Generalized Bilinear Differential Operators, Binary Bell Polynomials, and Exact Periodic Wave Solution of Boiti-Leon-Manna-Pempinelli Equation

Abstract

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 𝐷_𝑝-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 [3], Bäcklund transformation method [4, 5], Darboux transformation [6], inverse scattering transformation method [7], 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 𝐷_𝑝-operators in [14] which are used to create bilinear differential equations. Furthermore, different symbols are also used to furnish relations with Bell polynomials in [15], and even for trilinear equations in [16].

In this paper, we would like to explore the relations between multivariate binary Bell polynomials [17–19] and the 𝐷_𝑝-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 [22–25] 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 𝐷_𝑃-operators. In Section 3, we will explore the relations between multivariate binary Bell polynomials and the 𝐷_𝑝-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 𝐷-Operators and the Generalized 𝐷_𝑝-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 [14] as the following: 𝐷^𝑚_𝑥𝐷^𝑛_𝑡𝑎()𝑏()=𝜕𝑥,𝑡𝑥,𝑡^𝑚𝜕𝑦^𝑚𝜕^𝑛𝜕𝑦^𝑛×𝑎(𝑥+𝑦,𝑡+𝑠)𝑏(𝑥−𝑦,𝑡−𝑠)_{𝑠=0,𝑦=0}, (1) where 𝑚,𝑛=0,1,2,…. Generally, we have 𝐷^𝑛_𝑥𝐷^𝑚_𝑦𝐷^𝑠_𝑡𝑎𝑥,𝑦,𝑡𝑏𝑥^′,𝑦^′,𝑡^′=𝜕−𝜕𝜕𝑥𝜕𝑥^′𝑛𝜕−𝜕𝜕𝑦𝜕𝑦^′𝑚×𝜕−𝜕𝜕𝑡𝜕𝑡^′𝑠𝑎𝑥,𝑦,𝑡𝑏𝑥^′,𝑦^′,𝑡^′_{𝑥′=𝑥,𝑦′=𝑦,𝑡′=𝑡}, (2) where 𝑚,𝑛,𝑠=0,1,2,….

For instance, for the Boussinesq equation 𝑢_𝑡𝑡−𝑢_𝑥𝑥−3𝑢²_𝑥𝑥−𝑢_{𝑥𝑥𝑥𝑥}=0, (3) under 𝑢=2(ln𝐹)_𝑥𝑥, we have −𝐹²_𝑡+𝐹_𝑡𝑡𝐹+𝐹²_𝑥+3𝐹²𝐹⁴_𝑥−𝐹_𝑥𝑥𝐹−6𝐹𝐹²_𝑥𝐹_𝑥𝑥+4𝐹_𝑥𝐹_𝑥𝑥𝑥−𝐹_{𝑥𝑥𝑥𝑥}𝐹=0; (4) we can get its bilinear form with 𝐷-operators 𝐷²_𝑡−𝐷²_𝑥−𝐷⁴_𝑥𝐹⋅𝐹=0. (5)

However, based on the Hirota 𝐷-operators, Professor Ma put forward a kind of generalized bilinear 𝐷_𝑝-operators in [14]: 𝐷^𝑛_𝑝,𝑥𝐷^𝑚_𝑝,𝑦𝐷^𝑠_𝑝,𝑡𝑓𝑥,𝑦,𝑡𝑔𝑥^′,𝑦^′,𝑡^′=𝜕𝜕𝜕𝑥+𝛼𝜕𝑥^′𝑛𝜕𝜕𝜕𝑦+𝛼𝜕𝑦^′𝑚×𝜕𝜕𝜕𝑡+𝛼𝜕𝑡^′𝑠𝑓𝑥,𝑦,𝑡𝑔𝑥^′,𝑦^′,𝑡^′_{𝑥′=𝑥,𝑦′=𝑦,𝑡′=𝑡}, (6) where, for an integer 𝑘, the 𝑘th power of 𝛼 is defined by 𝛼^𝑘=(−1)^𝑟(𝑘),if𝑘≡𝑟(𝑘)mod𝑝 (7) with 0⩽𝑟(𝑘)<𝑝.

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

If 𝑝=5, we have 𝛼=−1,𝛼²=1,𝛼³=−1,𝛼⁴=𝛼⁵𝛼=1,₆=−1,𝛼⁷=1,𝛼⁸=−1,𝛼⁹=𝛼¹⁰⋮=1, (8)

By (6) and (8), it is clear to see that 𝐷_5,𝑥𝐷_5,𝑡𝑓⋅𝑔=𝑓_𝑥𝑡𝑔−𝑓_𝑥𝑔_𝑡−𝑓_𝑡𝑔_𝑥+𝑓𝑔_𝑥𝑡,𝐷²_5,𝑥𝑓⋅𝑔=𝑓_𝑥𝑥𝑔−2𝑓_𝑥𝑔_𝑥+𝑓𝑔_𝑥𝑥,𝐷⁴_5,𝑥𝑓⋅𝑔=𝑓_4𝑥−4𝑓_3𝑥𝑔_𝑥+6𝑓_𝑥𝑥𝑔_𝑥𝑥−4𝑓_𝑥𝑔_3𝑥+𝑓𝑔_4𝑥. (9)

Now, under 𝑢=2(ln𝐹)_𝑥𝑥, the generalized bilinear Boussinesq equation can be expressed as 𝐷²_5,𝑡−𝐷²_5,𝑥−𝐷⁴_5,𝑥𝐹⋅𝐹=0. (10)

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:𝑌_{𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}𝑦=𝑌_{𝑛1,…,𝑛𝑙}𝑦_𝑟1,…𝑦_𝑟𝑙=𝑒^−𝑦𝜕^𝑛1_𝑥1,…,𝜕^𝑛𝑙_𝑥𝑙𝑒^𝑦𝑛₁,…,𝑛_𝑙,𝒴≥0_{𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}(𝜐,𝜔)=𝑌_{𝑛1,…,𝑛𝑙}(𝑦)_{(𝑦𝑟1𝑥1𝑙𝑥𝑙,…,𝑟=(1/2)(1+(−1)𝑟1𝑙+⋯+𝑟)𝜐𝑟1𝑥11𝑥1,…,𝑟+(1/2)(1−(−1)𝑟1𝑙+⋯+𝑟)𝜔𝑟1𝑥11𝑥1,…,𝑟)}, (11)

in which 𝑦_{𝑟1𝑥1,…,𝑟𝑙𝑥𝑙}=𝜕^𝑟1_𝑥1,…,𝜕^𝑟𝑙_𝑥𝑙𝑦(𝑥₁,…,𝑥_𝑙).

In that way, we have 𝑦_𝑥(𝜐,𝜔)=𝜐_𝑥,𝑦_2𝑥(𝜐,𝜔)=𝜐²_𝑥+𝜔_𝑥𝑥,𝑦_𝑥,𝑦(𝜐,𝜔)=𝜔_𝑥𝑦+𝜐_𝑥𝜐_𝑦. (12)

For convenience, we assume that 𝐹=𝑒^{𝜉(𝑥1,…,𝑥𝑙)},𝐺=𝑒^{𝜂(𝑥1,…,𝑥𝑙)},𝜉=𝜔+𝜐2,𝜂=𝜔−𝜐2 (13) and read that (𝐹𝐺)⁻¹𝐷^𝑛1_𝑝,𝑥1,…,𝐷^𝑛𝑙_{𝑝,𝑥𝑙}𝐹⋅𝐺=𝐺⁻¹𝜂𝐹⁻¹(𝜉)𝐷^𝑛1_𝑝,𝑥1,…,𝐷^𝑛𝑙_{𝑝,𝑥𝑙}𝐹(𝜉)𝜂=⋅𝐺_𝑛1𝑘1=0⋯^𝑛𝑙_𝑘𝑙^𝑙_=0𝑖=1𝛼^{𝑘𝑖𝑛𝑖}_𝑘𝑖𝑒^−𝜉𝜕^{𝑛1−𝑘1}_𝑥1,…,𝜕^{𝑛𝑙−𝑘𝑙}_𝑥𝑙𝑒^𝜉×𝑒^−𝜂𝜕^{𝑛1−𝑘1}_𝑥1,…,𝜕^{𝑛𝑙−𝑘𝑙}_𝑥𝑙𝑒^𝜂=^𝑛1_𝑘1=0⋯^𝑛𝑙_𝑘𝑙^𝑙_=0𝑖=1^𝑛𝑖_𝑘𝑖𝑌_{𝑛1−𝑘1,…,𝑛𝑙−𝑘𝑙}(𝜉),𝑌_{𝑘1,…,𝑘𝑙}𝛼^{∑𝑟𝑙𝑟1𝑟𝑖}𝜂=𝑌_{𝑛1,…,𝑛𝑙}𝑦_{𝑟1,…,𝑟𝑙}=𝜉_{𝑟1,…,𝑟𝑙}+𝛼^{∑𝑟𝑙𝑟1𝑟𝑖}𝜂_{𝑟1,…,𝑟𝑙}=𝑌_{𝑛1,…,𝑛𝑙}𝑦_{𝑟1,…,𝑟𝑙}=121+𝛼^{∑𝑟𝑙𝑟1𝑟𝑖}𝜐_{𝑟1,…,𝑟𝑙}+121+𝛼^{∑𝑟𝑙𝑟1𝑟𝑖}𝜔_{𝑟1,…,𝑟𝑙}=𝒴_{𝑝;𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}(𝜐,𝜔). (14)

We find that the link between 𝒴-polynomials and the 𝐷_𝑝-operator can be given as the following through the above deduction: 𝒴_{𝑝;𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}𝜐=ln𝐹𝐺,𝜔=ln𝐹𝐺=𝑌_{𝑛1,…,𝑛𝑙}𝑦_{𝑟1,…,𝑟𝑙}=121+𝛼^{∑𝑟𝑙𝑟1𝑟𝑖}𝜐𝑟₁,…,𝑟_𝑙+121+𝛼^{∑𝑟𝑙𝑟1𝑟𝑖}𝜔𝑟₁,…,𝑟_𝑙. (15)

Particularly, when 𝐹=𝐺, we define 𝒫-polynomials by 𝒫_{𝑝;𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}𝑞=𝒴_{𝑝;𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}𝜐=0,𝜔=2ln𝐹=𝑞. (16)

When 𝑝=5, we can obtain that 𝑃_5;𝑦,𝑡=𝑞_𝑥𝑡,𝑃_5;2𝑥=𝑞_𝑥𝑥,𝑃_5;4𝑥=3𝑞²_2𝑥+𝑞_4𝑥,𝑃_5;3𝑥,𝑦=𝑞_3𝑥𝑦+3𝑞_𝑥𝑥𝑞_𝑥𝑦,𝑃_5;2𝑥,𝑡=0. (17)

Let us now utilize the 𝒫-polynomials given above to seek the bilinear form of BLMP equation with the 𝐷_𝑝-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 𝐷_𝑝-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 𝐷_𝑝-operators when acting on exponential functions.

4.1. Bilinear Form

BLMP equation can be written as 𝑢_𝑦,𝑡+𝑢_3𝑥,𝑦−3𝑢_2𝑥𝑢_𝑦−3𝑢_𝑥𝑢_𝑥,𝑦=0. (18)

Setting 𝑢=−𝑞_𝑥, inserting it into (18), and integrating with respect to 𝑥 yields 𝑞_𝑦,𝑡+𝑞_3𝑥,𝑦+3𝑞_𝑥𝑥𝑞_𝑥,𝑦−𝜆=0, (19) where 𝜆 is an integral constant.

Based on (17), (19) can be expressed as 𝑃_5;𝑦,𝑡𝑞+𝑃_5;3𝑥,𝑦𝑞−𝜆=0. (20)

From the above, we can get the bilinear form of (18): 𝐷_5,𝑦𝐷_5,𝑡+𝐷³_5,𝑥𝐷_5,𝑦𝐹⋅𝐹−𝜆⋅𝐹²=0 (21) with 𝑞=2ln𝐹.

4.2. Periodic Wave Solutions

When acting on exponential functions, we find that 𝐷_𝑝-operators have a good property 𝐻𝐷_𝑝,𝑥1,…,𝐷_{𝑝,𝑥𝑙}𝑒^𝜉1⋅𝑒^𝜉2=𝐻𝑘₁+𝛼𝑘₂,𝑙₁+𝛼𝑙₂,𝑤₁+𝛼𝑤₂𝑒^𝜉1+𝜉2, (22) if we assume that 𝜉_𝑖=𝑘_𝑖𝑥+𝑙_𝑖𝑦+𝑤_𝑖𝑡+𝜉_𝑖⁽⁰⁾𝑖=1,2,…. (23)

As a result of the property above, we consider Riemann’s theta function solution of (18): 𝐹=^∞_𝑛=−∞𝑒^{2𝜋𝑖𝑛𝜂+𝜋𝑖𝑛2𝜏}, (24) where 𝑛∈𝑍,𝜏∈𝐶,Im𝜏>0,𝜂=𝑘𝑥+𝑙𝑦+𝑤𝑡, with 𝑘, 𝑙, and 𝑤 being constants to be determined.

Then, we have 𝐻𝐷_𝑝,𝑥,𝐷_𝑝,𝑦,𝐷_𝑝,𝑡=𝐹⋅𝐹_{∞∞𝑛=−∞𝑚=−∞}𝐻𝐷_𝑝,𝑥,𝐷_𝑝,𝑦,𝐷_𝑝,𝑡𝑒^{2𝜋𝑖𝑛𝜂+𝜋𝑖𝑛2𝜏}𝑒^{2𝜋𝑖𝑚𝜂+𝜋𝑖𝑚2𝜏}=^∞∞_{𝑛=−∞𝑚=−∞}𝐻(()()()𝑤)𝑒2𝜋𝑖𝑛+𝛼𝑚𝑘,2𝜋𝑖𝑛+𝛼𝑚𝑙,2𝜋𝑖𝑛+𝛼𝑚^{2𝜋𝑖(𝑛+𝑚)𝜂+𝜋𝑖(𝑛2+𝑚2)𝜏}=^∞_𝑞=−∞^∞_𝑚=−∞𝐻2𝜋𝑖(1−𝛼)𝑛−𝛼²𝑞()𝑘,2𝜋𝑖1−𝛼𝑛−𝛼²𝑞()𝑙,2𝜋𝑖1−𝛼𝑛−𝛼²𝑞𝑤×𝑒^{𝜋𝑖(𝑛2+(𝑛+𝛼𝑞)2)𝜏}𝑒^{2𝜋𝑖(−𝛼𝑞)𝜂}=^∞_𝑞=−∞𝐻𝑞𝑒^{2𝜋𝑖(−𝛼𝑞)𝜂}, (25) 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 𝐻𝑞=^∞_𝑛=−∞𝐻2𝜋𝑖(1−𝛼)𝑛−𝛼²𝑞()𝑘,2𝜋𝑖1−𝛼𝑛−𝛼²𝑞()𝑙,2𝜋𝑖1−𝛼𝑛−𝛼²𝑞𝑤𝑒^{𝜋𝑖(𝑛2+(𝑛+𝛼𝑞)2)𝜏}=^∞_𝑛=−∞𝐻𝑤𝑒2𝜋𝑖2𝑛−𝑞𝑘,2𝜋𝑖2𝑛−𝑞𝑙,2𝜋𝑖2𝑛−𝑞^{𝜋𝑖(𝑛2+(𝑞−𝑛)2)𝜏}=^∞_ℎ=−∞𝐻𝑤2𝜋𝑖2ℎ−𝑞−2𝑘,2𝜋𝑖2ℎ−𝑞−2𝑙,2𝜋𝑖2ℎ−𝑞−2⋅𝑒^{𝜋𝑖(ℎ2+(𝑞−ℎ−2)2)𝜏}⋅𝑒^{2𝜋𝑖(𝑞−1)𝜏}=𝐻𝑞−2𝑒^{2𝜋𝑖(𝑞−1)𝜏}, (26) where ℎ=𝑛+𝛼.

From (26) we can infer that 𝐻𝑞=𝐻(0)𝑒^{𝜋𝑖𝑛𝑞𝜏},𝑞=2𝑛;𝐻(1)𝑒^{𝜋𝑖(2𝑛+2𝑛2)(𝑞+1)𝜏},𝑞=2𝑛+1. (27)

For (21), we may let 𝐻(0)=^∞_𝑛=−∞[2𝜋𝑖(1−𝛼)𝑛]²𝑙⋅𝑤+[2𝜋𝑖(1−𝛼)𝑛𝑘]³⋅2𝜋𝑖(1−𝛼)𝑛𝑙−𝜆𝑒^{2𝜋𝑖𝑛2𝜏}=^∞_𝑛=−∞−16𝑛²𝜋²𝑙𝑤+25𝑛⁴𝜋⁴𝑘³𝑙−𝜆𝑒^{2𝜋𝑖𝑛3𝜏}=0,𝐻(1)=^∞_𝑛=−∞2𝜋𝑖(1−𝛼)𝑛−𝛼²𝑙⋅2𝜋𝑖(1−𝛼)𝑛−𝛼²𝑤+2𝜋𝑖(1−𝛼)𝑛−𝛼²𝑘³⋅2𝜋𝑖(1−𝛼)𝑛−𝛼²𝑙−𝜆×𝑒^{2𝜋𝑖(𝑛2−2𝑛+1)𝜏}=^∞_𝑛=−∞−4(2𝑛−1)²𝜋²𝑙𝑤+16(2𝑛−1)⁴𝜋⁴𝑘³𝑙−𝜆×𝑒^{𝜋𝑖(2𝑛2−2𝑛+1)𝜏}=0. (28)

Also, the powers of 𝛼 obey rule (7). For the sake of computational convenience, we denote that 𝑔₁(𝑛)=𝑒^{2𝜋𝑖𝑛2𝜏},𝑎₁₁=^∞_𝑛=−∞−16𝑛²𝜋²𝑙𝑔₁(𝑛),𝑎₁₂=^∞_𝑛=−∞256𝑛⁴𝜋⁴𝑘³𝑙𝑔₁(𝑛),𝑎₁₃=^∞_𝑛=−∞𝑔₁(𝑛);𝑔₂(𝑛)=𝑒^{𝜋𝑖(2𝑛2−2𝑛+1)𝜏},𝑎₂₁=^∞_𝑛=−∞−4(2𝑛−1)²𝜋²𝑙𝑔₂(𝑛),𝑎₂₂=^∞_𝑛=−∞16(2𝑛−1)⁴𝜋⁴𝑘³𝑙𝑔₂(𝑛),𝑎₂₃=^∞_𝑛=−∞𝑔₂(𝑛). (29)

By (28), (29), and (30), we can get that 𝑎₁₁𝜔+𝑎₁₂−𝜆𝑎₁₃𝑎=0,₂₁𝜔+𝑎₂₂−𝜆𝑎₂₃=0. (30)

In view of (30), it is easy to see that 𝑎𝜔=¹³𝑎₂₂−𝑎₂₃𝑎₁₂𝑎₁₁𝑎₂₃−𝑎₁₃𝑎₂₁,𝑎𝜆=¹²𝑎₂₁−𝑎₁₁𝑎₂₂𝑎₁₁𝑎₂₃−𝑎₁₃𝑎₂₁. (31)

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

Then, assuming 𝑒^𝜋𝑖𝜏=𝛾, based on (29), we may obtain that 𝑎₁₁=^∞_𝑛=−∞−16𝑛²𝜋²𝑙⋅𝑒^{2𝜋𝑖𝑛2𝜏}=−32𝜋²𝑙𝛾²+4𝛾⁸+9𝛾¹⁸,𝑎+⋯₁₂=^∞_𝑛=−∞256𝑛⁴𝜋⁴𝑘³𝑙⋅𝑒^{2𝜋𝑖𝑛2𝜏}=2×256𝜋⁴𝑘³𝑙𝛾²+4𝛾⁸+9𝛾¹⁸,𝑎+⋯₁₃=^∞_𝑛=−∞𝑒^{2𝜋𝑖𝑛2𝜏}=1+2𝛾²+2𝛾⁸+2𝛾¹⁸𝑎+⋯,₂₁=^∞_𝑛=−∞−4(2𝑛−1)²𝜋²𝑙⋅𝑒^{𝜋𝑖(2𝑛2−2𝑛+1)𝜏}=−8𝜋²𝑙𝛾+9𝛾⁵+25𝛾¹³,𝑎+⋯₂₂=^∞_𝑛=−∞16(2𝑛−1)⁴𝜋⁴𝑘³𝑙⋅𝑒^{𝜋𝑖(2𝑛2−2𝑛+1)𝜏}=32𝜋⁴𝑘³𝑙𝛾+3⁴𝛾⁵+5⁴𝛾¹³,𝑎+⋯₂₃=^∞_𝑛=−∞𝑒^{𝜋𝑖(2𝑛2−2𝑛+1)𝜏}=1+2𝛾+2𝛾⁵+2𝛾¹³+⋯ (33) which lead to 𝑎₁₁𝑎₂₃−𝑎₂₁𝑎₁₃=8𝜋²𝛾,𝑎𝑙𝛾+𝑜₁₃𝑎₂₂−𝑎₁₂𝑎₂₃=32𝜋⁴𝑘³𝛾,𝑎𝑙𝛾+𝑜₁₁𝑎₂₂−𝑎₁₂𝑎₂₁=𝑜𝛾. (34) So, we have 𝜔→4𝜋²𝑘³ and 𝜆→0, as 𝛾→0.

It is interesting that if we set 𝑘₁=2𝜋𝑖𝑘,𝑙₁=2𝜋𝑖𝑙,𝑤₁𝜂=2𝜋𝑖𝑤,₁=𝑘₁𝑥+𝑙₁𝑦+𝑤₁𝑡+𝜋𝑖𝜏, (35) which can infer that 𝐹=^∞_𝑛=−∞𝑒^{2𝜋𝑖𝜂+𝜋𝑖𝑛2𝜏},=1+𝑒^𝜋𝑖𝜂𝑒^{2𝜋𝑖𝜂}+𝑒^{−2𝜋𝑖𝜂}+𝑒^{4𝜋𝑖𝜂}𝑒^{4𝜋𝑖𝜂}+𝑒^{−4𝜋𝑖𝜂}+⋯,=1+𝑒^𝜂1+𝛾²𝑒^𝜂1+𝑒^2𝜂1+𝛾⁶𝑒^−2𝜂1+𝑒^3𝜂1+⋯,⟶1+𝑒^𝜂1𝜂₁⟶0. (36)

From all the above, it can be proved that the periodic wave solution (32) just goes to the soliton solution 𝑤₁=2𝜋𝑖𝑤⟶−𝑘³₁. (37)

Thus, if we assume that 𝑘=0.01, 𝑙=0.01, and 𝜏=𝑖 to the solution 𝐹=1+𝑒^{2𝜋𝑖(𝑘𝑥+𝑙𝑦+4𝜋2𝑘3𝑡+𝜏)}, (38)solution of (18) can be shown in Figure 1.

Figure 1 

Solution of (18).

5. Conclusions and Remarks

In this paper, we obtain the bilinear form of bilinear differential equations by applying the 𝐷_𝑝-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).