Exact Solutions of Coupled Sine-Gordon Equations Using the Simplest Equation Method

have been introduced by Khusnutdinova and Pelinovsky [1]. The coupled sine-Gordon equations generalize the FrenkelKontorova dislocation model [2, 3]. System (1) with α = 1 was also proposed to describe the open states in DNAmodel [4]. Very recently, system (1) was studied bymany researchers and various methods. It was studied by Salas, using a special rational exponential ansatz [5]. Zhao et al. obtained some new solutions including Jacobi elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions by the Jacobi elliptic function expansion method [6], the hyperbolic auxiliary function method [7], and the symbolic computation method [8]. In the past four decades, the study of nonlinear partial differential equations (NLEEs) modelling physical phenomena has become an important research topic. Seeking exact solutions of NLEEs has long been one of the central themes of perpetual interest in mathematics and physics. With the development of symbolic computation packages like Maple and Mathematica, many powerful methods for finding exact solutions have been proposed, such as the homogeneous balancemethod [9, 10], the auxiliary equationmethod [11, 12], the Exp-function method [13, 14], the Darboux transformation [15, 16], the tanh-function method [17], and the (G/G)expansion method [18, 19]. The simplest equation method is a very powerful mathematical technique for finding exact solutions of nonlinear ordinary differential equations. It has been developed by Kudryashov [20, 21] and used successfully by many authors for finding exact solutions of ODEs in mathematical physics [22, 23]. In this paper, we will apply the simplest equation method [24] to obtain some new and more general explicit exact solutions of the coupled sine-Gordon equations.

Very recently, system (1) was studied by many researchers and various methods.It was studied by Salas, using a special rational exponential ansatz [5].Zhao et al. obtained some new solutions including Jacobi elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions by the Jacobi elliptic function expansion method [6], the hyperbolic auxiliary function method [7], and the symbolic computation method [8].
In the past four decades, the study of nonlinear partial differential equations (NLEEs) modelling physical phenomena has become an important research topic.Seeking exact solutions of NLEEs has long been one of the central themes of perpetual interest in mathematics and physics.With the development of symbolic computation packages like Maple and Mathematica, many powerful methods for finding exact solutions have been proposed, such as the homogeneous balance method [9,10], the auxiliary equation method [11,12], the Exp-function method [13,14], the Darboux transformation [15,16], the tanh-function method [17], and the (  /)expansion method [18,19].
The simplest equation method is a very powerful mathematical technique for finding exact solutions of nonlinear ordinary differential equations.It has been developed by Kudryashov [20,21] and used successfully by many authors for finding exact solutions of ODEs in mathematical physics [22,23].
In this paper, we will apply the simplest equation method [24] to obtain some new and more general explicit exact solutions of the coupled sine-Gordon equations.

The Simplest Equation Method
In this section, we will give the detailed description of the simplest equation method.
Step 1. Suppose that we have a nonlinear partial differential equation (PDE) for (, ) in the form where  is a polynomial in its arguments.
The Bernoulli equation we consider in this paper is where  and  are constants.Its solutions can be written as where  1 ,  2 , and  are constants.
For the Riccati equation where , , and  are constants, we will use the solutions where  2 =  2 − 4.
Solving the algebraic equations by symbolic computation, we can determine those parameters explicitly.

Exact Solutions of the Coupled Sine-Gordon Equations
In this section, we solve the coupled sine-Gordon equations by the simplest equation method.
In order to solve (1), we introduce a new unknown function  = (, ) by the formula so that (, ) = (, ) − (, ).According to (1), we have then Substitution ( 14)-( 15) into (13), we get the following coupled system of nonlinear differential equations: According to the first equation of ( 16), we have Substituting (17) into the second equation of ( 16), we obtain a single nonlinear second-order differential equation in the unknown  = (): As we can see, it suffices to find analytic solutions to (18).Observe that if () is a solution of (18), then −() is also a solution.(18) Using the Bernoulli Equation as the Simplest Equation.The balancing procedure yields  = 1.Thus, the solution of ( 18) is of the form
The equations in (30) are the same as those in (31) of [7].
Remark 2. Compared with [7], the exact solutions of this paper are more general, such that when  1 = ,  = 1, and  = 0 in (23), the solutions become as those in (36) of [7].
When  1 = 1,  = 1, and  = 0 in (23), the solutions become as those in (32) of [7].When  = 2 and  = 0 of (29), the solutions become as those in (31) of [7].There are many such examples; thus, it is easy to see that the study of [7] is a special case in this paper.So the exact solutions of this paper are more general, and all the solutions are new solutions which are not reported in the relevant literature reported.

Conclusions
In this paper, we obtained some exact solutions of the coupled sine-Gordon equations by using the simplest equation method.The Bernoulli equation and Riccati equation have been used as the simplest equation.The solutions obtained may be significant and important for the explanation of some practical physical problems.The method may also be applied to other nonlinear partial differential equations.Also, we have verified that the solutions that we have found are indeed solutions to the original nonlinear evolution equations.