Abstract
The simplest equation method has been used for finding the exact solutions of coupled sine-Gordon equations. Such equations have some useful applications in physics and biology, so finding their exact solutions is of great importance.
1. Introduction
Recently, the coupled sine-Gordon equations have been introduced by Khusnutdinova and Pelinovsky [1]. The coupled sine-Gordon equations generalize the Frenkel-Kontorova dislocation model [2, 3]. System (1) with was also proposed to describe the open states in DNA model [4].
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.
2. 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.
Step 2. By taking , , we look for traveling wave solutions of (2) and transform it to the ordinary differential equation (ODE)
Step 3. Suppose that the solution of (3) can be expressed as a finite series in the form
where satisfies the Bernoulli or Riccati equation, is a positive integer that can be determined by balancing procedure [21], and are parameters to be determined.
The Bernoulli equation we consider in this paper is
where and are constants. Its solutions can be written as
where , , and are constants.
For the Riccati equation
where , , and are constants, we will use the solutions
where .
Step 4. Substituting (4) into (3) with (5) (or (7)), then the left hand side of (3) is converted into a polynomial in , and equating each coefficient of the polynomial to zero yields a set of algebraic equations for , , . Solving the algebraic equations by symbolic computation, we can determine those parameters explicitly.
Step 5. Assuming that the constants , , can be obtained in Step 4 and substituting the results into (4), then we obtain the exact traveling wave solutions for (2).
Remark 1. In (5), when and we obtain the Bernoulli equation
Equation (9) admits the following exact solutions:
when , and
when .
3. 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 Let 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.
3.1. Solutions of (18) Using the Bernoulli Equation as the Simplest Equation
The balancing procedure yields . Thus, the solution of (18) is of the form
Substituting (19) into (18) and making use of the Bernoulli equation (5) and then equating the coefficients of the functions to zero, we obtain an algebraic system of equations in terms of , , and . Solving this system of algebraic equations, with the aid of Maple, one possible set of values of , , and is
Therefore, using solutions (6) of (5), ansatz (19), we obtain the following exact solution of (18):
Substituting (21) into (17) with (12), the exact traveling wave solution to (1) can be written as where , , and , , , , are arbitrary parameters.
Now, to obtain some special cases of the above solutions, we set , , , “” take “+”; we have where , .
The equations in (24) are the same as those in of [7].
If we set , , , we have where , .
The equations in (25) are the same as those in of [7].
Substituting (22) into (17) with (12), the exact traveling wave solution to (1) can be written as where , .
Substituting (19) along with (9) into (18) and setting all the coefficients of powers to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain
Therefore, using solutions (10) and (11) of (9), ansatz (19), we obtain the following exact solution of (18):
Then the exact solution to (1) can be written as where , , and , , , are arbitrary parameters.
Now, to obtain some special cases of the above solutions, we set , ; we have where , .
The equations in (30) are the same as those in of [7].
3.2. Solutions of (18) Using Riccati Equation as the Simplest Equation
The balancing procedure yields . Thus, the solution of (18) is of the form
Substituting (31) into (18) and making use of the Riccati Equation (7) and then equating the coefficients of the functions to zero, we obtain an algebraic system of equations in terms of , , , and . Solving this system of algebraic equations, with the aid of Maple, one possible set of values of , , , and is
Therefore, using solutions (8) of (7), ansatz (31), we obtain the following exact solution of (18):
Substituting (33) into (17) with (12), the exact traveling wave solution to (1) can be written as where , .
Substituting (34) into (17) with (12), the exact traveling wave solution to (1) can be written as where , .
Remark 2. Compared with [7], the exact solutions of this paper are more general, such that when , , and in (23), the solutions become as those in (36) of [7]. When , , and in (23), the solutions become as those in (32) of [7]. When and 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.
4. 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.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the National Natural Science Foundation of China (11161020 and 11361023), the Natural Science Foundation of Yunnan Province (2011FZ193 and 2013FZ117), and the Natural Science Foundation of Education Committee of Yunnan Province (2012Y452 and 2013C079).