Double Periodic Wave Solutions of the (2 + 1)-Dimensional Sawada-Kotera Equation

and Applied Analysis 3 In general, for a polynomial operator F(D x , D y , D t ) with respect toD x ,D y , andD t , one has the following useful formula: F (D x , D y , D t ) θ [ ε󸀠 0 ] (ξ, τ) ⋅ θ [ ε 0 ] (ξ, τ)


Introduction
It is always important to investigate the exact solutions for nonlinear evolution equations, which play an important role in the study of nonlinear models of natural and social phenomena.Nonlinear wave phenomena appears in various scientific and engineering areas, such as fluid mechanics, theory of solitons, hydrodynamics, and theory of turbulence, optical fibers, chaos theory, biology, and chemical physics.In the last three decades, various powerful methods have been presented, such as extended tanh method [1], homogeneous balance method [2], Lie group method [3], Wronskian technique [4,5], Darboux transformation method [6], Hirota's bilinear method [7][8][9][10], and algebro-geometrical approach [11].
The Hirota's bilinear method provides a powerful way to derive soliton solutions to nonlinear integrable equations and its basis is the Hirota bilinear formulation.Once the corresponding bilinear forms are obtained, multisoliton solutions and rational solutions to nonlinear differential equations can be computed in quite a systematic way.In 1980s, based on Hirota bilinear forms, Nakamura proposed a comprehensive method to construct a kind of multiperiodic solutions of nonlinear equations in his papers [12,13], such a method of solution does not need any Lax pairs and their induced Riemann surfaces for the considered equations.The advantage of the method is that it only relies on the existed Hirota bilinear forms.Moreover, all parameters appearing in Riemann matrices are completely arbitrary, whereas algebrogeometric solutions involve specific Riemann constants, which are usually difficult to compute.
In recent years, Hon et al. have extended this method to investigate the discrete Toda lattice [14], (2 + 1)-dimensional modified Bogoyavlenskii-Schiff equation [15], and the Supersymmetric KdV-Sawada-Kotera-Ramani equation [16].Ma et al. constructed one-periodic and two-periodic wave solutions to a class of (2 + 1)-dimensional Hirota bilinear equations [17].Tian and Zhang gave the exact periodic solutions for some evolution equations with the aid of the Hirota bilinear method and theta functions identities [18,19].
Our aim in the present work is to improve the main steps of the existing methods of Fan and Chow in [15] into the case of three dimensions.We propose a theorem, which actually provides us a direct and unifying way for applying in a class of (2+1)-dimensional nonlinear partial differential equations.Once such an equation is written in a bilinear form, its double periodic wave solutions can be directly obtained by using this theorem.
The organization of this paper is as follows.In Section 2, we briefly introduce the Hirota bilinear operator and the Riemann theta function.In particular, we present a theorem for

Hirota Bilinear Operator and Riemann Theta Function
In this section we briefly present the notation that will be used in this paper.Here the bilinear operators where   =    1 +    2 + ⋅ ⋅ ⋅ +     +    +   ,  = 1, 2, with   ,   , . . .,   ,   ,   being constants.More generally, one has where (  1 ,   2 , . . .,    ,   ) is a polynomial about operators   1 ,   2 , . . .,    ,   .These properties are useful in deriving Hirota's bilinear form and constructing periodic wave solutions of nonlinear equations.Then, one would like to consider a general Riemann theta function and discuss its periodicity; the Riemann theta function reads where  ∈ Z, complex parameter ,  ∈ C, and complex phase variables  ∈ C,  > 0 which is called the period matrix of the Riemann theta function.
In the definition of the theta function (4), for the case  =  = 0, hereafter, one uses the notation of (, ) =  [ In particular, () is called double periodic as  = 2, and it becomes periodic with the period  if and only if   =   .Proposition 3. The theta function (, ) has the periodic properties: One regards the vectors 1 and  as periods of the theta function (, ) with multipliers 1 and exp(−2 + ), respectively.
Proposition 4. The meromorphic functions () on C is as follows: then it holds that that is, () is a double periodic function with 1 and .

Theorem 5. Assuming that 𝜗 [ 𝜀 󸀠
0 ] (, ) and  [  0 ] (, ) are two Riemann theta functions with  =  +  +  + , then Hirota bilinear operators   ,   , and   exhibit the following perfect properties when they act on a pair of theta functions: where the notation ∑ =0,1 represents two different transformations corresponding to  = 0, 1.The bilinear formulas for ,  are the same as (12) by replacing   with   and   .
Proof.Making use of the formula (2), we obtain the relation Shifting summation index as  =   −   , then Formula ( 13) follows from (12).Formulas ( 13) and ( 14) imply that if the following equations are satisfied then  [   0 ] (, ) and  [  0 ] (, ) are periodic wave solutions of the bilinear equation: Remark 6. Formula ( 17) actually provides us an unified approach to construct periodic wave solutions for nonlinear equations.Once an equation is written bilinear forms, then its periodic wave solutions can be directly obtained by solving system (17).
In the framework of Bell-polynomial manipulations, the Bellpolynomial expression and Bell-polynomial-typed BT for (19) have been given in [26].Here we construct its double periodic wave solution and show that the one-soliton solution can be obtained as limiting case of the double periodic wave solution..We consider a variable transformation  = 6(ln  (, , ))  .

Construct Double Periodic Wave Solutions of the (2 + 1)DSK Equation
Substituting ( 21) into (19) and integrating with respect to , we then get the following Hirota's bilinear form: where  is an integration constant.
Remark 7. The constant  may be taken to be zero in the construction of soliton solutions.However, in our double periodic wave case, the nonzero constant  plays an important role and cannot be dropped.

Feature and Asymptotic Property of Double Periodic
Waves.The double periodic wave solution (30) possesses a simple characterization as follows.
(i) It has a single phase variable ; that is, it is onedimensional.
(ii) It has two fundamental periods 1 and  in the phase variable .
(iii) The speed parameter of  is given by  = (− ( 6  (6)  1 + 5 3  (4) 1 − 5 (iv) It has only one wave pattern for all time and it can be viewed as a parallel superposition of overlapping onesoliton waves, placed one period apart.Now, we further consider the asymptotic properties of the double periodic wave solution.The relation between the periodic wave solution (30) and the one-soliton solution (23) can be established as follows.