Generalized Bilinear Differential Operators Application in a (3+1)-Dimensional Generalized Shallow Water Equation

The relations between Dp-operators and multidimensional binary Bell polynomials are explored and applied to construct the bilinear forms withDp-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.


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 [1], Bäcklund transformation method [2,3], Darboux transformation [4], inverse scattering transformation method [5], 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 [12]; 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 [15] 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 -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   -operators in [18], which are used to create bilinear differential equations.Furthermore, different symbols are also used to furnish relations with Bell polynomials in [19] and even for trilinear equations in [20].In this paper, we would like to explore how to construct the bilinear forms 2 Advances in Mathematical Physics with   -operators and how to obtain the exact solutions of nonlinear equation with the help of   bilinear operators method.
The paper is structured as follows.In Section 2, we will give a brief introduction about the bilinear   -operators.In Section 3, we explore the relations between multivariate binary Bell polynomials and the   -operators.The   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  operators of the (3+1)-dimensional generalized shallow water equation and then take advantage of the   -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.

Bilinear 𝐷 𝑝 -Operators
It is known to us that Hirota bilinear -operators play a significant role in Hirota direct method.The -operators are defined as follows: where the right-hand side is computed in According to the definition of Hirota bilinear -operators, we have Based on the Hirota -operators, Professor Ma put forward a kind of bilinear   -operators in [18]: where the powers of  are determined by where  = () mod  with 0 ≤ () < ;  ≥ 0.
Obviously, the case of  = 1 gives the normal derivatives, and the cases of  = 2,  ∈ , reduce to Hirota bilinear operators.
In particular, when  = 0, we have According to the definition of   -operator, when  = 3, we have when  = 5, we have Now, under  = 2(ln )  , for Kdv equation, we have we can get its bilinear form with   -operators: In fact, if we seek the bilinear form with   -operators of nonlinear integrable differential equations according to the definition of   -operators, this needs some special skills and complex computations.So we would like to explore the relations between   -operators and multivariate binary Bell polynomials.The bilinear forms with   -operators of nonlinear integrable differential equations are obtained quickly and easily by applying the relations.

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: with For example, For the sake of computational convenience, we assume that  =  ( 1 ,...,  ) ,  =  ( 1 ,...,  ) ; we have where We find that the link between -polynomials and the  operator can be given in the following through the above deduction: In particular, when  = , (17) becomes Equation (18) give the relations between   -operators and multivariate binary Bell polynomials.
Then we have From ( 13) and ( 18), we have Setting  =  2 , substituting it into (21), and integrating with respect to  yield where  1 is an arbitrary function of .
From ( 19) and ( 23), we get the bilinear form with  operators of ( 21) Example 2 ((2+1)-dimensional Kdv equation).Consider Setting  =  2 , substituting it into (25), and integrating with respect to  yield where  2 is an arbitrary function of , .Based on ( 20) and ( 26), (25) can be written as follows: From ( 19) and ( 27), we get the bilinear form with  operators of (25): Example 3 ((2+1)-dimensional Sawada-Kotera equation).Consider Setting  = 3 2 , substituting it into (29), and integrating with respect to  yield where  3 is an arbitrary function of , .Based on ( 20) and ( 30), ( 29) can be written as follows: From ( 19) and (31), we get the bilinear form with  operators of (29): From the above computation process for seeking the bilinear forms of three nonlinear equation, we can find that the bilinear forms with   -operators of nonlinear integrable differential equations are obtained quickly and easily by appling the relations between   -operators and multidimensional bilinear Bell polynomials.

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 -polynomials and the   -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,   -operators, and the special property of the   -operators when acting on exponential functions.
To this end, the soliton solution of (33) can be obtained when we consider limit of the periodic solution (50).Then, assuming   = , we can obtain that

𝑒 2𝜋𝑖𝑛 2 Figure 1 :
Figure 1: A one-periodic wave (50) of the (3+1)-dimensional shallow water wave equation (33) with parameters  = 0.01,  = 0.01, ℎ = 0.01, and  = .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() on -plane.(b) Perspective view of the periodic wave Abs() on -plane.(c) Perspective view of the periodic wave Abs() on -plane.(d) Wave propagation pattern of the wave along the -axis.(e) Wave propagation pattern of the wave along the -axis.(f) Wave propagation pattern of the wave along the -axis.