiplier Λ πΌ (π₯, π’, π’(1) , . . .) has the property that
Λ πΌ πΈπΌ = π·π ππ
(22)
Λ2 = −
π¦2 π1σΈ
(π‘) + 2π₯π1 (π‘) + π¦
3
(−π2σΈ
where ππ , π = 1, 2, . . . , 8 are arbitrary functions of π‘.
Corresponding to the above multipliers we have the
following eight local conserved vectors of ((3a) and (3b)):
π1π‘ =
1
{−2π1 (π‘) π£ + 2π₯π1 (π‘) π£π₯ − π¦2 π1σΈ (π‘) π£π₯ } ,
2
π1π₯ =
1
{−8π¦2 π1σΈ (π‘) π’π₯ π£ − 8π¦2 π1σΈ (π‘) π£π₯ π’
4
+ 16π₯π1 (π‘) π’π₯ π£ + 16π₯π1 (π‘) π£π₯ π’
πΏ (Λ πΌ πΈπΌ )
= 0.
πΏπ’πΌ
+ 7π¦2 π1σΈ (π‘) π£π₯ π£ − 14π₯π1 (π‘) π£π₯ π£
Once the multipliers are obtained the conserved vectors
are calculated via a homotopy formula [13, 14].
3.2. Construction of Conservation Laws for ((3a) and (3b)).
We now construct conservation laws for the coupled KP
system ((3a) and (3b)) using the multiplier method. For the
(π‘))
+ 6π₯π¦π2 (π‘) + π5 (π‘) + π¦π6 (π‘) ,
holds identically. In this paper, we will consider multipliers
of the zeroth order, that is, Λ πΌ = Λ πΌ (π‘, π₯, π¦, π’, π£). The
determining equations for the multiplier Λ πΌ are
(23)
(24)
+ 5π¦2 π1σΈ (π‘) π’π₯ π’ − 10π₯π1 (π‘) π’π₯ π’
− 16π1 (π‘) π’π£ + 5π1 (π‘) π’2 + 2π¦2 π1σΈ σΈ (π‘) π£
− 4π₯π1σΈ (π‘) π£ + 7π1 (π‘) π£2 − 8π1 (π‘) π£π₯π₯
+ 8π₯π1 (π‘) π£π₯π₯π₯ + 4π₯π1 (π‘) π£π‘
− 2π¦2 π1σΈ (π‘) π£π‘ − 4π¦2 π1σΈ (π‘) π£π₯π₯π₯ } ,
6
Journal of Applied Mathematics
π¦
π1 = 2π¦π1σΈ (π‘) π£ + 2π₯π1 (π‘) π£π¦ − π¦2 π1σΈ (π‘) π£π¦ ,
π2π‘ =
π6π₯ =
1
{−6π¦π2 (π‘) π£ + 6π₯π¦π2 (π‘) π£π₯ + π¦3 (−π2σΈ (π‘)) π£π₯ } ,
2
π2π₯ =
1
{8π¦π6 (π‘) π’π₯ π£ + 8π¦π6 (π‘) π£π₯ π’
4
− 5π¦π6 (π‘) π’π₯ π’ − 7π¦π6 (π‘) π£π₯ π£
1
{−8π¦3 π2σΈ (π‘) π’π₯ π£ − 8π¦3 π2σΈ (π‘) π£π₯ π’
4
− 2π¦π6σΈ (π‘) π£ + 4π¦π6 (π‘) π£π₯π₯π₯
+ 2π¦π6 (π‘) π£π‘ } ,
+ 48π₯π¦π2 (π‘) π’π₯ π£ + 48π₯π¦π2 (π‘) π£π₯ π’
π¦
π6 = π¦π6 (π‘) π£π¦ − π6 (π‘) π£,
+ 5π¦3 π2σΈ (π‘) π’π₯ π’ − 30π₯π¦π2 (π‘) π’π₯ π’
+ 7π¦3 π2σΈ (π‘) π£π₯ π£ − 42π₯π¦π2 (π‘) π£π₯ π£
π7π‘ =
− 48π¦π2 (π‘) π’π£ + 15π¦π2 (π‘) π’2
+
2π¦3 π2σΈ σΈ
(π‘) π£ −
12π₯π¦π2σΈ
1
{−2π7 (π‘) π’ + 2π₯π7 (π‘) π’π₯ − π¦2 π7σΈ (π‘) π’π₯ } ,
2
π7π₯ =
(π‘) π£
1
{−5π¦2 π7σΈ (π‘) π’π₯ π£ − 5π¦2 π7σΈ (π‘) π£π₯ π’
4
+ 21π¦π2 (π‘) π£2 − 24π¦π2 (π‘) π£π₯π₯
+ 10π₯π7 (π‘) π’π₯ π£ + 10π₯π7 (π‘) π£π₯ π’
+ 24π₯π¦π2 (π‘) π£π₯π₯π₯ + 12π₯π¦π2 (π‘) π£π‘
+ 7π¦2 π7σΈ (π‘) π’π₯ π’ − 14π₯π7 (π‘) π’π₯ π’
− 2π¦3 π2σΈ (π‘) π£π‘ − 4π¦3 π2σΈ (π‘) π£π₯π₯π₯ } ,
+ 4π¦2 π7σΈ (π‘) π£π₯ π£ − 8π₯π7 (π‘) π£π₯ π£
π¦
π2 = 3π¦2 π2σΈ (π‘) π£ − 6π₯π2 (π‘) π£
− 10π7 (π‘) π’π£ + 2π¦2 π7σΈ σΈ (π‘) π’
+ 6π₯π¦π2 (π‘) π£π¦ − π¦3 π2σΈ (π‘) π£π¦ ,
π3π‘ =
π3π₯ =
− 4π₯π7σΈ (π‘) π’ + 7π7 (π‘) π’2
1
π (π‘) π’π₯ ,
2 3
+ 4π7 (π‘) π£2 − 8π7 (π‘) π’π₯π₯
1
{5π3 (π‘) π’π₯ π£ + 5π3 (π‘) π£π₯ π’
4
+ 8π₯π7 (π‘) π’π₯π₯π₯ + 4π₯π7 (π‘) π’π‘
− 2π¦2 π7σΈ (π‘) π’π‘ − 4π¦2 π7σΈ (π‘) π’π₯π₯π₯ } ,
− 7π3 (π‘) π’π₯ π’ − 4π3 (π‘) π£π₯ π£
− 2π3σΈ (π‘) π’ + 4π3 (π‘) π’π₯π₯π₯ + 2π3 (π‘) π’π‘ } ,
π¦
π3 = π3 (π‘) π’π¦ ,
1
π4π‘ = π¦π4 (π‘) π’π₯ ,
2
π4π₯ =
1
{5π¦π4 (π‘) π’π₯ π£ + 5π¦π4 (π‘) π£π₯ π’
4
π¦
π7 = 2π¦π7σΈ (π‘) π’ + 2π₯π7 (π‘) π’π¦ − π¦2 π7σΈ (π‘) π’π¦ ,
π8π‘ =
1
{−6π¦π8 (π‘) π’ + 6π₯π¦π8 (π‘) π’π₯ − π¦3 π8σΈ (π‘) π’π₯ } ,
2
π8π₯ =
1
{−5π¦3 π8σΈ (π‘) π’π₯ π£ − 5π¦3 π8σΈ (π‘) π£π₯ π’
4
+ 30π₯π¦π8 (π‘) π’π₯ π£ + 30π₯π¦π8 (π‘) π£π₯ π’
− 7π¦π4 (π‘) π’π₯ π’ − 4π¦π4 (π‘) π£π₯ π£ − 2π¦π4σΈ π’
+ 7π¦3 π8σΈ (π‘) π’π₯ π’ − 42π₯π¦π8 (π‘) π’π₯ π’
+ 4π¦π4 (π‘) π’π₯π₯π₯ + 2π¦π4 (π‘) π’π‘ } ,
π¦
π4
+ 4π¦3 π8σΈ (π‘) π£π₯ π£ − 24π₯π¦π8 (π‘) π£π₯ π£
= π¦π4 (π‘) π’π¦ − π4 (π‘) π’,
− 30π¦π8 (π‘) π’π£ + 2π¦3 π8σΈ σΈ π’
1
π5π‘ = π5 (π‘) π£π₯ ,
2
π5π₯ =
− 12π₯π¦π8σΈ (π‘) π’ + 21π¦π8 (π‘) π’2
1
{8π5 (π‘) π’π₯ π£ + 8π5 (π‘) π£π₯ π’
4
+ 12π¦π8 (π‘) π£2 − 24π¦π8 (π‘) π’π₯π₯
+ 24π₯π¦π8 (π‘) π’π₯π₯π₯ + 12π₯π¦π8 (π‘) π’π‘
− 5π5 (π‘) π’π₯ π’ − 7π5 (π‘) π£π₯ π£ − 2π5σΈ (π‘) π£
+ 4π5 (π‘) π£π₯π₯π₯ + 2π5 (π‘) π£π‘ } ,
π¦
π5 = π5 (π‘) π£π¦ ,
1
π6π‘ = π¦π6 (π‘) π£π₯ ,
2
− 2π¦3 π8σΈ (π‘) π’π‘ − 4π¦3 π8σΈ (π‘) π’π₯π₯π₯ } ,
π¦
π8 = 3π¦2 π8σΈ (π‘) π’ − 6π₯π8 (π‘) π’
+ 6π₯π¦π8 (π‘) π’π¦ − π¦3 π8σΈ (π‘) π’π¦ .
(25)
Journal of Applied Mathematics
We note that because of the arbitrary functions ππ , π =
1, 2, . . . , 8 in the multipliers, we obtain an infinitely many
conservation laws for the coupled KP system ((3a) and (3b)).
4. Concluding Remarks
The coupled Kadomtsev-Petviashvili system ((3a) and (3b))
was studied in this paper. The simplest equation method
was used to obtain travelling wave solutions of the coupled
KP system ((3a) and (3b)). The simplest equations that were
used in the solution process were the Bernoulli and Riccati
equations. However, it should be noted that the solutions
((13a) and (13b)), ((16a) and (16b)), and ((17a) and (17b))
obtained by using these simplest equations are not connected
to each other. We have checked the correctness of the
solutions obtained here by substituting them back into the
coupled KP system ((3a) and (3b)). Furthermore, infinitely
many conservation laws for the coupled KP system ((3a)
and (3b)) were derived by employing the multiplier method.
The importance of constructing the conservation laws was
discussed in the introduction.
Acknowledgments
C. M. Khalique would like to thank the Organizing Committee of Symmetries, Differential Equations, and Applications:
Galois Bicentenary (SDEA2012) Conference for their kind
hospitality during the conference.
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