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

governs the dynamics of solitary waves. Firstly, it was derived to describe shallowwater 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]


Introduction
The well-known Korteweg-de Vries (KdV) equation [1]   + 6  +   = 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] (  + 6  +   )  +   = 0. ( This equation models shallow long waves in the -direction with some mild dispersion in the -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][5][6][7][8][9].
In this paper, we study a new coupled KP equation [10]: 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][18][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.

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 where  is a real constant, transforms ((3a) and (3b)) to the following nonlinear coupled ordinary differential equations (ODEs): 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 Here () satisfies the Bernoulli and Riccati equations,  is a positive integer that can be determined by balancing procedure, and A 0 , . . ., A  , B 0 , . . ., B  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: where  and  are constants.Its solution can be written as Secondly, for the Riccati equation: (, , and  are constants), we shall use the solutions where  2 =  2 − 4 > 0 and  is a constant of integration.

Solutions of ((3a) and (3b)) Using the Bernoulli Equation as the Simplest Equation.
In this case the balancing procedure yields  = 2 so the solutions of ((5a) and (5b)) are of the form Substituting (11) into ((5a) and (5b)) and making use of the Bernoulli equation (7) and then equating all coefficients of the functions   to zero, we obtain an algebraic system of equations in terms of A 0 , A 1 , A 2 , B 0 , B 1 , and B 2 .
Solving the system of algebraic equations, with the aid of Mathematica, we obtain  = 1,  = 3, where  is any root of Consequently, a solution of ((3a) and (3b)) is given by  (, , ) (, , ) where  =  −  + ( − 1) and  is a constant of integration.

Solutions of ((3a) and (3b)) Using Riccati Equation as the Simplest Equation.
The balancing procedure gives  = 2 so the solutions of ((5a) and (5b)) are of the form Substituting ( 14) into ((5a) and (5b)) and making use of the Riccati equation ( 9), we obtain algebraic system of equations in terms of A 0 , A 1 , A 2 , B 0 , B 1 , and B 2 by equating all coefficients of the functions   to zero.
Solving the algebraic equations one obtains

𝜌 = −1,
A 0 =  (8 +  2 + 5) , where  is any root of 469 3 − 416 2 + 304 − 256 and hence solutions of ((3a) and (3b)) are where  =  −  + ( − 1) and  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.

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.

Preliminaries.
We briefly present the notation and pertinent results which we utilize below.For details the reader is referred to [20].
Equation ( 21) defines a local conservation law of system (18).A multiplier Λ  (, ,  (1) , . ..) has the property that holds identically.In this paper, we will consider multipliers of the zeroth order, that is, Λ  = Λ  (, , , , ).The determining equations for the multiplier Λ  are Once the multipliers are obtained the conserved vectors are calculated via a homotopy formula [13,14].

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.