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

and Applied Analysis 3 yx (υ, ω) = υx, y2x (υ, ω) = υ 2 x + ωxx, yx,y (υ, ω) = ωxy + υxυy. (12) For convenience, we assume that F = e ξ(x 1 ,...,x l ) , G = e η(x 1 ,...,x l ) ,


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 -operators.Recently, Ma put forward generalized bilinear differential operators named   -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][18][19] and the   -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][23][24][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 -operators and the generalized  operators.In Section 3, we will explore the relations between multivariate binary Bell polynomials and the   -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.
If  = 5, we have By ( 6) and ( 8), it is clear to see that Now, under  = 2(ln )  , the generalized bilinear Boussinesq equation can be expressed as Then, we would like to discuss how to use the -operator to seek the bilinear differential form of other nonlinear integrable differential equations with the help of binary Bell polynomial.

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

(17)
Let us now utilize the P-polynomials given above to seek the bilinear form of BLMP equation with the   -operators.

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  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   -operators when acting on exponential functions.

Bilinear Form. BLMP equation can be written as
Setting  = −  , inserting it into (18), and integrating with respect to  yields where  is an integral constant.
From the above, we can get the bilinear form of ( 18): with  = 2 ln .

Periodic Wave Solutions.
When acting on exponential functions, we find that   -operators have a good property if we assume that As a result of the property above, we consider Riemann's theta function solution of ( 18): where  ∈ ,  ∈ , Im  > 0,  =  +  + , with , , and  being constants to be determined.Then, we have where  = −(1/)( + ).
To the bilinear form of BLMP equation, () satisfies the period characters when  = 5.The powers of  obey rule (7), noting that where ℎ =  + .
It is interesting that if we set which can infer that From all the above, it can be proved that the periodic wave solution (32) just goes to the soliton solution  1 = 2 → − 3  1 . (37) Thus, if we assume that  = 0.01,  = 0.01, and  =  to the solution solution of ( 18) can be shown in Figure 1.

Conclusions and Remarks
In this paper, we obtain the bilinear form of bilinear differential equations by applying the   -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  = 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.