AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/738609 738609 Research Article Generalized Bilinear Differential Operators, Binary Bell Polynomials, and Exact Periodic Wave Solution of Boiti-Leon-Manna-Pempinelli Equation Dong Huanhe 1 Zhang Yanfeng 1 Zhang Yongfeng 1 Yin Baoshu 2,3 Xia Tiecheng 1 College of Mathematics and System Science Shandong University of Science and Technology Qingdao 266590 China sdust.edu.cn 2 Institute of Oceanology China Academy of Sciences Qingdao 266071 China cas.cn 3 Key Laboratory of Ocean Circulation and Wave Chinese Academy of Sciences Qingdao 266071 China cas.cn 2014 772014 2014 05 05 2014 20 06 2014 8 7 2014 2014 Copyright © 2014 Huanhe Dong 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.

We introduce how to obtain the bilinear form and the exact periodic wave solutions of a class of (2+1)-dimensional nonlinear integrable differential equations directly and quickly with the help of the generalized Dp-operators, binary Bell polynomials, and a general Riemann theta function in terms of the Hirota method. As applications, we solve the periodic wave solution of BLMP equation and it can be reduced to soliton solution via asymptotic analysis when the value of p is 5.

1. Introduction

It is significantly important to research nonlinear evolution equations in exploring physical phenomena in depth [1, 2]. Since the soliton theory has been proposed, the research on seeking the exact solutions of the soliton equations has attracted great attention and made great progress. A series of methods have been proposed, such as Painléve test , Bäcklund transformation method [4, 5], Darboux transformation , inverse scattering transformation method , Lie group method [8, 9], and Hamiltonian method [10, 11]. Particularly, Hirota direct method [12, 13] provides a direct approach to solve a kind of specific bilinear differential equations among the exciting methods. As we all know, once the bilinear forms of nonlinear differential equations are obtained, we can construct the multisoliton solutions, the bilinear Bäcklund transformation, and Lax pairs easily. It is clear that the key of Hirota direct method is to find the bilinear forms of the given differential equations by the Hirota differential D-operators. 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 the relations between multivariate binary Bell polynomials  and the Dp-operators and to find the bilinear form of Boiti-Leon-Manna-Pempinelli (BLMP) equation [20, 21]. Then, we can obtain the exact periodic wave solution  of the BLMP equation with the help of a general Riemann theta function in terms of Hirota method.

The paper is structured as follows. In Section 2, we will give a brief introduction about the difference between the Hirota differential D-operators and the generalized DP-operators. In Section 3, we will explore the relations between multivariate binary Bell polynomials and the Dp-operators. In Section 4, we will use the relation in Section 2 to seek the differential form of the BLMP equation and then take advantage of the Riemann theta function [26, 27] and Hirota method to obtain its exact periodic wave solution which can be reduced to the soliton solution via asymptotic analysis.

2. Hirota Bilinear <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M10"><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:math></inline-formula>-Operators and the Generalized <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M11"><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 in  as the following: (1)DxmDtna(x,t)b(x,t)=mymnyn×a(x+y,t+s)b(x-y,t-s)|s=0,y=0, where m,n=0,1,2,. Generally, we have (2)DxnDymDtsa(x,y,t)b(x,y,t)=(x-x)n(y-y)m×(t-t)sa(x,y,t)b(x,y,t)|x=x,y=y,t=t, where m,n,s=0,1,2,.

For instance, for the Boussinesq equation (3)utt-uxx-3(u2)xx-uxxxx=0, under u=2(lnF)xx, we have (4)-Ft2+FttF+Fx2+3F2Fx4-FxxF-6FFx2Fxx+4FxFxxx-FxxxxF=0; we can get its bilinear form with D-operators (5)(Dt2-Dx2-Dx4)F·F=0.

However, based on the Hirota D-operators, Professor Ma put forward a kind of generalized bilinear Dp-operators in : (6)Dp,xnDp,ymDp,tsf(x,y,t)g(x,y,t)=(x+αx)n(y+αy)m×(t+αt)sf(x,y,t)g(x,y,t)|x=x,y=y,t=t, where, for an integer k, the kth power of α is defined by (7)αk=(-1)r(k),if  kr(k)modp with 0r(k)<p.

For example, if p=2k(kN), the D-operators are Hirota operators.

If p=5, we have (8)α=-1,α2=1,α3=-1,α4=α5=1,α6=-1,α7=1,α8=-1,α9=α10=1,

By (6) and (8), it is clear to see that (9)D5,xD5,tf·g=fxtg-fxgt-ftgx+fgxt,D5,x2f·g=fxxg-2fxgx+fgxx,D5,x4f·g=f4x-4f3xgx+6fxxgxx-4fxg3x+fg4x.

Now, under u=2(lnF)xx, the generalized bilinear Boussinesq equation can be expressed as (10)(D5,t2-D5,x2-D5,x4)F·F=0.

Then, we would like to discuss how to use the D-operator to seek the bilinear differential form of other nonlinear integrable differential equations with the help of binary Bell polynomial.

3. Binary Bell Polynomial

As we all know, Bell proposed three kinds of exponent-form polynomials. Later, Lambert, Gilson, and their partners generalized the third type of Bell polynomials in [28, 29] which is used mainly in this paper.

The multidimensional binary Bell polynomials which we will use are defined as follows:(11)Yn1x1,,nlxl(y)=Yn1,,nl(yr1,yrl)=e-yx1n1,,xlnley(n1,,nl0),Yn1x1,,nlxl(υ,ω)=Yn1,,nl(y)|(yr1x1,,rlxl=(1/2)(1+(-1)r1++rl)υr1x1,,r1x1+(1/2)(1-(-1)r1++rl)ω  r1x1,,r1x1),

in which yr1x1,,rlxl=x1r1,,xlrly(x1,,xl).

In that way, we have (12)yx(υ,ω)=υx,y2x(υ,ω)=υx2+ωxx,yx,y(υ,ω)=ωxy+υxυy.

For convenience, we assume that (13)F=eξ(x1,,xl),G=eη(x1,,xl),ξ=ω+υ2,η=ω-υ2 and read that (14)(FG)-1Dp,x1n1,,Dp,xlnlF·G=G-1(η)F-1(ξ)Dp,x1n1,,Dp,xlnlF(ξ)·G(η)=k1=0n1kl=0nli=1lαki(kini)(e-ξx1n1-k1,,xlnl-kleξ)×(e-ηx1n1-k1,,xlnl-kleη)=k1=0n1kl=0nli=1l(kini)Yn1-k1,,nl-kl(ξ),Yk1,,kl(αr1rlriη)=Yn1,,nl(yr1,,rl=ξr1,,rl+αr1rlriηr1,,rl)=Yn1,,nl(yr1,,rl=12(1+αr1rlri)υr1,,rllllllllllllllllllllll+12(1+αr1rlri)ωr1,,rl)=Yp;n1x1,,nlxl(υ,ω).

We find that the link between Y-polynomials and the Dp-operator can be given as the following through the above deduction: (15)Yp;n1x1,,nlxl(υ=lnFG,ω=lnFG)=Yn1,,nl(yr1,,rl=12(1+αr1rlri)υr1,,rlllllllllllllllllllll+12(1+αr1rlri)ωr1,,rl).

Particularly, when F=G, we define P-polynomials by (16)Pp;n1x1,,nlxl(q)=Yp;n1x1,,nlxl(υ=0,ω=2lnF=q).

When p=5, we can obtain that (17)P5;y,t=qxt,P5;2x=qxx,P5;4x=3q2x2+q4x,P5;3x,y=q3xy+3qxxqxy,P5;2x,t=0.

Let us now utilize the P-polynomials given above to seek the bilinear form of BLMP equation with the Dp-operators.

4. Boiti-Leon-Manna-Pempinelli Equation

In this section, firstly, we will give the bilinear form of BLMP equation with the help of P-polynomials and the Dp-operators. And then, we construct the exact periodic wave solution of BLMP equation with the aid of the Riemann theta function, Hirota direct method, and the special property of the Dp-operators when acting on exponential functions.

4.1. Bilinear Form

BLMP equation can be written as (18)uy,t+u3x,y-3u2xuy-3uxux,y=0.

Setting u=-qx, inserting it into (18), and integrating with respect to x yields (19)qy,t+q3x,y+3qxxqx,y-λ=0, where λ is an integral constant.

Based on (17), (19) can be expressed as (20)P5;y,t(q)+P5;3x,y(q)-λ=0.

From the above, we can get the bilinear form of (18): (21)(D5,yD5,t+D5,x3D5,y)F·F-λ·F2=0 with q=2lnF.

4.2. Periodic Wave Solutions

When acting on exponential functions, we find that Dp-operators have a good property (22)H(Dp,x1,,Dp,xl)eξ1·eξ2=H(k1+αk2,l1+αl2,w1+αw2)eξ1+ξ2, if we assume that (23)ξi=kix+liy+wit+ξi(0)i=1,2,.

As a result of the property above, we consider Riemann’s theta function solution of (18): (24)F=n=-e2πinη+πin2τ, where nZ,τC,Imτ>0,η=kx+ly+wt, with k, l, and w being constants to be determined.

Then, we have (25)H(Dp,x,Dp,y,Dp,t)F·F=n=-m=-H(Dp,x,Dp,y,Dp,t)e2πinη+πin2τe2πimη+πim2τ=n=-m=-H(2πi(n+αm)k,2πi(n+αm)l,2πi(n+αm)w)e2πi(n+m)η+πi(n2+m2)τ=q=-{m=-H((2πi(1-α)n-α2q)k,lllllllllllllllllllllllllllllllllll(2πi(1-α)n-α2q)l,lllllllllllllllllllllllllllllllllll(2πi(1-α)n-α2q)w)llllllllllllllllllllllllll×eπi(n2+(n+αq)2)τm=-H((2πi(1-α)n-α2q)k,}e2πi(-αq)η=q=-H-(q)e2πi(-αq)η, where q=-(1/α)(m+n).

To the bilinear form of BLMP equation, H-(q) satisfies the period characters when p=5. The powers of α obey rule (7), noting that (26)H-(q)=n=-H((2πi(1-α)n-α2q)k,llllllllllllllllllllllll(2πi(1-α)n-α2q)l,lllllllllllllllllllllllll(2πi(1-α)n-α2q)w)eπi(n2+(n+αq)2)τ=n=-H(2πi(2n-q)k,2πi(2n-q)l,llllllllllllllllll2πi(2n-q)w)eπi(n2+(q-n)2)τ=h=-H(2πi(2h-(q-2))k,lllllllllllllllll2πi(2h-(q-2))l,2πi(2h-(q-2))w)llllllllllllll·eπi(h2+(q-h-2)2)τ·e2πi(q-1)τ=H-(q-2)e2πi(q-1)τ, where h=n+α.

From (26) we can infer that (27)H-(q)={H-(0)eπinqτ,q=2n;H-(1)eπi(2n+2n2)(q+1)τ,q=2n+1.

For (21), we may let (28)H-(0)=n=-{[2πi(1-α)n]2l·w+[2πi(1-α)nk]3llllllllllllll·2πi(1-α)nl-λ[2πi(1-α)n]2l·w+[2πi(1-α)nk]3}e2πin2τ=n=-(-16n2π2lw+25n4π4k3l-λ)e2πin3τ=0,H-(1)=n=-{2π(i(1-α)n-α2)llllllllllllll·2π(i(1-α)n-α2)wlllllllllllll+[2π(i(1-α)n-α2)k]3lllllllllllll·2π(i(1-α)n-α2)l-λ}lllllllllllll×e2πi(n2-2n+1)τ=n=-[-4(2n-1)2π2lw+16(2n-1)4π4k3l-λ]lllllllllllll×eπi(2n2-2n+1)τ=0.

Also, the powers of α obey rule (7). For the sake of computational convenience, we denote that (29)g1(n)=e2πin2τ,a11=n=--16n2π2lg1(n),a12=n=-(256n4π4k3l)g1(n),a13=n=-g1(n);g2(n)=eπi(2n2-2n+1)τ,a21=n=--4(2n-1)2π2lg2(n),a22=n=-(16(2n-1)4π4k3l)g2(n),a23=n=-g2(n).

By (28), (29), and (30), we can get that (30)a11ω+a12-λa13=0,a21ω+a22-λa23=0.

In view of (30), it is easy to see that (31)ω=a13a22-a23a12a11a23-a13a21,λ=a12a21-a11a22a11a23-a13a21.

Thus, we obtain the periodic wave solution of BLMP equation: (32)u=-2(lnF)x, where F is given by (24) and ω,λ are satisfied with (31).

Then, assuming eπiτ=γ, based on (29), we may obtain that (33)a11=n=--16n2π2l·e2πin2τ=-32π2l(γ2+4γ8+9γ18+),a12=n=-256n4π4k3l·e2πin2τ=2×256π4k3l(γ2+4γ8+9γ18+),a13=n=-e2πin2τ=1+2γ2+2γ8+2γ18+,a21=n=--4(2n-1)2π2l·eπi(2n2-2n+1)τ=-8π2l(γ+9γ5+25γ13+),a22=n=-16(2n-1)4π4k3l·eπi(2n2-2n+1)τ=32π4k3l(γ+34γ5+54γ13+),a23=n=-eπi(2n2-2n+1)τ=1+2γ+2γ5+2γ13+ which lead to (34)a11a23-a21a13=8π2lγ+o(γ),a13a22-a12a23=32π4k3lγ+o(γ),a11a22-a12a21=o(γ). So, we have ω4π2k3 and λ0, as γ0.

It is interesting that if we set (35)k1=2πik,l1=2πil,w1=2πiw,η1=k1x+l1y+w1t+πiτ,       which can infer that (36)F=n=-e2πiη+πin2τ,=1+eπiη(e2πiη+e-2πiη)+e4πiη(e4πiη+e-4πiη)+,=1+eη1+γ2(eη1+e2η1)+γ6(e-2η1+e3η1)+,1+eη1(η10).

From all the above, it can be proved that the periodic wave solution (32) just goes to the soliton solution (37)w1=2πiw-k13.

Thus, if we assume that k=0.01, l=0.01, and τ=i to the solution (38)F=1+e2πi(kx+ly+4π2k3t+τ),solution of (18) can be shown in Figure 1.

Solution of (18).

5. Conclusions and Remarks

In this paper, we obtain the bilinear form of bilinear differential equations by applying the Dp-operators and binary Bell polynomials, which has proved to be a quick and direct method. Furthermore, together with Riemann theta function and Hirota method, we successfully get the exact periodic wave solution and figure of BLMP equation when p=5.

There are many other interesting questions on bilinear differential equations, for example, how to apply the generalized operators into the discrete equations; it is known that researches on the discrete and differential equations are also significant. Besides, we will try to explore other operators to construct more nonlinear evolution equations simply and directly in the near future.

Conflict of Interests

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

Acknowledgments

This work was supported by National Natural Science Foundation of China (no. 11271007), Nature Science Foundation of Shandong Province of China (no. ZR2013AQ017), the Strategic Pioneering Program of Chinese Academy of Sciences (no. XDA 10020104), SDUST Research Fund (no. 2012 KYTD105), Open Fund of the Key Laboratory of Ocean Circulation and Waves, Chinese Academy of Science (no. KLOCAW1401), and Graduate Innovation Foundation from Shandong University of Science and Technology (no. YC130321).

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