JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 741780 10.1155/2013/741780 741780 Research Article On the Solutions and Conservation Laws of a Coupled Kadomtsev-Petviashvili Equation 0000-0002-1986-4859 Khalique Chaudry Masood Qadir Asghar 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 nwu.ac.za 2013 17 2 2013 2013 06 10 2012 02 12 2012 2013 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

The well-known Korteweg-de Vries (KdV) equation  (1)ut+6uux+uxxx=0 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,  and references therein.

An essential extension of the KdV equation is the Kadomtsev-Petviashvili (KP) equation given by  (2)(ut+6uux+uxxx)x+uyy=0. 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 .

In this paper, we study a new coupled KP equation :(3a)(ut+uxxx-74uux-vvx+54(uv)x)x+uyy=0,(3b)(vt+vxxx-54uux-74vvx+2(uv)x)x+vyy=0,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  and later modified by Vitanov . 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 . 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 . Also recently, conservation laws have been employed to find solutions of the certain PDEs .

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 ((<xref ref-type="disp-formula" rid="EEq1.3a">3a</xref>) and (<xref ref-type="disp-formula" rid="EEq1.3b">3b</xref>)) 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 (4)u=F(z),v=G(z),z=t-ρx+(ρ-1)y, where ρ is a real constant, transforms ((3a) and (3b)) to the following nonlinear coupled ordinary differential equations (ODEs): (5a)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)2=0,(5b)ρ4G′′′′(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.

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 (6)F(z)=i=0M𝒜i(H(z))i,G(z)=i=0Mi(H(z))i. Here H(z) satisfies the Bernoulli and Riccati equations, M is a positive integer that can be determined by balancing procedure, and 𝒜0,,𝒜M, 0,,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: (7)H(z)=aH(z)+bH2(z), where a and b are constants. Its solution can be written as (8)H(z)=a{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}.

Secondly, for the Riccati equation: (9)H(z)=aH2(z)+bH(z)+c (a, b, and c are constants), we shall use the solutions (10)H(z)=-b2a-θ2atanh[12θ(z+C)],H(z)=-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2), where θ2=b2-4ac>0 and C is a constant of integration.

2.1. Solutions of ((<xref ref-type="disp-formula" rid="EEq1.3a">3a</xref>) and (<xref ref-type="disp-formula" rid="EEq1.3b">3b</xref>)) 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 (11)F(z)=𝒜0+𝒜1H+𝒜2H2,G(z)=0+1H+2H2. 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, 𝒜2,0,1, and 2.

Solving the system of algebraic equations, with the aid of Mathematica, we obtain (12)a=1,b=3,𝒜0=k(ρ4+ρ2-3ρ+1)ρ2,𝒜1=36𝒜0ρ4ρ4+ρ2-3ρ+1,𝒜2=3𝒜1,0=121312×{-49248𝒜0ρ9-8208𝒜0ρ8+67392𝒜02ρ7-98496𝒜0ρ7+11232𝒜02ρ6+279072𝒜0ρ6-75978𝒜03ρ5+67392𝒜02ρ5-98496𝒜0ρ5-12663𝒜03ρ4-190944𝒜02ρ4+270864𝒜0ρ4-67608𝒜02ρ3-492480𝒜0ρ3+2814𝒜0𝒜1ρ3-22536𝒜02ρ2+205200𝒜0ρ2+938𝒜0𝒜1ρ2+2814𝒜0𝒜1ρ+34704𝒜0-7504𝒜0𝒜1+41472ρ11+6912ρ10+124416ρ9-352512ρ8+186624ρ7-705024ρ6+1285632ρ5-884736ρ4+1181952ρ3-1645056ρ2+933120ρ-1728009248𝒜0ρ9},1=𝒜11022976×{9849𝒜12ρ3-3245184𝒜0ρ3-354564𝒜0𝒜1ρ3+90144𝒜1ρ3+3752𝒜12ρ2-1081728𝒜0ρ2-135072𝒜0𝒜1ρ2+30048𝒜1ρ2+11256𝒜12ρ-50652𝒜0𝒜1ρ+90144𝒜1ρ-25326𝒜12-16884𝒜0𝒜1-240384𝒜1+1665792𝒜12ρ3},2=𝒜1340992×{9849𝒜12ρ3-3245184𝒜0ρ3-354564𝒜0𝒜1ρ3+90144𝒜1ρ3+3752𝒜12ρ2-1081728𝒜0ρ2-135072𝒜0𝒜1ρ2+30048𝒜1ρ2+11256𝒜12ρ-50652𝒜0𝒜1ρ+90144𝒜1ρ-25326𝒜12-16884𝒜0𝒜1-240384𝒜1+1665792𝒜12ρ3}, where k is any root of 469k3-416k2+304k-256=0. Consequently, a solution of ((3a) and (3b)) is given by(13a)u(t,x,y)=A0+A1a{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}+A2a2{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}2,(13b)v(t,x,y)=B0+B1a{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}    +B2a2{cosh[a(z+C)]+sinh[a(z+C)]1-bcosh[a(z+C)]-bsinh[a(z+C)]}2,where z=t-ρx+(ρ-1)y and C is a constant of integration.

2.2. Solutions of ((<xref ref-type="disp-formula" rid="EEq1.3a">3a</xref>) and (<xref ref-type="disp-formula" rid="EEq1.3b">3b</xref>)) Using Riccati Equation as the Simplest Equation

The balancing procedure gives M=2 so the solutions of ((5a) and (5b)) are of the form (14)F(z)=𝒜0+𝒜1H+𝒜2H2,G(z)=0+1H+2H2. 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,𝒜2,0,1,and  2 by equating all coefficients of the functions Hi to zero.

Solving the algebraic equations one obtains (15)ρ=-1,𝒜0=k(8ac+b2+5),𝒜1=12a𝒜0b8ac+b2+5,𝒜2=a𝒜1b,0=3(-2048a2bc-256ab3+208a𝒜0b-1280ab+15𝒜0𝒜1)×(74𝒜1)-1,1=𝒜1(336ab-29𝒜1)192ab-6𝒜1,2=117760000×{-270144a2𝒜02𝒜1bc+1080576a2𝒜02𝒜2c2+50652a𝒜02𝒜1b3-84420a𝒜02𝒜1b+5784000a𝒜1b-751200a𝒜0𝒜1b-3552000ab1+1350720a𝒜02𝒜2c+6009600a𝒜0𝒜2c+70350𝒜0𝒜12+313000𝒜12-422100𝒜02𝒜2-3756000𝒜0𝒜2+28920000𝒜2-4221𝒜0𝒜12b4-938𝒜13b3c-14070𝒜0𝒜12b2+62600𝒜12b2-480256𝒜23c4-4006400𝒜22c2-1800960𝒜0𝒜22c2+112560𝒜12𝒜2c2}, where k is any root of 469k3-416k2+304k-256 and hence solutions of ((3a) and (3b)) are (16a)u(t,x,y)=A0+A1{-b2a-θ2atanh[12θ(z+C)]}+A2{-b2a-θ2atanh[12θ(z+C)]}2,(16b)v(t,x,y)=B0+B1{-b2a-θ2atanh[12θ(z+C)]}+B2{-b2a-θ2atanh[12θ(z+C)]}2,(17a)u(t,x)=A0+A1{-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)}+A2{-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)}2,(17b)v(t,x)=B0+B1{-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)}+B2{-b2a-θ2atanh(12θz)+sech(θz/2)Ccosh(θz/2)-(2a/θ)sinh(θz/2)}2,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.

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

3. Conservation Laws of ((<xref ref-type="disp-formula" rid="EEq1.3a">3a</xref>) and (<xref ref-type="disp-formula" rid="EEq1.3b">3b</xref>))

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 .

Consider a kth-order system of PDEs of n-independent variables x=(x1,x2,,xn) and m-dependent variables u=(u1,u2,,um): (18)Eα(x,u,u(1),,u(k))=0,α=1,,m, where u(1),u(2),,u(k) denote the collections of all first, second,, kth-order partial derivatives, that is, uiα=Di(uα),uijα=DjDi(uα),, respectively, with the total derivative operator with respect to xi given by (19)Di=xi+uiαuα+uijαujα+,i=1,,n, where the summation convention is used whenever appropriate.

The Euler-Lagrange operator, for each α, is given by (20)δδuα=uα+s1(-1)sDi1,,Disui1i2,,isα,α=1,,m.

The n-tuple vector T=(T1,T2,,Tn),Tj𝒜,j=1,,n, where 𝒜 is the space of differential functions, is a conserved vector of (18) if Ti satisfies (21)DiTi|(18)=0. Equation (21) defines a local conservation law of system (18).

A multiplier Λα(x,u,u(1),) has the property that (22)ΛαEα=DiTi holds identically. In this paper, we will consider multipliers of the zeroth order, that is, Λα=Λα(t,x,y,u,v). The determining equations for the multiplier Λα are (23)δ(ΛαEα)δuα=0.

Once the multipliers are obtained the conserved vectors are calculated via a homotopy formula [13, 14].

3.2. Construction of Conservation Laws for ((<xref ref-type="disp-formula" rid="EEq1.3a">3a</xref>) and (<xref ref-type="disp-formula" rid="EEq1.3b">3b</xref>))

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 ), Λ1(t,x,y,u,v), Λ2(t,x,y,u,v) that are given by (24)Λ1=f3(t)+yf4(t)-y2f7(t)+2xf7(t)+y3(-f8(t))+6xyf8(t),Λ2=-y2f1(t)+2xf1(t)+y3(-f2(t))+6xyf2(t)+f5(t)+yf6(t), 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)): (25)T1t=12{-2f1(t)v+2xf1(t)vx-y2f1(t)vx},T1x=14{-8y2f1(t)uxv-8y2f1(t)vxu+16xf1(t)uxv+16xf1(t)vxu+5y2f1(t)uxu-10xf1(t)uxu+7y2f1(t)vxv-14xf1(t)vxv-16f1(t)uv+5f1(t)u2+2y2f1′′(t)v-4xf1(t)v+7f1(t)v2-8f1(t)vxx+8xf1(t)vxxx+4xf1(t)vt-2y2f1(t)vt-4y2f1(t)vxxx},T1y=2yf1(t)v+2xf1(t)vy-y2f1(t)vy,T2t=12{-6yf2(t)v+6xyf2(t)vx+y3(-f2(t))vx},T2x=14{-8y3f2(t)uxv-8y3f2(t)vxu+48xyf2(t)uxv+48xyf2(t)vxu+5y3f2(t)uxu-30xyf2(t)uxu+7y3f2(t)vxv-42xyf2(t)vxv-48yf2(t)uv+15yf2(t)u2+2y3f2(t)v-12xyf2(t)v+21yf2(t)v2-24yf2(t)vxx+24xyf2(t)vxxx+12xyf2(t)vt-2y3f2(t)vt-4y3f2(t)vxxx},T2y=3y2f2(t)v-6xf2(t)v+6xyf2(t)vy-y3f2(t)vy,T3t=12f3(t)ux,T3x=14{f3(t)uxxx2f3(t)u2f3(t)ut5f3(t)uxv+5f3(t)vxu-7f3(t)uxu-4f3(t)vxv-2f3(t)u+4f3(t)uxxx+2f3(t)ut},T3y=f3(t)uy,T4t=12yf4(t)ux,T4x=14{5yf4(t)uxv+5yf4(t)vxu-7yf4(t)uxu-4yf4(t)vxv-2yf4u+4yf4(t)uxxx+2yf4(t)ut},T4y=yf4(t)uy-f4(t)u,T5t=12f5(t)vx,T5x=14{8f5(t)uxv+8f5(t)vxu-5f5(t)uxu-7f5(t)vxv-2f5(t)v+4f5(t)vxxx+2f5(t)vt},T5y=f5(t)vy,T6t=12yf6(t)vx,T6x=14{8yf6(t)uxv+8yf6(t)vxu-5yf6(t)uxu-7yf6(t)vxv-2yf6(t)v+4yf6(t)vxxx+2yf6(t)vt},T6y=yf6(t)vy-f6(t)v,T7t=12{-2f7(t)u+2xf7(t)ux-y2f7(t)ux},T7x=14{-5y2f7(t)uxv-5y2f7(t)vxu+10xf7(t)uxv+10xf7(t)vxu+7y2f7(t)uxu-14xf7(t)uxu+4y2f7(t)vxv-8xf7(t)vxv-10f7(t)uv+2y2f7(t)u-4xf7(t)u+7f7(t)u2+4f7(t)v2-8f7(t)uxx+8xf7(t)uxxx+4xf7(t)ut-2y2f7(t)ut-4y2f7(t)uxxx},T7y=2yf7(t)u+2xf7(t)uy-y2f7(t)uy,T8t=12{-6yf8(t)u+6xyf8(t)ux-y3f8(t)ux},T8x=14{-5y3f8(t)uxv-5y3f8(t)vxu+30xyf8(t)uxv+30xyf8(t)vxu+7y3f8(t)uxu-42xyf8(t)uxu+4y3f8(t)vxv-24xyf8(t)vxv-30yf8(t)uv+2y3f8u-12xyf8(t)u+21yf8(t)u2+12yf8(t)v2-24yf8(t)uxx+24xyf8(t)uxxx+12xyf8(t)ut-2y3f8(t)ut-4y3f8(t)uxxx},T8y=3y2f8(t)u-6xf8(t)u+6xyf8(t)uy-y3f8(t)uy. 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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