On the Solutions and Conservation Laws of a Coupled Kadomtsev-Petviashvili Equation

Abstract

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

The well-known Korteweg-de Vries (KdV) equation [1] 𝑢_𝑡+6𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥=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] (𝑢_𝑡+6𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥)_𝑥+𝑢_𝑦𝑦=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].

In this paper, we study a new coupled KP equation [10]:𝑢_𝑡+𝑢_𝑥𝑥𝑥−74𝑢𝑢_𝑥−𝑣𝑣_𝑥+54(𝑢𝑣)_𝑥𝑥+𝑢_𝑦𝑦𝑣=0,_𝑡+𝑣_𝑥𝑥𝑥−54𝑢𝑢_𝑥−74𝑣𝑣_𝑥+2(𝑢𝑣)_𝑥𝑥+𝑣_𝑦𝑦=0, (3a)(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]. 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.

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−1𝑦, (4) where 𝜌 is a real constant, transforms ((3a) and (3b)) to the following nonlinear coupled ordinary differential equations (ODEs): 𝜌⁴𝐹^′′′′(𝑧)+54𝜌²𝐺(𝑧)𝐹^′′(𝑧)−74𝜌²𝐹(𝑧)𝐹^′′(𝑧)+𝜌²𝐹^′′(𝑧)−3𝜌𝐹^′′(𝑧)+𝐹^′′(𝑧)+52𝜌²𝐹^′(𝑧)𝐺^′(𝑧)−74𝜌²𝐹^′(𝑧)²+54𝜌²𝐹(𝑧)𝐺^′′(𝑧)−𝜌²𝐺(𝑧)𝐺^′′(𝑧)−𝜌²𝐺^′(𝑧)²𝜌=0,₄𝐺^′′′′(𝑧)+2𝜌²𝐺(𝑧)𝐹^′′(𝑧)−54𝜌²𝐹(𝑧)𝐹^′′(𝑧)+4𝜌²𝐹^′(𝑧)𝐺^′(𝑧)−54𝜌²𝐹^′(𝑧)²+2𝜌²𝐹(𝑧)𝐺^′′(𝑧)−74𝜌²𝐺(𝑧)𝐺^′′(𝑧)+𝜌²𝐺^′′(𝑧)−3𝜌𝐺^′′(𝑧)+𝐺^′′(𝑧)−74𝜌²𝐺^′(𝑧)²=0. (5a)(5b)

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 𝐹(𝑧)=^𝑀_𝑖=0𝒜_𝑖(𝐻(𝑧))^𝑖,𝐺(𝑧)=^𝑀_𝑖=0ℬ_𝑖(𝐻(𝑧))^𝑖. (6) Here 𝐻(𝑧) satisfies the Bernoulli and Riccati equations, 𝑀 is a positive integer that can be determined by balancing procedure, and 𝒜₀,…,𝒜_𝑀, ℬ₀,…,ℬ_𝑀 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), (7) where 𝑎 and 𝑏 are constants. Its solution can be written as 𝐻(𝑧)[𝑎([𝑎(=𝑎cosh𝑧+𝐶)]+sinh𝑧+𝐶)][𝑎([𝑎(.1−𝑏cosh𝑧+𝐶)]−𝑏sinh𝑧+𝐶)] (8)

Secondly, for the Riccati equation: 𝐻^′(𝑧)=𝑎𝐻²(𝑧)+𝑏𝐻(𝑧)+𝑐 (9) (𝑎, 𝑏, and 𝑐 are constants), we shall use the solutions 𝐻(𝑧)𝑏=−−𝜃2𝑎12𝑎tanh2𝜃(,𝐻(𝑧)𝑏𝑧+𝐶)=−−𝜃2𝑎12𝑎tanh2+()𝜃𝑧sech𝜃𝑧/2()−()(),𝐶cosh𝜃𝑧/22𝑎/𝜃sinh𝜃𝑧/2 (10) 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+𝒜₁𝐻+𝒜₂𝐻²,𝐺(𝑧)=ℬ₀+ℬ₁𝐻+ℬ₂𝐻². (11) Substituting (11) into ((5a) and (5b)) and making use of the Bernoulli equation (7) and then equating all coefficients of the functions 𝐻^𝑖 to zero, we obtain an algebraic system of equations in terms of 𝒜₀,𝒜₁, 𝒜₂,ℬ₀,ℬ₁, and ℬ₂.

Solving the system of algebraic equations, with the aid of Mathematica, we obtain 𝒜𝑎=1,𝑏=3,₀=𝑘𝜌⁴+𝜌²−3𝜌+1𝜌²,𝒜₁=36𝒜₀𝜌⁴𝜌⁴+𝜌²,𝒜−3𝜌+1₂=3𝒜₁,ℬ₀=1×21312−49248𝒜₀𝜌⁹−8208𝒜₀𝜌⁸+67392𝒜²₀𝜌⁷−98496𝒜₀𝜌⁷+11232𝒜²₀𝜌⁶+279072𝒜₀𝜌⁶−75978𝒜³₀𝜌⁵+67392𝒜²₀𝜌⁵−98496𝒜₀𝜌⁵−12663𝒜³₀𝜌⁴−190944𝒜²₀𝜌⁴+270864𝒜₀𝜌⁴−67608𝒜²₀𝜌³−492480𝒜₀𝜌³+2814𝒜₀𝒜₁𝜌³−22536𝒜²₀𝜌²+205200𝒜₀𝜌²+938𝒜₀𝒜₁𝜌²+2814𝒜₀𝒜₁𝜌+34704𝒜₀−7504𝒜₀𝒜₁+41472𝜌¹¹+6912𝜌¹⁰+124416𝜌⁹−352512𝜌⁸+186624𝜌⁷−705024𝜌⁶+1285632𝜌⁵−884736𝜌⁴+1181952𝜌³−1645056𝜌²,ℬ+933120𝜌−172800₁=𝒜₁×10229769849𝒜²₁𝜌³−3245184𝒜₀𝜌³−354564𝒜₀𝒜₁𝜌³+90144𝒜₁𝜌³+3752𝒜²₁𝜌²−1081728𝒜₀𝜌²−135072𝒜₀𝒜₁𝜌²+30048𝒜₁𝜌²+11256𝒜²₁𝜌−50652𝒜₀𝒜₁𝜌+90144𝒜₁𝜌−25326𝒜²₁−16884𝒜₀𝒜₁−240384𝒜₁,ℬ+1665792₂=𝒜₁×3409929849𝒜²₁𝜌³−3245184𝒜₀𝜌³−354564𝒜₀𝒜₁𝜌³+90144𝒜₁𝜌³+3752𝒜²₁𝜌²−1081728𝒜₀𝜌²−135072𝒜₀𝒜₁𝜌²+30048𝒜₁𝜌²+11256𝒜²₁𝜌−50652𝒜₀𝒜₁𝜌+90144𝒜₁𝜌−25326𝒜²₁−16884𝒜₀𝒜₁−240384𝒜₁+1665792, (12) where 𝑘 is any root of 469𝑘³−416𝑘²+304𝑘−256=0. Consequently, a solution of ((3a) and (3b)) is given by𝑢𝑡,𝑥,𝑦=𝐴₀+𝐴₁𝑎cosh[𝑎(𝑧+𝐶)]+sinh[𝑎(𝑧+𝐶)]1−𝑏cosh[𝑎(𝑧+𝐶)]−𝑏sinh[𝑎(𝑧+𝐶)]+𝐴₂𝑎²cosh[𝑎(𝑧+𝐶)]+sinh[𝑎(𝑧+𝐶)]1−𝑏cosh[𝑎(𝑧+𝐶)]−𝑏sinh[𝑎(𝑧+𝐶)]²,𝑣𝑡,𝑥,𝑦=𝐵₀+𝐵₁𝑎cosh[𝑎(𝑧+𝐶)]+sinh[𝑎(𝑧+𝐶)]1−𝑏cosh[𝑎(𝑧+𝐶)]−𝑏sinh[𝑎(𝑧+𝐶)]+𝐵₂𝑎²cosh[𝑎(𝑧+𝐶)]+sinh[𝑎(𝑧+𝐶)]1−𝑏cosh[𝑎(𝑧+𝐶)]−𝑏sinh[𝑎(𝑧+𝐶)]², (13a)(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+𝒜₁𝐻+𝒜₂𝐻²,𝐺(𝑧)=ℬ₀+ℬ₁𝐻+ℬ₂𝐻². (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,andℬ₂ by equating all coefficients of the functions 𝐻^𝑖 to zero.

Solving the algebraic equations one obtains 𝒜𝜌=−1,₀=𝑘8𝑎𝑐+𝑏²,𝒜+5₁=12𝑎𝒜₀𝑏8𝑎𝑐+𝑏²,𝒜+5₂=𝑎𝒜₁𝑏,ℬ₀=3−2048𝑎²𝑏𝑐−256𝑎𝑏³+208𝑎𝒜₀𝑏−1280𝑎𝑏+15𝒜₀𝒜₁×74𝒜₁⁻¹,ℬ₁=𝒜₁336𝑎𝑏−29𝒜₁192𝑎𝑏−6𝒜₁,ℬ₂=1×17760000−270144𝑎²𝒜²₀𝒜₁𝑏𝑐+1080576𝑎²𝒜²₀𝒜₂𝑐²+50652𝑎𝒜²₀𝒜₁𝑏³−84420𝑎𝒜²₀𝒜₁𝑏+5784000𝑎𝒜₁𝑏−751200𝑎𝒜₀𝒜₁𝑏−3552000𝑎𝑏ℬ₁+1350720𝑎𝒜²₀𝒜₂𝑐+6009600𝑎𝒜₀𝒜₂𝑐+70350𝒜₀𝒜²₁+313000𝒜²₁−422100𝒜²₀𝒜₂−3756000𝒜₀𝒜₂[phone]𝒜₂−4221𝒜₀𝒜²₁𝑏⁴−938𝒜³₁𝑏³𝑐−14070𝒜₀𝒜²₁𝑏²+62600𝒜²₁𝑏²−480256𝒜³₂𝑐⁴−4006400𝒜²₂𝑐²−1800960𝒜₀𝒜²₂𝑐²+112560𝒜²₁𝒜₂𝑐², (15) where 𝑘 is any root of 469𝑘³−416𝑘²+304𝑘−256 and hence solutions of ((3a) and (3b)) are 𝑢𝑡,𝑥,𝑦=𝐴₀+𝐴₁−𝑏−𝜃2𝑎12𝑎tanh2𝜃(𝑧+𝐶)+𝐴₂−𝑏−𝜃2𝑎12𝑎tanh2𝜃(𝑧+𝐶)²,𝑣𝑡,𝑥,𝑦=𝐵₀+𝐵₁−𝑏−𝜃2𝑎12𝑎tanh2𝜃(𝑧+𝐶)+𝐵₂−𝑏−𝜃2𝑎12𝑎tanh2𝜃(𝑧+𝐶)², (16a)(16b)𝑢(𝑡,𝑥)=𝐴₀+𝐴₁−𝑏−𝜃2𝑎12𝑎tanh2+()𝜃𝑧sech𝜃𝑧/2𝐶cosh(𝜃𝑧/2)−(2𝑎/𝜃)sinh(𝜃𝑧/2)+𝐴₂−𝑏−𝜃2𝑎12𝑎tanh2+()𝜃𝑧sech𝜃𝑧/2𝐶cosh(𝜃𝑧/2)−(2𝑎/𝜃)sinh(𝜃𝑧/2)²,𝑣(𝑡,𝑥)=𝐵₀+𝐵₁−𝑏−𝜃2𝑎12𝑎tanh2+()𝜃𝑧sech𝜃𝑧/2𝐶cosh(𝜃𝑧/2)−(2𝑎/𝜃)sinh(𝜃𝑧/2)+𝐵₂−𝑏−𝜃2𝑎12𝑎tanh2+()𝜃𝑧sech𝜃𝑧/2𝐶cosh(𝜃𝑧/2)−(2𝑎/𝜃)sinh(𝜃𝑧/2)², (17a)(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.

(a)

(b)

Figure 1 

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

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

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.

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,…,𝑥^𝑛) and 𝑚-dependent variables 𝑢=(𝑢¹,𝑢²,…,𝑢^𝑚): 𝐸_𝛼𝑥,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, 𝑢^𝛼_𝑖=𝐷_𝑖(𝑢^𝛼),𝑢^𝛼_𝑖𝑗=𝐷_𝑗𝐷_𝑖(𝑢^𝛼),…, respectively, with the total derivative operator with respect to 𝑥^𝑖 given by 𝐷_𝑖=𝜕𝜕𝑥^𝑖+𝑢^𝛼_𝑖𝜕𝜕𝑢^𝛼+𝑢^𝛼_𝑖𝑗𝜕𝜕𝑢^𝛼_𝑗+⋯,𝑖=1,…,𝑛, (19) where the summation convention is used whenever appropriate.

The Euler-Lagrange operator, for each 𝛼, is given by 𝛿𝛿𝑢^𝛼=𝜕𝜕𝑢^𝛼+_𝑠≥1(−1)^𝑠𝐷_𝑖1,…,𝐷_𝑖𝑠𝜕𝜕𝑢^𝛼_{𝑖1𝑖2,…,𝑖𝑠},𝛼=1,…,𝑚. (20)

The 𝑛-tuple vector 𝑇=(𝑇¹,𝑇²,…,𝑇^𝑛),𝑇^𝑗∈𝒜,𝑗=1,…,𝑛, where 𝒜 is the space of differential functions, is a conserved vector of (18) if 𝑇^𝑖 satisfies 𝐷_𝑖𝑇^𝑖₍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_𝛼𝐸_𝛼=𝐷_𝑖𝑇^𝑖 (22) 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 𝛿Λ_𝛼𝐸_𝛼𝛿𝑢^𝛼=0. (23)

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 coupled KP system ((3a) and (3b)), we obtain the zeroth-order multipliers (with the aid of GeM [21]), Λ₁(𝑡,𝑥,𝑦,𝑢,𝑣), Λ₂(𝑡,𝑥,𝑦,𝑢,𝑣) that are given by Λ₁=𝑓₃(𝑡)+𝑦𝑓₄(𝑡)−𝑦²𝑓^′₇(𝑡)+2𝑥𝑓₇(𝑡)+𝑦³−𝑓^′₈(𝑡)+6𝑥𝑦𝑓₈(𝑡),Λ₂=−𝑦²𝑓^′₁(𝑡)+2𝑥𝑓₁(𝑡)+𝑦³−𝑓^′₂(𝑡)+6𝑥𝑦𝑓₂(𝑡)+𝑓₅(𝑡)+𝑦𝑓₆(𝑡), (24) where 𝑓_𝑖, 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)): 𝑇^𝑡₁=12−2𝑓₁(𝑡)𝑣+2𝑥𝑓₁(𝑡)𝑣_𝑥−𝑦²𝑓^′₁(𝑡)𝑣_𝑥,𝑇^𝑥₁=14−8𝑦²𝑓^′₁(𝑡)𝑢_𝑥𝑣−8𝑦²𝑓^′₁(𝑡)𝑣_𝑥𝑢+16𝑥𝑓₁(𝑡)𝑢_𝑥𝑣+16𝑥𝑓₁(𝑡)𝑣_𝑥𝑢+5𝑦²𝑓^′₁(𝑡)𝑢_𝑥𝑢−10𝑥𝑓₁(𝑡)𝑢_𝑥𝑢+7𝑦²𝑓^′₁(𝑡)𝑣_𝑥𝑣−14𝑥𝑓₁(𝑡)𝑣_𝑥𝑣−16𝑓₁(𝑡)𝑢𝑣+5𝑓₁(𝑡)𝑢²+2𝑦²𝑓₁^′′(𝑡)𝑣−4𝑥𝑓^′₁(𝑡)𝑣+7𝑓₁(𝑡)𝑣²−8𝑓₁(𝑡)𝑣_𝑥𝑥+8𝑥𝑓₁(𝑡)𝑣_𝑥𝑥𝑥+4𝑥𝑓₁(𝑡)𝑣_𝑡−2𝑦²𝑓^′₁(𝑡)𝑣_𝑡−4𝑦²𝑓^′₁(𝑡)𝑣_𝑥𝑥𝑥,𝑇^𝑦₁=2𝑦𝑓^′₁(𝑡)𝑣+2𝑥𝑓₁(𝑡)𝑣_𝑦−𝑦²𝑓^′₁(𝑡)𝑣_𝑦,𝑇^𝑡₂=12−6𝑦𝑓₂(𝑡)𝑣+6𝑥𝑦𝑓₂(𝑡)𝑣_𝑥+𝑦³−𝑓^′₂(𝑡)𝑣_𝑥,𝑇^𝑥₂=14−8𝑦³𝑓^′₂(𝑡)𝑢_𝑥𝑣−8𝑦³𝑓^′₂(𝑡)𝑣_𝑥𝑢+48𝑥𝑦𝑓₂(𝑡)𝑢_𝑥𝑣+48𝑥𝑦𝑓₂(𝑡)𝑣_𝑥𝑢+5𝑦³𝑓^′₂(𝑡)𝑢_𝑥𝑢−30𝑥𝑦𝑓₂(𝑡)𝑢_𝑥𝑢+7𝑦³𝑓^′₂(𝑡)𝑣_𝑥𝑣−42𝑥𝑦𝑓₂(𝑡)𝑣_𝑥𝑣−48𝑦𝑓₂(𝑡)𝑢𝑣+15𝑦𝑓₂(𝑡)𝑢²+2𝑦³𝑓₂^′′(𝑡)𝑣−12𝑥𝑦𝑓^′₂(𝑡)𝑣+21𝑦𝑓₂(𝑡)𝑣²−24𝑦𝑓₂(𝑡)𝑣_𝑥𝑥+24𝑥𝑦𝑓₂(𝑡)𝑣_𝑥𝑥𝑥+12𝑥𝑦𝑓₂(𝑡)𝑣_𝑡−2𝑦³𝑓^′₂(𝑡)𝑣_𝑡−4𝑦³𝑓^′₂(𝑡)𝑣_𝑥𝑥𝑥,𝑇^𝑦₂=3𝑦²𝑓^′₂(𝑡)𝑣−6𝑥𝑓₂(𝑡)𝑣+6𝑥𝑦𝑓₂(𝑡)𝑣_𝑦−𝑦³𝑓^′₂(𝑡)𝑣_𝑦,𝑇^𝑡₃=12𝑓₃(𝑡)𝑢_𝑥,𝑇^𝑥₃=145𝑓₃(𝑡)𝑢_𝑥𝑣+5𝑓₃(𝑡)𝑣_𝑥𝑢−7𝑓₃(𝑡)𝑢_𝑥𝑢−4𝑓₃(𝑡)𝑣_𝑥𝑣−2𝑓^′₃(𝑡)𝑢+4𝑓₃(𝑡)𝑢_𝑥𝑥𝑥+2𝑓₃(𝑡)𝑢_𝑡,𝑇^𝑦₃=𝑓₃(𝑡)𝑢_𝑦,𝑇^𝑡₄=12𝑦𝑓₄(𝑡)𝑢_𝑥,𝑇^𝑥₄=145𝑦𝑓₄(𝑡)𝑢_𝑥𝑣+5𝑦𝑓₄(𝑡)𝑣_𝑥𝑢−7𝑦𝑓₄(𝑡)𝑢_𝑥𝑢−4𝑦𝑓₄(𝑡)𝑣_𝑥𝑣−2𝑦𝑓^′₄𝑢+4𝑦𝑓₄(𝑡)𝑢_𝑥𝑥𝑥+2𝑦𝑓₄(𝑡)𝑢_𝑡,𝑇^𝑦₄=𝑦𝑓₄(𝑡)𝑢_𝑦−𝑓₄(𝑡)𝑇𝑢,_𝑡5=12𝑓₅(𝑡)𝑣_𝑥,𝑇^𝑥₅=148𝑓₅(𝑡)𝑢_𝑥𝑣+8𝑓₅(𝑡)𝑣_𝑥𝑢−5𝑓₅(𝑡)𝑢_𝑥𝑢−7𝑓₅(𝑡)𝑣_𝑥𝑣−2𝑓^′₅(𝑡)𝑣+4𝑓₅(𝑡)𝑣_𝑥𝑥𝑥+2𝑓₅(𝑡)𝑣_𝑡,𝑇^𝑦₅=𝑓₅(𝑡)𝑣_𝑦,𝑇^𝑡₆=12𝑦𝑓₆(𝑡)𝑣_𝑥,𝑇^𝑥₆=148𝑦𝑓₆(𝑡)𝑢_𝑥𝑣+8𝑦𝑓₆(𝑡)𝑣_𝑥𝑢−5𝑦𝑓₆(𝑡)𝑢_𝑥𝑢−7𝑦𝑓₆(𝑡)𝑣_𝑥𝑣−2𝑦𝑓^′₆(𝑡)𝑣+4𝑦𝑓₆(𝑡)𝑣_𝑥𝑥𝑥+2𝑦𝑓₆(𝑡)𝑣_𝑡,𝑇^𝑦₆=𝑦𝑓₆(𝑡)𝑣_𝑦−𝑓₆(𝑡)𝑇𝑣,_𝑡7=12−2𝑓₇(𝑡)𝑢+2𝑥𝑓₇(𝑡)𝑢_𝑥−𝑦²𝑓^′₇(𝑡)𝑢_𝑥,𝑇^𝑥₇=14−5𝑦²𝑓^′₇(𝑡)𝑢_𝑥𝑣−5𝑦²𝑓^′₇(𝑡)𝑣_𝑥𝑢+10𝑥𝑓₇(𝑡)𝑢_𝑥𝑣+10𝑥𝑓₇(𝑡)𝑣_𝑥𝑢+7𝑦²𝑓^′₇(𝑡)𝑢_𝑥𝑢−14𝑥𝑓₇(𝑡)𝑢_𝑥𝑢+4𝑦²𝑓^′₇(𝑡)𝑣_𝑥𝑣−8𝑥𝑓₇(𝑡)𝑣_𝑥𝑣−10𝑓₇(𝑡)𝑢𝑣+2𝑦²𝑓₇^′′(𝑡)𝑢−4𝑥𝑓^′₇(𝑡)𝑢+7𝑓₇(𝑡)𝑢²+4𝑓₇(𝑡)𝑣²−8𝑓₇(𝑡)𝑢_𝑥𝑥+8𝑥𝑓₇(𝑡)𝑢_𝑥𝑥𝑥+4𝑥𝑓₇(𝑡)𝑢_𝑡−2𝑦²𝑓^′₇(𝑡)𝑢_𝑡−4𝑦²𝑓^′₇(𝑡)𝑢_𝑥𝑥𝑥,𝑇^𝑦₇=2𝑦𝑓^′₇(𝑡)𝑢+2𝑥𝑓₇(𝑡)𝑢_𝑦−𝑦²𝑓^′₇(𝑡)𝑢_𝑦,𝑇^𝑡₈=12−6𝑦𝑓₈(𝑡)𝑢+6𝑥𝑦𝑓₈(𝑡)𝑢_𝑥−𝑦³𝑓^′₈(𝑡)𝑢_𝑥,𝑇^𝑥₈=14−5𝑦³𝑓^′₈(𝑡)𝑢_𝑥𝑣−5𝑦³𝑓^′₈(𝑡)𝑣_𝑥𝑢+30𝑥𝑦𝑓₈(𝑡)𝑢_𝑥𝑣+30𝑥𝑦𝑓₈(𝑡)𝑣_𝑥𝑢+7𝑦³𝑓^′₈(𝑡)𝑢_𝑥𝑢−42𝑥𝑦𝑓₈(𝑡)𝑢_𝑥𝑢+4𝑦³𝑓^′₈(𝑡)𝑣_𝑥𝑣−24𝑥𝑦𝑓₈(𝑡)𝑣_𝑥𝑣−30𝑦𝑓₈(𝑡)𝑢𝑣+2𝑦³𝑓₈^′′𝑢−12𝑥𝑦𝑓^′₈(𝑡)𝑢+21𝑦𝑓₈(𝑡)𝑢²+12𝑦𝑓₈(𝑡)𝑣²−24𝑦𝑓₈(𝑡)𝑢_𝑥𝑥+24𝑥𝑦𝑓₈(𝑡)𝑢_𝑥𝑥𝑥+12𝑥𝑦𝑓₈(𝑡)𝑢_𝑡−2𝑦³𝑓^′₈(𝑡)𝑢_𝑡−4𝑦³𝑓^′₈(𝑡)𝑢_𝑥𝑥𝑥,𝑇^𝑦₈=3𝑦²𝑓^′₈(𝑡)𝑢−6𝑥𝑓₈(𝑡)𝑢+6𝑥𝑦𝑓₈(𝑡)𝑢_𝑦−𝑦³𝑓^′₈(𝑡)𝑢_𝑦. (25) We note that because of the arbitrary functions 𝑓_𝑖, 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.