On the Solutions and Conservation Laws of a Coupled Kadomtsev

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; [email protected] 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] 𝑢𝑡 + 6𝑢𝑢𝑥 + 𝑢𝑥𝑥𝑥 = 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] (𝑢𝑡 + 6𝑢𝑢𝑥 + 𝑢𝑥𝑥𝑥 )𝑥 + 𝑢𝑦𝑦 = 0.

(2)

This equation models shallow long waves in the 𝑥-direction with some mild dispersion in the 𝑦-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 (𝑢𝑡 + 𝑢𝑥𝑥𝑥 − 𝑢𝑢𝑥 − 𝑣𝑣𝑥 + (𝑢𝑣)𝑥 ) + 𝑢𝑦𝑦 = 0, 4 4 𝑥

(3a)

5 7 (𝑣𝑡 + 𝑣𝑥𝑥𝑥 − 𝑢𝑢𝑥 − 𝑣𝑣𝑥 + 2(𝑢𝑣)𝑥 ) + 𝑣𝑦𝑦 = 0, 4 4 𝑥

(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 𝐻(𝑧) satisfies the Bernoulli and Riccati equations, 𝑀 is a positive integer that can be determined by balancing procedure, and A0 , . . . , A𝑀, B0 , . . . , B𝑀 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:

𝐻󸀠 (𝑧) = 𝑎𝐻 (𝑧) + 𝑏𝐻2 (𝑧) ,

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 𝑢 = 𝐹 (𝑧) ,

𝑣 = 𝐺 (𝑧) ,

𝑧 = 𝑡 − 𝜌𝑥 + (𝜌 − 1) 𝑦, (4)

where 𝜌 is a real constant, transforms ((3a) and (3b)) to the following nonlinear coupled ordinary differential equations (ODEs):

where 𝑎 and 𝑏 are constants. Its solution can be written as

𝐻 (𝑧) = 𝑎 {

cosh [𝑎 (𝑧 + 𝐶)] + sinh [𝑎 (𝑧 + 𝐶)] }. 1 − 𝑏 cosh [𝑎 (𝑧 + 𝐶)] − 𝑏 sinh [𝑎 (𝑧 + 𝐶)] (8)

Secondly, for the Riccati equation:

5 𝜌4 𝐹󸀠󸀠󸀠󸀠 (𝑧) + 𝜌2 𝐺 (𝑧) 𝐹󸀠󸀠 (𝑧) 4

𝐻󸀠 (𝑧) = 𝑎𝐻2 (𝑧) + 𝑏𝐻 (𝑧) + 𝑐

7 − 𝜌2 𝐹 (𝑧) 𝐹󸀠󸀠 (𝑧) + 𝜌2 𝐹󸀠󸀠 (𝑧) 4 5 − 3𝜌𝐹󸀠󸀠 (𝑧) + 𝐹󸀠󸀠 (𝑧) + 𝜌2 𝐹󸀠 (𝑧) 𝐺󸀠 (𝑧) 2

(5a)

7 5 − 𝜌2 𝐹󸀠 (𝑧)2 + 𝜌2 𝐹 (𝑧) 𝐺󸀠󸀠 (𝑧) 4 4 − 𝜌2 𝐺 (𝑧) 𝐺󸀠󸀠 (𝑧) − 𝜌2 𝐺󸀠 (𝑧)2 = 0, 4

󸀠󸀠󸀠󸀠

𝜌𝐺

2

󸀠󸀠

(𝑧) + 2𝜌 𝐺 (𝑧) 𝐹 (𝑧)

5 − 𝜌2 𝐹 (𝑧) 𝐹󸀠󸀠 (𝑧) + 4𝜌2 𝐹󸀠 (𝑧) 𝐺󸀠 (𝑧) 4 5 − 𝜌2 𝐹󸀠 (𝑧)2 + 2𝜌2 𝐹 (𝑧) 𝐺󸀠󸀠 (𝑧) 4 7 − 𝜌2 𝐺 (𝑧) 𝐺󸀠󸀠 (𝑧) + 𝜌2 𝐺󸀠󸀠 (𝑧) 4 7 − 3𝜌𝐺󸀠󸀠 (𝑧) + 𝐺󸀠󸀠 (𝑧) − 𝜌2 𝐺󸀠 (𝑧)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)

(𝑎, 𝑏, and 𝑐 are constants), we shall use the solutions

𝐻 (𝑧) = −

𝑏 𝜃 1 − tanh [ 𝜃 (𝑧 + 𝐶)] , 2𝑎 2𝑎 2

𝐻 (𝑧) = −

𝜃 1 𝑏 − tanh ( 𝜃𝑧) 2𝑎 2𝑎 2

+

(5b)

(7)

(10)

sech (𝜃𝑧/2) , 𝐶 cosh (𝜃𝑧/2) − (2𝑎/𝜃) sinh (𝜃𝑧/2)

where 𝜃2 = 𝑏2 − 4𝑎𝑐 > 0 and 𝐶 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 𝑀 = 2 so the solutions of ((5a) and (5b)) are of the form

𝐹 (𝑧) = A0 + A1 𝐻 + A2 𝐻2 , 𝐺 (𝑧) = B0 + B1 𝐻 + B2 𝐻2 .

(11)

𝑀

𝐹 (𝑧) = ∑A𝑖 (𝐻 (𝑧))𝑖 , 𝑖=0 𝑀

𝐺 (𝑧) = ∑B𝑖 (𝐻 (𝑧))𝑖 . 𝑖=0

(6)

Substituting (11) into ((5a) and (5b)) and making use of the Bernoulli equation (7) and then equating all coefficients of the functions 𝐻𝑖 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 𝑎 = 1, 𝑏 = 3, A0 =

𝑘 (𝜌4 + 𝜌2 − 3𝜌 + 1) 𝜌2

A1 340992

B2 =

× {9849A21 𝜌3 − 3245184A0 𝜌3 − 354564A0 A1 𝜌3

,

+ 90144A1 𝜌3 + 3752A21 𝜌2

36A0 𝜌4 A1 = 4 , 𝜌 + 𝜌2 − 3𝜌 + 1

− 1081728A0 𝜌2 − 135072A0 A1 𝜌2

A2 = 3A1 , B0 =

+ 30048A1 𝜌2 + 11256A21 𝜌

1 21312

− 50652A0 A1 𝜌 + 90144A1 𝜌 − 25326A21 − 16884A0 A1

× {−49248A0 𝜌9 − 8208A0 𝜌8

− 240384A1 + 1665792} ,

+ 67392A20 𝜌7 − 98496A0 𝜌7

(12)

+ 11232A20 𝜌6 + 279072A0 𝜌6 − 75978A30 𝜌5 + 67392A20 𝜌5

where 𝑘 is any root of 469𝑘3 − 416𝑘2 + 304𝑘 − 256 = 0. Consequently, a solution of ((3a) and (3b)) is given by

− 98496A0 𝜌5 − 12663A30 𝜌4 − 190944A20 𝜌4 + 270864A0 𝜌4 − 67608A20 𝜌3 − 492480A0 𝜌3 + 2814A0 A1 𝜌3 − 22536A20 𝜌2 + 205200A0 𝜌2 + 938A0 A1 𝜌2 + 2814A0 A1 𝜌 + 34704A0 − 7504A0 A1 + 41472𝜌11 10

+ 6912𝜌 + 124416𝜌

− 352512𝜌 + 186624𝜌

+ 𝐵2 𝑎2 {

− 884736𝜌4 + 1181952𝜌3 − 1645056𝜌2 + 933120𝜌 − 172800} , A1 B1 = 1022976 ×

− 3245184A0 𝜌

3

− 354564A0 A1 𝜌3 + 90144A1 𝜌3

− 135072A0 A1 𝜌 + 30048A1 𝜌

cosh [𝑎 (𝑧 + 𝐶)] + sinh [𝑎 (𝑧 + 𝐶)] } 1 − 𝑏 cosh [𝑎 (𝑧 + 𝐶)] − 𝑏 sinh [𝑎 (𝑧 + 𝐶)]

2 cosh [𝑎 (𝑧 + 𝐶)] + sinh [𝑎 (𝑧 + 𝐶)] }, 1 − 𝑏 cosh [𝑎 (𝑧 + 𝐶)] − 𝑏 sinh [𝑎 (𝑧 + 𝐶)] (13a)

cosh [𝑎 (𝑧 + 𝐶)] + sinh [𝑎 (𝑧 + 𝐶)] } 1 − 𝑏 cosh [𝑎 (𝑧 + 𝐶)] − 𝑏 sinh [𝑎 (𝑧 + 𝐶)]

2 cosh [𝑎 (𝑧 + 𝐶)] + sinh [𝑎 (𝑧 + 𝐶)] }, 1 − 𝑏 cosh [𝑎 (𝑧 + 𝐶)] − 𝑏 sinh [𝑎 (𝑧 + 𝐶)] (13b)

where 𝑧 = 𝑡 − 𝜌𝑥 + (𝜌 − 1)𝑦 and 𝐶 is a constant of integration. 2.2. Solutions of ((3a) and (3b)) Using Riccati Equation as the Simplest Equation. The balancing procedure gives 𝑀 = 2 so the solutions of ((5a) and (5b)) are of the form 𝐹 (𝑧) = A0 + A1 𝐻 + A2 𝐻2 ,

+ 3752A21 𝜌2 − 1081728A0 𝜌2 2

+ 𝐴 2 𝑎2 {

= 𝐵0 + 𝐵1 𝑎 {

7

− 705024𝜌6 + 1285632𝜌5

{9849A21 𝜌3

= 𝐴 0 + 𝐴 1𝑎 {

𝑣 (𝑡, 𝑥, 𝑦)

9

8

𝑢 (𝑡, 𝑥, 𝑦)

2

+ 11256A21 𝜌 − 50652A0 A1 𝜌 + 90144A1 𝜌 − 25326A21 − 16884A0 A1 − 240384A1 + 1665792} ,

𝐺 (𝑧) = B0 + B1 𝐻 + B2 𝐻2 .

(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 𝐻𝑖 to zero.

4

Journal of Applied Mathematics 𝑢 (𝑡, 𝑥) =𝐴 0 + 𝐴 1 {−

Solving the algebraic equations one obtains 𝜌 = −1,

+

2

A0 = 𝑘 (8𝑎𝑐 + 𝑏 + 5) , A1 =

12𝑎A0 𝑏 , 8𝑎𝑐 + 𝑏2 + 5

+ 𝐴 2 {−

𝑎A1 A2 = , 𝑏

+

B0 = 3 (−2048𝑎2 𝑏𝑐 − 256𝑎𝑏3 + 208𝑎A0 𝑏

B1 = B2 =

A1 (336𝑎𝑏 − 29A1 ) , 192𝑎𝑏 − 6A1

2 sech (𝜃𝑧/2) }, 𝐶 cosh (𝜃𝑧/2)−(2𝑎/𝜃) sinh (𝜃𝑧/2) (17a)

+

1 17760000

+ 𝐵2 {−

× {−270144𝑎2 A20 A1 𝑏𝑐

+

(15)

+ 1080576𝑎2 A20 A2 𝑐2 + 50652𝑎A20 A1 𝑏3 − 84420𝑎A20 A1 𝑏 + 5784000𝑎A1 𝑏

sech (𝜃𝑧/2) } 𝐶 cosh (𝜃𝑧/2) − (2𝑎/𝜃) sinh (𝜃𝑧/2)

𝑏 𝜃 1 − tanh ( 𝜃𝑧) 2𝑎 2𝑎 2

𝑣 (𝑡, 𝑥) =𝐵0 + 𝐵1 {−

−1

− 1280𝑎𝑏 + 15A0 A1 ) × (74A1 ) ,

𝑏 𝜃 1 − tanh ( 𝜃𝑧) 2𝑎 2𝑎 2

𝑏 𝜃 1 − tanh ( 𝜃𝑧) 2𝑎 2𝑎 2 sech (𝜃𝑧/2) } 𝐶 cosh (𝜃𝑧/2)−(2𝑎/𝜃) sinh (𝜃𝑧/2)

𝑏 𝜃 1 − tanh ( 𝜃𝑧) 2𝑎 2𝑎 2 2 sech (𝜃𝑧/2) }, 𝐶 cosh (𝜃𝑧/2) − (2𝑎/𝜃) sinh (𝜃𝑧/2) (17b)

where 𝑧 = 𝑡 − 𝑝𝑥 + (𝑝 − 1)𝑦 and 𝐶 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.

− 751200𝑎A0 A1 𝑏 − 3552000𝑎𝑏B1 + 1350720𝑎A20 A2 𝑐 + 6009600𝑎A0 A2 𝑐 + 70350A0 A21 + 313000A21 − 422100A20 A2 − 3756000A0 A2

3. Conservation Laws of ((3a) and (3b))

+ 28920000A2 − 4221A0 A21 𝑏4

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 𝑏3 𝑐 − 14070A0 A21 𝑏2 + 62600A21 𝑏2 − 480256A32 𝑐4 − 4006400A22 𝑐2 − 1800960A0 A22 𝑐2 + 112560A21 A2 𝑐2 } , where 𝑘 is any root of 469𝑘3 − 416𝑘2 + 304𝑘 − 256 and hence solutions of ((3a) and (3b)) are 𝑢 (𝑡, 𝑥, 𝑦) =𝐴 0 + 𝐴 1 {− + 𝐴 2 {−

𝑏 𝜃 1 − tanh [ 𝜃 (𝑧 + 𝐶)]} 2𝑎 2𝑎 2 2

𝑏 𝜃 1 − tanh [ 𝜃 (𝑧 + 𝐶)]} , 2𝑎 2𝑎 2

(16a) 𝑏 𝜃 1 tanh [ 𝜃 (𝑧 + 𝐶)]} 𝑣 (𝑡, 𝑥, 𝑦) = 𝐵0 + 𝐵1 {− − 2𝑎 2𝑎 2 + 𝐵2 {−

2 𝑏 𝜃 1 − tanh [ 𝜃 (𝑧 + 𝐶)]} , 2𝑎 2𝑎 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 𝑘th-order system of PDEs of 𝑛-independent variables 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) and 𝑚-dependent variables 𝑢 = (𝑢1 , 𝑢2 , . . . , 𝑢𝑚 ): 𝐸𝛼 (𝑥, 𝑢, 𝑢(1) , . . . , 𝑢(𝑘) ) = 0,

𝛼 = 1, . . . , 𝑚,

(18)

where 𝑢(1) , 𝑢(2) , . . . , 𝑢(𝑘) denote the collections of all first, second,. . ., 𝑘th-order partial derivatives, that is, 𝑢𝑖𝛼 = 𝐷𝑖 (𝑢𝛼 ), 𝑢𝑖𝑗𝛼 = 𝐷𝑗 𝐷𝑖 (𝑢𝛼 ), . . ., respectively, with the total derivative operator with respect to 𝑥𝑖 given by 𝐷𝑖 =

𝜕 𝜕 𝜕 + 𝑢𝑖𝛼 𝛼 + 𝑢𝑖𝑗𝛼 𝛼 + ⋅ ⋅ ⋅ , 𝜕𝑥𝑖 𝜕𝑢 𝜕𝑢𝑗

𝑖 = 1, . . . , 𝑛,

(19)

where the summation convention is used whenever appropriate.

Journal of Applied Mathematics

𝑥

5

−2 0

2 25 20 𝑢 15 10

6

𝑣 4 2

5 −2

−2 −2

𝑦

0

0

𝑦 0

2

2

(a)

𝑥

2

(b)

Figure 1: Profile of the travelling wave solution ((13a) and (13b)).

The Euler-Lagrange operator, for each 𝛼, is given by 𝜕 𝛿 𝜕 = + ∑(−1)𝑠 𝐷𝑖1 , . . . , 𝐷𝑖𝑠 𝛼 , 𝛿𝑢𝛼 𝜕𝑢𝛼 𝑠≥1 𝜕𝑢𝑖1 𝑖2 ,...,𝑖𝑠

coupled KP system ((3a) and (3b)), we obtain the zerothorder multipliers (with the aid of GeM [21]), Λ 1 (𝑡, 𝑥, 𝑦, 𝑢, 𝑣), Λ 2 (𝑡, 𝑥, 𝑦, 𝑢, 𝑣) that are given by (20)

Λ 1 = 𝑓3 (𝑡) + 𝑦𝑓4 (𝑡) − 𝑦2 𝑓7󸀠 (𝑡) + 2𝑥𝑓7 (𝑡)

𝛼 = 1, . . . , 𝑚.

+ 𝑦3 (−𝑓8󸀠 (𝑡)) + 6𝑥𝑦𝑓8 (𝑡) ,

The 𝑛-tuple vector 𝑇 = (𝑇1 , 𝑇2 , . . . , 𝑇𝑛 ), 𝑇𝑗 ∈ A, 𝑗 = 1, . . . , 𝑛, where A is the space of differential functions, is a conserved vector of (18) if 𝑇𝑖 satisfies 󵄨 𝐷𝑖 𝑇𝑖 󵄨󵄨󵄨󵄨(18) = 0.

(21)

Equation (21) defines a local conservation law of system (18). A multiplier Λ 𝛼 (𝑥, 𝑢, 𝑢(1) , . . .) has the property that Λ 𝛼 𝐸𝛼 = 𝐷𝑖 𝑇𝑖

(22)

Λ2 = −

𝑦2 𝑓1󸀠

(𝑡) + 2𝑥𝑓1 (𝑡) + 𝑦

3

(−𝑓2󸀠

where 𝑓𝑖 , 𝑖 = 1, 2, . . . , 8 are arbitrary functions of 𝑡. Corresponding to the above multipliers we have the following eight local conserved vectors of ((3a) and (3b)): 𝑇1𝑡 =

1 {−2𝑓1 (𝑡) 𝑣 + 2𝑥𝑓1 (𝑡) 𝑣𝑥 − 𝑦2 𝑓1󸀠 (𝑡) 𝑣𝑥 } , 2

𝑇1𝑥 =

1 {−8𝑦2 𝑓1󸀠 (𝑡) 𝑢𝑥 𝑣 − 8𝑦2 𝑓1󸀠 (𝑡) 𝑣𝑥 𝑢 4 + 16𝑥𝑓1 (𝑡) 𝑢𝑥 𝑣 + 16𝑥𝑓1 (𝑡) 𝑣𝑥 𝑢

𝛿 (Λ 𝛼 𝐸𝛼 ) = 0. 𝛿𝑢𝛼

+ 7𝑦2 𝑓1󸀠 (𝑡) 𝑣𝑥 𝑣 − 14𝑥𝑓1 (𝑡) 𝑣𝑥 𝑣

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

(𝑡))

+ 6𝑥𝑦𝑓2 (𝑡) + 𝑓5 (𝑡) + 𝑦𝑓6 (𝑡) ,

holds identically. In this paper, we will consider multipliers of the zeroth order, that is, Λ 𝛼 = Λ 𝛼 (𝑡, 𝑥, 𝑦, 𝑢, 𝑣). The determining equations for the multiplier Λ 𝛼 are (23)

(24)

+ 5𝑦2 𝑓1󸀠 (𝑡) 𝑢𝑥 𝑢 − 10𝑥𝑓1 (𝑡) 𝑢𝑥 𝑢

− 16𝑓1 (𝑡) 𝑢𝑣 + 5𝑓1 (𝑡) 𝑢2 + 2𝑦2 𝑓1󸀠󸀠 (𝑡) 𝑣 − 4𝑥𝑓1󸀠 (𝑡) 𝑣 + 7𝑓1 (𝑡) 𝑣2 − 8𝑓1 (𝑡) 𝑣𝑥𝑥 + 8𝑥𝑓1 (𝑡) 𝑣𝑥𝑥𝑥 + 4𝑥𝑓1 (𝑡) 𝑣𝑡 − 2𝑦2 𝑓1󸀠 (𝑡) 𝑣𝑡 − 4𝑦2 𝑓1󸀠 (𝑡) 𝑣𝑥𝑥𝑥 } ,

6

Journal of Applied Mathematics 𝑦

𝑇1 = 2𝑦𝑓1󸀠 (𝑡) 𝑣 + 2𝑥𝑓1 (𝑡) 𝑣𝑦 − 𝑦2 𝑓1󸀠 (𝑡) 𝑣𝑦 , 𝑇2𝑡 =

𝑇6𝑥 =

1 {−6𝑦𝑓2 (𝑡) 𝑣 + 6𝑥𝑦𝑓2 (𝑡) 𝑣𝑥 + 𝑦3 (−𝑓2󸀠 (𝑡)) 𝑣𝑥 } , 2 𝑇2𝑥 =

1 {8𝑦𝑓6 (𝑡) 𝑢𝑥 𝑣 + 8𝑦𝑓6 (𝑡) 𝑣𝑥 𝑢 4 − 5𝑦𝑓6 (𝑡) 𝑢𝑥 𝑢 − 7𝑦𝑓6 (𝑡) 𝑣𝑥 𝑣

1 {−8𝑦3 𝑓2󸀠 (𝑡) 𝑢𝑥 𝑣 − 8𝑦3 𝑓2󸀠 (𝑡) 𝑣𝑥 𝑢 4

− 2𝑦𝑓6󸀠 (𝑡) 𝑣 + 4𝑦𝑓6 (𝑡) 𝑣𝑥𝑥𝑥 + 2𝑦𝑓6 (𝑡) 𝑣𝑡 } ,

+ 48𝑥𝑦𝑓2 (𝑡) 𝑢𝑥 𝑣 + 48𝑥𝑦𝑓2 (𝑡) 𝑣𝑥 𝑢 𝑦

𝑇6 = 𝑦𝑓6 (𝑡) 𝑣𝑦 − 𝑓6 (𝑡) 𝑣,

+ 5𝑦3 𝑓2󸀠 (𝑡) 𝑢𝑥 𝑢 − 30𝑥𝑦𝑓2 (𝑡) 𝑢𝑥 𝑢 + 7𝑦3 𝑓2󸀠 (𝑡) 𝑣𝑥 𝑣 − 42𝑥𝑦𝑓2 (𝑡) 𝑣𝑥 𝑣

𝑇7𝑡 =

− 48𝑦𝑓2 (𝑡) 𝑢𝑣 + 15𝑦𝑓2 (𝑡) 𝑢2 +

2𝑦3 𝑓2󸀠󸀠

(𝑡) 𝑣 −

12𝑥𝑦𝑓2󸀠

1 {−2𝑓7 (𝑡) 𝑢 + 2𝑥𝑓7 (𝑡) 𝑢𝑥 − 𝑦2 𝑓7󸀠 (𝑡) 𝑢𝑥 } , 2

𝑇7𝑥 =

(𝑡) 𝑣

1 {−5𝑦2 𝑓7󸀠 (𝑡) 𝑢𝑥 𝑣 − 5𝑦2 𝑓7󸀠 (𝑡) 𝑣𝑥 𝑢 4

+ 21𝑦𝑓2 (𝑡) 𝑣2 − 24𝑦𝑓2 (𝑡) 𝑣𝑥𝑥

+ 10𝑥𝑓7 (𝑡) 𝑢𝑥 𝑣 + 10𝑥𝑓7 (𝑡) 𝑣𝑥 𝑢

+ 24𝑥𝑦𝑓2 (𝑡) 𝑣𝑥𝑥𝑥 + 12𝑥𝑦𝑓2 (𝑡) 𝑣𝑡

+ 7𝑦2 𝑓7󸀠 (𝑡) 𝑢𝑥 𝑢 − 14𝑥𝑓7 (𝑡) 𝑢𝑥 𝑢

− 2𝑦3 𝑓2󸀠 (𝑡) 𝑣𝑡 − 4𝑦3 𝑓2󸀠 (𝑡) 𝑣𝑥𝑥𝑥 } ,

+ 4𝑦2 𝑓7󸀠 (𝑡) 𝑣𝑥 𝑣 − 8𝑥𝑓7 (𝑡) 𝑣𝑥 𝑣

𝑦

𝑇2 = 3𝑦2 𝑓2󸀠 (𝑡) 𝑣 − 6𝑥𝑓2 (𝑡) 𝑣

− 10𝑓7 (𝑡) 𝑢𝑣 + 2𝑦2 𝑓7󸀠󸀠 (𝑡) 𝑢

+ 6𝑥𝑦𝑓2 (𝑡) 𝑣𝑦 − 𝑦3 𝑓2󸀠 (𝑡) 𝑣𝑦 , 𝑇3𝑡 = 𝑇3𝑥 =

− 4𝑥𝑓7󸀠 (𝑡) 𝑢 + 7𝑓7 (𝑡) 𝑢2

1 𝑓 (𝑡) 𝑢𝑥 , 2 3

+ 4𝑓7 (𝑡) 𝑣2 − 8𝑓7 (𝑡) 𝑢𝑥𝑥

1 {5𝑓3 (𝑡) 𝑢𝑥 𝑣 + 5𝑓3 (𝑡) 𝑣𝑥 𝑢 4

+ 8𝑥𝑓7 (𝑡) 𝑢𝑥𝑥𝑥 + 4𝑥𝑓7 (𝑡) 𝑢𝑡 − 2𝑦2 𝑓7󸀠 (𝑡) 𝑢𝑡 − 4𝑦2 𝑓7󸀠 (𝑡) 𝑢𝑥𝑥𝑥 } ,

− 7𝑓3 (𝑡) 𝑢𝑥 𝑢 − 4𝑓3 (𝑡) 𝑣𝑥 𝑣 − 2𝑓3󸀠 (𝑡) 𝑢 + 4𝑓3 (𝑡) 𝑢𝑥𝑥𝑥 + 2𝑓3 (𝑡) 𝑢𝑡 } , 𝑦

𝑇3 = 𝑓3 (𝑡) 𝑢𝑦 , 1 𝑇4𝑡 = 𝑦𝑓4 (𝑡) 𝑢𝑥 , 2 𝑇4𝑥 =

1 {5𝑦𝑓4 (𝑡) 𝑢𝑥 𝑣 + 5𝑦𝑓4 (𝑡) 𝑣𝑥 𝑢 4

𝑦

𝑇7 = 2𝑦𝑓7󸀠 (𝑡) 𝑢 + 2𝑥𝑓7 (𝑡) 𝑢𝑦 − 𝑦2 𝑓7󸀠 (𝑡) 𝑢𝑦 , 𝑇8𝑡 =

1 {−6𝑦𝑓8 (𝑡) 𝑢 + 6𝑥𝑦𝑓8 (𝑡) 𝑢𝑥 − 𝑦3 𝑓8󸀠 (𝑡) 𝑢𝑥 } , 2 𝑇8𝑥 =

1 {−5𝑦3 𝑓8󸀠 (𝑡) 𝑢𝑥 𝑣 − 5𝑦3 𝑓8󸀠 (𝑡) 𝑣𝑥 𝑢 4 + 30𝑥𝑦𝑓8 (𝑡) 𝑢𝑥 𝑣 + 30𝑥𝑦𝑓8 (𝑡) 𝑣𝑥 𝑢

− 7𝑦𝑓4 (𝑡) 𝑢𝑥 𝑢 − 4𝑦𝑓4 (𝑡) 𝑣𝑥 𝑣 − 2𝑦𝑓4󸀠 𝑢

+ 7𝑦3 𝑓8󸀠 (𝑡) 𝑢𝑥 𝑢 − 42𝑥𝑦𝑓8 (𝑡) 𝑢𝑥 𝑢

+ 4𝑦𝑓4 (𝑡) 𝑢𝑥𝑥𝑥 + 2𝑦𝑓4 (𝑡) 𝑢𝑡 } , 𝑦 𝑇4

+ 4𝑦3 𝑓8󸀠 (𝑡) 𝑣𝑥 𝑣 − 24𝑥𝑦𝑓8 (𝑡) 𝑣𝑥 𝑣

= 𝑦𝑓4 (𝑡) 𝑢𝑦 − 𝑓4 (𝑡) 𝑢,

− 30𝑦𝑓8 (𝑡) 𝑢𝑣 + 2𝑦3 𝑓8󸀠󸀠 𝑢

1 𝑇5𝑡 = 𝑓5 (𝑡) 𝑣𝑥 , 2 𝑇5𝑥 =

− 12𝑥𝑦𝑓8󸀠 (𝑡) 𝑢 + 21𝑦𝑓8 (𝑡) 𝑢2

1 {8𝑓5 (𝑡) 𝑢𝑥 𝑣 + 8𝑓5 (𝑡) 𝑣𝑥 𝑢 4

+ 12𝑦𝑓8 (𝑡) 𝑣2 − 24𝑦𝑓8 (𝑡) 𝑢𝑥𝑥 + 24𝑥𝑦𝑓8 (𝑡) 𝑢𝑥𝑥𝑥 + 12𝑥𝑦𝑓8 (𝑡) 𝑢𝑡

− 5𝑓5 (𝑡) 𝑢𝑥 𝑢 − 7𝑓5 (𝑡) 𝑣𝑥 𝑣 − 2𝑓5󸀠 (𝑡) 𝑣 + 4𝑓5 (𝑡) 𝑣𝑥𝑥𝑥 + 2𝑓5 (𝑡) 𝑣𝑡 } , 𝑦

𝑇5 = 𝑓5 (𝑡) 𝑣𝑦 , 1 𝑇6𝑡 = 𝑦𝑓6 (𝑡) 𝑣𝑥 , 2

− 2𝑦3 𝑓8󸀠 (𝑡) 𝑢𝑡 − 4𝑦3 𝑓8󸀠 (𝑡) 𝑢𝑥𝑥𝑥 } , 𝑦

𝑇8 = 3𝑦2 𝑓8󸀠 (𝑡) 𝑢 − 6𝑥𝑓8 (𝑡) 𝑢 + 6𝑥𝑦𝑓8 (𝑡) 𝑢𝑦 − 𝑦3 𝑓8󸀠 (𝑡) 𝑢𝑦 . (25)

Journal of Applied Mathematics We note that because of the arbitrary functions 𝑓𝑖 , 𝑖 = 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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