Exact Traveling Wave Solutions to the ( 2 + 1 )-Dimensional Jaulent-Miodek Equation

Nonlinear differential equations widely describe many important dynamical systems in various fields of science, especially in nonlinear optics, plasma physics, solid state physics, and fluid mechanics. It has aroused widespread attention in the study of nonlinear differential equations [1–28]. Exact solutions of nonlinear differential equations play an important role in the study of mathematical physics phenomena. Hence, seeking explicit solutions of physics equations is an interesting and significant subject. In 2001, Geng et al. [29] developed some (2 + 1)dimensional models from the Jaulent-Miodek hierarchy [30]. Over the past few years, many research results for the (2 + 1)dimensional Jaulent-Miodek equations have been generated [31–34], such as the algebraic-geometrical solutions, the bifurcation and exact solutions, the N-soliton solution, and Multiple kink solutions for the (2 + 1)-dimensional JaulentMiodek equations. In 2012, Zhang et al. [35] studied the following (2 + 1)dimensional Jaulent-Miodek equation: a1uxt + a2u2 xuxx − uxxxx − a3uxxuy − a4uxuxy + a5uyy = 0, (1)


Introduction and Main Results
Nonlinear differential equations widely describe many important dynamical systems in various fields of science, especially in nonlinear optics, plasma physics, solid state physics, and fluid mechanics.It has aroused widespread attention in the study of nonlinear differential equations .Exact solutions of nonlinear differential equations play an important role in the study of mathematical physics phenomena.Hence, seeking explicit solutions of physics equations is an interesting and significant subject.

Substituting traveling wave transform
(, , ) = V () , into (1), and then integrating it we get where  =  3 + 4 ,  and  are constants, and  is the integration constant.Setting  = V  , (3) becomes We say that a meromorphic function  belongs to the class  if  is an elliptic function, or a rational function of , or a rational function of   ,  ∈ C.Only these functions can satisfy an algebraic addition theorem which was proved by Weierstrass, so the letter  was utilized [36].In 2006, Eremenko [36] proved that all meromorphic solutions of the Kuramoto-Sivashinsky algebraic differential equation belong to the class .Recently, Kudryashov et al. [37,38] used Laurent series to seek meromorphic exact solutions of some nonlinear differential equations.Following their work, the complex method was proposed by Yuan et al. [39,40].

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They employed the Nevanlinna value distribution theory to investigate the existence of meromorphic solutions to some differential equations and then obtain the representations of meromorphic solutions to these differential equations [41,42].It shows that the complex method has a strong theoretical basis which can proof that meromorphic solutions of certain differential equations belong to the class  and obtain exact solutions by the indeterminate forms of the solutions.Besides, this method can be applied to get all traveling wave exact solutions or general solutions of related differential equations [43,44].In this article, we would like to use the complex method to obtain exact traveling wave solutions to the (2 + 1)-dimensional Jaulent-Miodek equation.
= 0, then the meromorphic solutions  of (4) belong to the class .In addition, (4) has the following classes of solutions.
(i) The rational function solutions where in the latter case.
(i) The rational function solutions where  0 ∈ C,  1 ̸ = 0,  1 and  2 are integral constants.(ii) The simply periodic solutions where  0 ∈ C,  1 ̸ = 0,  3 and  4 are integral constants.(iii) The elliptic function solutions where The rest of this paper is organized as follows.Section 2 introduces some preliminary theory and the complex method.In Section 3, we will give the proof of Theorems 1 and 2. Some computer simulations will be given to illustrate our main results in Section 4. Conclusions are presented at the end of the paper.

Preliminary Theory and the Complex Method
At first, we give some notations and definitions, and then we introduce some lemmas and the complex method.
Consider the following differential equation: where  ∈ N,  ̸ = 0,  are constants.Set ,  ∈ N, and meromorphic solutions  of ( 13) have at least one pole.If (13) has exactly  distinct meromorphic solutions, and their multiplicity of the pole at  = 0 is , then (13) is said to satisfy the ⟨, ⟩ condition.It could be not easy to show that the ⟨, ⟩ condition of (13) holds, so we need the weak ⟨, ⟩ condition as follows.
Each simply periodic solution  fl () is a rational function of  =   ( ∈ C) and is expressed as which has (≤ ) distinct poles of multiplicity .

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When  2 =  3 = 0, Weierstrass elliptic functions can be degenerated to rational functions according to When Δ( 2 ,  3 ) = 0, it can also be degenerated to simple periodic functions according to By the above definitions and lemmas, we now present the complex method as below for the convenience of readers.
Step 2. Substitute ( 14) into the ODE to determine whether the weak ⟨, ⟩ condition holds.
Step 3. Find out meromorphic solutions () of the ODE with a pole at  = 0, in which we have −1 integral constants.
Step 5. Substituting the inverse transformation  −1 into the meromorphic solutions, we get the exact solutions for the original PDE.
By (21), we infer that the indeterminate rational solutions of (4) are with pole at  = 0.
Substituting  1 () into (4), we have where where So the rational solutions of (4) are where in the latter case.
To obtain simply periodic solutions, let  =   , and substitute  = () into Eq.( 4), then Substituting into (30), we obtain that where 3  2 in the latter case.
Inserting  =   into (32), we can get simply periodic solutions to (4) with pole at  = 0 where 2 in the latter case.
So simply periodic solutions of (4) are where 2 in the latter case.
Proof of Theorem 2. By Theorem 1, we can obtain the rational function solutions of (3) which are where  0 ∈ C,  1 ̸ = 0,  1 and  2 are integral constants.

Computer Simulations
In this section, we illustrate our main results by some computer simulations.We carry out further analysis to the properties of the new solutions as in the following figures.