AMP Advances in Mathematical Physics 1687-9139 1687-9120 Hindawi Publishing Corporation 10.1155/2015/291804 291804 Research Article Generalized Bilinear Differential Operators Application in a (3+1)-Dimensional Generalized Shallow Water Equation Wu Jingzhu 1 Xing Xiuzhi 1 Geng Xianguo 2 Hounkonnou Mahouton N. 1 Department of Mathematics Zhoukou Normal University Zhoukou 466000 China zknu.edu.cn 2 Department of Mathematics Zhengzhou University Zhengzhou 450052 China zzu.edu.cn 2015 1092015 2015 14 04 2015 26 06 2015 29 06 2015 1092015 2015 Copyright © 2015 Jingzhu Wu et al. 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.

The relations between Dp-operators and multidimensional binary Bell polynomials are explored and applied to construct the bilinear forms with Dp-operators of nonlinear equations directly and quickly. Exact periodic wave solution of a (3+1)-dimensional generalized shallow water equation is obtained with the help of the Dp-operators and a general Riemann theta function in terms of the Hirota method, which illustrate that bilinear Dp-operators can provide a method for seeking exact periodic solutions of nonlinear integrable equations. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.

1. Introduction

The studies of exact solutions of nonlinear partial differential equations (NPDEs) have received considerable attention in connection with the important problems that arise in scientific applications. Many powerful methods have been proposed to obtain exact solutions of (NPDEs); a series of methods have been proposed, such as Painléve test , Bäcklund transformation method [2, 3], Darboux transformation , inverse scattering transformation method , Lie group method [6, 7], Hamiltonian method [8, 9], and the Hirota method [10, 11].

In order to seek the periodic solutions of nonlinear evolution equations, Porubov and Parker proposed Weierstrass elliptic function expansion method ; Liu et al. proposed Jacobi elliptic sine function expansion methods [13, 14] and obtained some exact periodic solutions of some nonlinear evolution equations. They pointed out that their method can be applied to solve the nonlinear evolution equations in which the odd- and even-order derivative terms do not coexist. Zhang  developed Jacobi elliptic function expansion method to solve some nonlinear evolution equations in which the odd- and even-order derivative term coexist and obtained some exact periodic solutions of the equations. The bilinear method developed by Hirota have proved to be particularly powerful in obtaining the soliton solutions, quasiperiodic wave solutions, and periodic wave solutions [16, 17]. As we all know, once the bilinear forms of nonlinear differential equations are obtained, the multisoliton solutions, the bilinear Bäcklund transformation, and Lax pairs of NPDEs can be constructed easily. It is clear that the key of Hirota direct method is finding the bilinear forms of the given differential equations by the Hirota differential D-operators. However, Hirota bilinear equations are special and there are many other bilinear differential equations which are not written in the Hirota bilinear form.

In fact, solving nonlinear equations (especially nonlinear partial differential equations) is very difficult, and there is no unified method. The present methods can only be applied to a certain equation or some equations. So the work of continuing to find some effective method of solving nonlinear equations is important and meaningful. Recently, Ma put forward generalized bilinear differential operators named Dp-operators in , which are used to create bilinear differential equations. Furthermore, different symbols are also used to furnish relations with Bell polynomials in  and even for trilinear equations in . In this paper, we would like to explore how to construct the bilinear forms with Dp-operators and how to obtain the exact solutions of nonlinear equation with the help of Dp bilinear operators method.

The paper is structured as follows. In Section 2, we will give a brief introduction about the bilinear Dp-operators. In Section 3, we explore the relations between multivariate binary Bell polynomials and the Dp-operators. The Dp bilinear forms of some nonlinear evolutions are given quickly and easily from the relations. In Section 4, we will use the relation in Section 2 to seek the bilinear form with Dp-operators of the (3+1)-dimensional generalized shallow water equation and then take advantage of the Dp-operators and the Riemann theta function [21, 22] to obtain its exact periodic wave solution which can be reduced to the soliton solution via asymptotic analysis.

2. Bilinear <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M14"><mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>-Operators

It is known to us that Hirota bilinear D-operators play a significant role in Hirota direct method. The D-operators are defined as follows: (1)DxmDykf·g=x1-x2my1-y2kfx1,y1gx2,y2,where the right-hand side is computed in (2)x1=x2=x,y1=y2=y.According to the definition of Hirota bilinear D-operators, we have(3)Dxf·g=fxg-fgx,Dx2f·g=fxxg-2fxgx+fgxx,DxDtf·g=fxtg-fxgt-ftgx+fgxt,Dx3f·g=fxxxg-3fxxgx+3fxgxx-fgxxx,DtDx3f·g=f3xtg-f3xgt-3f2xtgx+3f2xgxt+3fxtg2x-3fxg2xt-ftg3x+fg3xt.

Based on the Hirota D-operators, Professor Ma put forward a kind of bilinear Dp-operators in : (4)Dp,xmDp,ykfx1,y1gx2,y2=x1+αx2my1+αy2kx1=x2=x,y1=y2=y,where the powers of α are determined by(5)αi=-1ri,where i=rimodp with 0r(i)<p; i0.

Obviously, the case of p=1 gives the normal derivatives, and the cases of p=2k, kN, reduce to Hirota bilinear operators.

In particular, when m=0, we have (6)Dp,xnf·gx=x+αxnfxgxx=x=i=0nαiCnixn-ifxig.According to the definition of Dp-operator, when p=3, we have (7)α0=1,α=-1,α2=α3=1,α4=-1,α5=α6=1,α7=-1,α8=α9=1,,D3,x4f·g=i=04αiC4ix4-ifxig=f4xg-4f3xgx+6f2xg2x+4fxg3x-fg4x;when p=5, we have(8)α0=1,α1=-1,α2=1,α3=-1,α4=α5=1,α6=-1,α7=1,α8=-1,,D5,xf·g=fxg-fgx,D5,tD5,xf·g=D5,tfxg-fgx=fxtg-fxgt-ftgx+fgxt,D5,x2f·g=i=02αiC2ix2-ifxig=fxxg-2fxgx+fgxx,D5,x4f·g=i=04αiC4ix4-ifxig=f4xg-4f3xgx+6fxxgxx-4fxgxx+fg4x,D5x5f·g=i=05αiC5ix5-ifxig=f5xg-5f4xgx+10f3xg2x-10f2xg3x+5fxg4x+fg5x.

Now, under u=2(lnf)xx, for Kdv equation, (9)ut+6uux+uxxx=0,we have (10)xfxtf-fxft+f4xf-4f3xfx+3f2x2f2=0;we can get its bilinear form with Dp-operators:(11)D5,xD5,t+D5x4f·f=0.

In fact, if we seek the bilinear form with Dp-operators of nonlinear integrable differential equations according to the definition of Dp-operators, this needs some special skills and complex computations. So we would like to explore the relations between Dp-operators and multivariate binary Bell polynomials. The bilinear forms with Dp-operators of nonlinear integrable differential equations are obtained quickly and easily by applying the relations.

3. Relations with Bell Exponential Polynomials 3.1. Relations with Bell Exponential Polynomials

As we all know, Bell proposed three kinds of exponent form polynomials. Later, Wang and Chen generalized the third type of Bell polynomials in [23, 24] which is used mainly in this paper. The multidimensional binary Bell polynomials which we will use are defined as follows:(12)Yn1x1,,nlxly=Yn1,,nlyr1x1,,yrlxl=e-yx1n1xlnleyn1,,nl0,with yr1x1,,rlxl=x1n1xlnl, r1=0,,n1,,rl=0,,nl.

For example,(13)Yx=yx,Y2x=yx2+y2x,Y3x=yx3+3yxy2x+y3x,Y4x=yx4+4yxy3x+6yx2y2x+3y2x2+y4x,Y5x=yx5+5yxy4x+15yxy2x2+10yx2y3x+10y2xy3x+10yx3yxx+y5x,Y6x=yx5+5yxy4x+15yxy2x2+10yx2y3x+10y2xy3x+10yx3yxx+y5xx+yx5+5yxy4x+15yxy2x2+10yx2y3x+10y2xy3x+10yx3yxx+y5xxyx;Yx,t=yx2+yxt,Y2x,t=2yxyxt+y2x,t+yx2yt+y2xyt,Y3x,t=3yx2yt+3yxty2x+3yxy2x,t+y3x,t+yx3+3yxy2x+y3xyt.For the sake of computational convenience, we assume that(14)f=eξx1,,xl,g=eηx1,,xl;we have(15)fg-1Dp,x1n1,,Dp,xlnlf·g=k1=0n1kl=0n1i=1lαkinikie-ξx1n1-k1xlnl-kleξe-ηx1n1-k1xlnl-kleη=k1=0n1kl=0n1i=1lαkinikiYn1-k1x1,,nl-klxlξYk1x1,,klxlη=k1=0n1kl=0n1i=1lαkinikiYn1-k1x1,,nl-klxlξr1,,rlYk1x1,,klxlαr1++rlηr1,,rl=Yn1,,nlyr1,,rl=ξr1,,rl+αr1++rlηr1,,rl=Y¯p;n1x1,,nlxlv,w,where(16)w=ξ+η,v=ξ-η.We find that the link between Y¯-polynomials and the Dp-operator can be given in the following through the above deduction:(17)fg-1Dp,x1n1,,Dp,xlnlf·g=Y¯p;n1x1,,nlxlv=lnfg,w=lnfg=Yn1,,nlyr1,,rl=ξr1,,rl+αr1++rlηr1,,rl.In particular, when f=g, (17) becomes(18)f-2Dp,x1n1,,Dp,xlnlf·f=P¯p;n1x1,,nlxlq=Y¯p;n1x1,,nlxlv=0,w=2lnf=q.Equation (18) give the relations between Dp-operators and multivariate binary Bell polynomials.

Then we have(19)Dp,x1n1,,Dp,xlnlf·f=f2P¯p;n1x1,,nlxlq=f2Y¯p;n1x1,,nlxlv=0,w=2lnf=q.

From (13) and (18), we have(20)P¯3;2x=P¯5;2x=P¯7;2x=qxx,P¯3;4x=3q2x2,P¯3;x,t=P¯5;x,t=P¯7;x,t=qxt,P¯5;4x=P¯7;4x=q4x+3q2x2,P¯5;3x,y=P¯7;3x,y=q3x,y+3qxxqxy,P¯5;2x,t=0,P¯5;6x=15q2x3+15q2xq4x,P¯7;6x=15q2x3+15q2xq4x+q6x.

3.2. Bilinear Form with <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M63"><mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>-Operators

In this section, we will construct the bilinear forms for Kdv equation, (2+1)-dimensional Kdv equation, and (2+1)-dimensional Sawada-Kotera equation with the Dp-operators quickly and easily by utilizing the relations between Dp-operators and multidimensional bilinear Bell polynomials.

Example 1 (Kdv equation).

Consider (21)ut+6uux+uxxx=0.

Setting u=q2x, substituting it into (21), and integrating with respect to x yield(22)qxt+3q2x2+q4x-λ1=0,where λ1 is an arbitrary function of t.

Based on (20) and (22), (21) can be written as follows:(23)P¯5,x,tq+P¯5,4xq-λ1=0.From (19) and (23), we get the bilinear form with Dp-operators of (21) (24)D5,xD5,t+D5,x4f·f-λ1f2=0.

Example 2 ((2+1)-dimensional Kdv equation).

Consider (25)ut+3uuy+uxxy+3uxuydx=0.

Setting u=q2x, substituting it into (25), and integrating with respect to x yield(26)qxt+3qxyq2x2+q3x,y-λ2=0,where λ2 is an arbitrary function of y,t. Based on (20) and (26), (25) can be written as follows:(27)P¯5,x,tq+P¯5,3x,yq-λ2=0.From (19) and (27), we get the bilinear form with Dp-operators of (25): (28)D5,xD5,t+D5,x3D5,yf·f-λ2f2=0.

Consider (29)ut-u4x+5uu2x+53u3+5uxyx+5u2ydx-5uuy-5uxuydx=0.Setting u=3q2x, substituting it into (29), and integrating with respect to x yield (30)qxt+5q2y-q6x+15q2xq4x+15q2x3-5q3xy+3q2xqxy-λ3=0,where λ3 is an arbitrary function of y,t. Based on (20) and (30), (29) can be written as follows:(31)P¯7,x,tq+5P¯7,2yq-5P¯7,3x,yq-P¯7,6xq-λ3=0.From (19) and (31), we get the bilinear form with Dp-operators of (29): (32)D7,xD7,t+5D7,y2-5D7,x3D7,y-D7,x6f·f-λ3f2=0.

From the above computation process for seeking the bilinear forms of three nonlinear equation, we can find that the bilinear forms with Dp-operators of nonlinear integrable differential equations are obtained quickly and easily by appling the relations between Dp-operators and multidimensional bilinear Bell polynomials.

4. Periodic Wave Solution of the (3+1)-Dimensional Generalized Shallow Water Equation

In this section, firstly, we will give the bilinear form of a (3+1)-dimensional generalized shallow water equation with the help of P¯-polynomials and the Dp-operators. And then, we construct the exact periodic wave solution of the (3+1)-dimensional generalized shallow water equation with the aid of the Riemann theta function, Dp-operators, and the special property of the Dp-operators when acting on exponential functions.

The following is (3+1)-dimensional generalized shallow water equation:(33)uxxxy+3uxxuy+3uxuxy-uyt-uxz=0.Setting u=qx, inserting it into (33), and integrating with respect to x yield(34)q3x,y+3q2xqx,y-qy,t-qx,z-λ=0,where λ is an arbitrary function of y,z,t. Based on (20) and (34), (33) can be expressed as(35)-P¯5;y,t-P¯5;x,z+P¯5;3x,y-λ=0.From the above, we can get the bilinear form of (33):(36)-D5;yD5;t-D5;xD5;z+D5;x3D5;yf·f-λ·f2=0with q=2lnf. When acting on exponential functions, we find that Dp-operators have a good property: (37)GDp,x1,,Dp,xleξ1·eξ2=Gk1+αk2,l1+αl2,h1+αh2,ω1+αω2eξ1+ξ2,(38)ξi=kix+liy+hiz+ωit+ξi0,i=1,2,In order to construct periodic wave solutions of (33), we study the multidimensional Riemann theta function with genus N given by(39)fξ=fξ,τ=nzNe-πiτn,n+2πiξ,nin which n=(n1,n2,,nN)TzN denotes the integer value vector and ξ=(ξ1,ξ2,,ξN) is complex phase variable. In addition, for the given two vectors h=(h1,h2,,hN) and g=(g1,g2,,gN) their inner product can be written by(40)h,g=h1g1+h2g2++hNgN.-iτ=(-iτij) in (39) is a positive definite and real-valued symmetric N×N matrix, which can be called the period matrix of the theta function. The entries τij of τ are free parameters of the theta function (39); we consider that Riemann’s (39) converges to a real-valued function with an arbitrary vector ξCN.

In what follows we construct the one-periodic wave solutions of (33). For N=1, Riemann theta function (39) reduces Fourier series in n as follows:(41)f=n=-+eπin2τ+2πinη,where nZ, τC, Imτ>0, and η=kx+ly+hz+ωt, with k,l,h, and ω being constants to be determined.

Riemann theta function (41) satisfying the bilinear equation (36) yields the sufficient conditions for obtaining periodic waves. Substituting the theta function (41) into the left of (36) and using the property (37), we have (42)GDp,x,Dp,y,Dp,z,Dp,tf·f=n=-+m=-+GDp,x,Dp,y,Dp,z,Dp,te2πinη+πin2τe2πimη+πim2τ=n=-+m=-+G2πin+αmk,2πin+αml,2πin+αmh,2πin+αmωe2πin+mη+πin2+m2τ=δ=-+n=-+G2πi1-αn-α2δk,2πi1-αn-α2δl,2πi1-αn-α2δh,2πi1-αn-α2δωeπin2+n+αδ2τe2πi-αδη=δ=-+G¯δe2πi-αδη,where δ=-(1/α)(m+n). To the bilinear form of (33), G¯(δ) satisfies the period characters when p=5. The powers of α obey rule (5), noting that (43)G¯δ=n=-+G2πi1-αn-α2δk,2πi1-αn-α2δl,2πi1-αn-α2δh,2πi1-αn-α2δωeπin2+n+αδ2τ=n=-+G2πi2n-δk,2πi2n-δl,2πi2n-δh,2πi2n-δωeπin2+δ-n2τ=n=-+G2πi2s-δ-2k,2πi2s-δ-2l,2πi2s-δ-2h,2πi2s-δ-2ωeπis2+δ-s-22τe2πiδ-1τ=G¯δ-2e2πiδ-1τ,where s=n+α. From (43) we can infer that (44)G¯δ=G¯0eπinδτ,δ=2n;G¯1eπi2n+2n2δ+1τ,δ=2n+1,(45)G¯0=n=--2πi1-αn2lω-2πi1-αn2kh+2πi1-αnk32πi1-αnl-λe2πin2τ=n=-16π2n2lω+16π2n2kh+256π4n4k3l-λe2πin2τ=0,(46)G¯1=n=--2πi1-αn-α2l·2πi1-αn-α2ω-2πi1-αn-α2k2πi1-αn-α2h+2πi1-αn-α2k32πi1-αn-α2l-λe2πin2-2n+1τ=n=-4π22n-12lω+4π22n-12kh+16π42n-14k3l-λeπi2n2-2n+1τ=0.Also, the powers of α obey rule (5). For the sake of computational convenience, we denote that (47)ρ1n=e2πin2τ,a11=n=-16π2ln2ρ1n,a12=n=-ρ1n,b1=-n=-16π2n2kh+256π4n4k3lρ1n,ρ2n=eπi2n2-2n+1τ,a21=n=-n24π22n-12lρ2n,a22=n=-ρ2n,b2=-n=-4π22n-12n2kh+16π42n-14k3lρ2n. Then (45) and (46) can be written as (48)a11ω+a12λ-b1=0,a21ω+a22λ-b2=0.Solving this system, we get(49)ω=b1a22-b2a12a11a22-a12a21,λ=b2a11-b1a21a11a22-a12a21.Finally, we get one-periodic wave solution:(50)u=2lnfx,where f is given by (41) and ω, λ are satisfied with (49). And if we assume that k=0.01, l=0.01, h=0.01, and τ=i to (50), the solution (50) of (33) can be shown in Figure 1.

A one-periodic wave (50) of the (3+1)-dimensional shallow water wave equation (33) with parameters k=0.01, l=0.01, h=0.01, and τ=i. This figure shows that every one-periodic wave is one-dimensional, and it can be viewed as a superposition of overlapping solitary waves, placed one period apart. (a) Perspective view of the periodic wave Abs(u) on xot-plane. (b) Perspective view of the periodic wave Abs(u) on yot-plane. (c) Perspective view of the periodic wave Abs(u) on zot-plane. (d) Wave propagation pattern of the wave along the x-axis. (e) Wave propagation pattern of the wave along the y-axis. (f) Wave propagation pattern of the wave along the z-axis.

To this end, the soliton solution of (33) can be obtained when we consider limit of the periodic solution (50). Then, assuming eπiτ=γ, we can obtain that(51)a11=n=-16π2ln2e2πin2τ=32π2lγ2+4γ8+,a12=n=-e2πin2τ=1+2γ2+2γ8+2γ18+,a21=n=-n24π22n-12leπi2n2-2n+1τ=8π2lγ+9γ5+,a22=n=-eπi2n2-2n+1τ=γ+γ5+,b1=-n=-16π2n2kh+256π4n4k3le2πin2τ=-32π2kh+16π2k2lγ2+4h+16π2k2lγ8+,b2=-n=-4π22n-12n2kh+16π22n-14k3leπi2n2-2n+1τ=-8π2kh+4π2k2lγ+9h+4π2k2lγ5+,which lead to (52)a11a22-a12a21=8π2lγ+oγ,a22b1-a12b2=8π2kh+4π2k2lγ+oγ.So, we have ω(hk+4π2k3l)/l as Imτ+(γ0).

And we can write f as (53)f=n=-+eπin2τ+2πinη=n=-+eπin2τe2πinη=1+n=1+eπin2τe2πinη+e-2πinη=1+eπiτe2πiη+e-2πiη+e4πinτe4πiη+e-4πiη+e9πinτe6πiη+e-6πiη+=1+eπiτe2πiη+eπiτe-2πiη+e4πinτe4πiη+e4πinτe-4πiη+e9πinτe6πiη+=1+e2πiη+πiτ+e2πiτe-2πiη-πiτ+e4πiη+2πinτ+e6πinτe-4πiη-2πiτ+e6πiη+3πiτ+It is interesting that if we set η=2πiη+πiτ, (53) can be rewritten as(54)f=1+eη+γ2e-η+e2η+γ6e-2η+e3η+,From (54), we have f1+eη as (55)Imτ+γ0. Then the periodic wave solution (50) of (33) turns to the soliton(56)u=2lnfx,f=1+eη=1+eπi2kt+2ly+2hz+2ωt+τ.

5. Conclusions and Remarks

In this paper, we investigate a (3+1)-dimensional generalized shallow water wave equation (33). Its bilinear form is given by applying the relations Dp-operators and binary Bell polynomials, which has proved to be a quick and direct method. Then, we successfully get the exact periodic wave solution with the help of Dp-operators and Riemann theta function in terms of Hirota direct method. Furthermore, we obtain the corresponding soliton solutions via asymptotic analysis for their periodic wave solutions.

There are many other interesting questions on bilinear differential equations; for example, can the approach be generalized to solve trilinear equations with trilinear differential operators? How to apply the Dp-operators into the discrete equations? Besides, we will try to explore how to construct more nonlinear evolution equations with other operators simply and directly. We will continue to explore these problems in the near future.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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