In recent years, many approaches have been utilized for finding the exact solutions of nonlinear partial differential equations. One such method is known as the first integral method and was proposed by Feng. In this paper, we utilize this method and obtain exact solutions of two nonlinear partial differential equations, namely double sine-Gordon and Burgers equations. It is found that the method by Feng is a very efficient method which can be used to obtain exact solutions of a large number of nonlinear partial differential equations.
Keywords:first integral method; double sine-Gordon equation; Burgers equation; exact solutions
With the availability of symbolic computation packages like Maple or Mathematica, the search for obtaining exact solutions of nonlinear partial differential equations (PDEs) has become more and more stimulating for mathematicians and scientists. Having exact solutions of nonlinear PDEs makes it possible to study nonlinear physical phenomena thoroughly and facilitates testing the numerical solvers as well as aiding the stability analysis of solutions. In recent years, many approaches to solve nonlinear PDEs such as the extended tanh function method [1-6], the modified extended tanh function method [7,8], the exp-function method [9-11], the Weierstrass elliptic function method , the Laplace decomposition method [13,14] and so on have been employed.
Among these, the first integral method, which is based on the ring theory of commutative algebra, due to Feng [15-19] has been applied by many authors to solve different types of nonlinear equations in science and engineering [20-23]. Therefore, in the present article, the first integral method is applied to analytic treatment of some important nonlinear of partial differential equations.
The rest of this article is arranged as follows. In Section 2, the basic ideas of the first integral method are expressed. In Section 3, the method is employed for obtaining the exact solutions of double sine-Gordon (SG) and Burgers equations, and finally conclusions are presented in Section 4.
2 The first integral method
where c is constant. Then we have
Thus PDE (1) is then transformed to the ordinary differential equation (ODE)
We now introduce a new transformation, namely
and this gives us the system of ODEs
If we can find the integrals of (6) under the same conditions, the qualitative theory of differential equations  tells us that the general solutions of (6) can be obtained directly. But in general, it is very difficult even for a single first integral. Since for a plane autonomous system, there is no methodical theory which gives us first integrals, we will therefore apply the division theorem to find one first integral (6), which will reduce (4) to a first-order integral for an ordinary differential equation. By solving this equation, exact solutions of (1) will be obtained. We recall the division theorem.
Theorem 2.1 (Division theorem, see )
The division theorem follows immediately from the Hilbert-Nullstellensatz theorem .
Theorem 2.2 (Hilbert-Nullstellensatz theorem)
LetKbe a field andLbe an algebraic closure ofK. Then:
(ii) Let, be two elements of; for the set of polynomials ofzero atxto be identical with the set of polynomials ofzero aty, it is necessary and sufficient that there exists a K-automorphismSofLsuch thatfor.
3.1 Exact solutions to the double sine-Gordon equation
In order to apply the first integral method described in Section 2, we first introduce the transformations
Using (2) and (3), Eq. (7) becomes
We next use the transformation
Substituting (20) in (14), we obtain
Combining Eq. (21) with (13), second-order differential Eq. (11) can be reduced to
Solving Eq. (22) directly and changing to the original variables, we obtain the exact solutions to Eq. (7):
Therefore, the exact solutions to the double sine-Gordon (SG) equation can be written as
3.2 Exact solutions to the Burgers equation
The Burgers equation 
is one of the most famous nonlinear diffusion equations. The positive parameter a refers to a dissipative effect.
Using (2) and (3), Eq. (27) becomes
We rewrite (28) as follows:
Since is a polynomial in X, from (33) we conclude that is a constant and . For simplicity, we take , and balancing the degrees of and , we conclude that or 1. If , suppose that , then from (34), we find
where B is an arbitrary integration constant.
Using (36) in (31), we obtain
Combining Eq. (37) with the first part of (30), we obtain the exact solutions of Eq. (29) as follows:
Therefore, the exact solutions to the Burgers equation can be written as
Since is a polynomial of X, from (41) we conclude that is a constant and . For simplicity, we take , and balancing the degrees of , and , we conclude that or 1. If , suppose that . Then from (42), we find
where B is an arbitrary constant of integration. From (43) we have
where D is an arbitrary constant of integration. Substituting and in (44) and setting all the coefficients of powers of X to zero, we obtain a system of nonlinear algebraic equations. Solving these equations, we obtain
Now using (45) in (31), we get
Combining Eq. (46) with the first part of (30), we obtain the exact solutions to Eq. (29) in the form
The first integral method was employed successfully to solve some important nonlinear partial differential equations, including the double sine-Gordon and Burgers equations, analytically. Some exact solutions for these equations were formally obtained by applying the first integral method. Due to the good performance of the first integral method, we feel that it is a powerful technique in handling a wide variety of nonlinear partial differential equations. Also, this method is computerizable, which permits us to accomplish difficult and tiresome algebraic calculations on a computer with ease.
The authors declare that they have no competing interests.
The authors declare that the manuscript was realized in collaboration with the same responsibility. All authors read and approved the final version of the manuscript.
Wazwaz, AM: New solitary wave and periodic wave solutions to the -dimensional Nizhnik-Novikov-Veselov system. Appl. Math. Comput.. 187, 1584–1591 (2007). Publisher Full Text
Abdou, MA: The extended tanh method and its applications for solving nonlinear physical models. Appl. Math. Comput.. 190, 988–996 (2007). Publisher Full Text
Shukri, S, Al-Khaled, K: The extended tanh method for solving systems of nonlinear wave equations. Appl. Math. Comput.. 217(5), 1997–2006 (2010). Publisher Full Text
Malfliet, W, Hereman, W: The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Phys. Scr.. 54, 563–568 (1996). Publisher Full Text
Malfliet, W, Hereman, W: The tanh method: II. Perturbation technique for conservative systems. Phys. Scr.. 54, 569–575 (1996). Publisher Full Text
Soliman, AA: The modified extended tanh-function method for solving Burgers-type equations. Physica A. 361, 394–404 (2006). Publisher Full Text
Abdou, MA, Soliman, AA: Modified extended tanh-function method and its application on nonlinear physical equations. Phys. Lett. A. 353, 487–492 (2006). Publisher Full Text
Wazwaz, AM: Single and multiple-soliton and solutions for the -dimensional KdV equation. Appl. Math. Comput.. 204, 20–26 (2008). Publisher Full Text
Wazwaz, AM, Mehanna, MS: A variety of exact travelling wave solutions for the -dimensional Boiti-Leon-Pempinelli equation. Appl. Math. Comput.. 217(4), 1484–1490 (2010). Publisher Full Text
Zhang, S, Zhang, HQ: An exp-function method for new N-soliton solutions with arbitrary functions of a -dimensional vcBK system. Comput. Math. Appl.. 61(8), 1923–1930 (2011). Publisher Full Text
Deng, X, Cao, J, Li, X: Travelling wave solutions for the nonlinear dispersion Drinfeld-Sokolov system. Commun. Nonlinear Sci. Numer. Simul.. 15, 281–290 (2010). Publisher Full Text
Lu, B, Zhang, HQ, Xie, FD: Travelling wave solutions of nonlinear partial equations by using the first integral method. Appl. Math. Comput.. 216, 1329–1336 (2010). Publisher Full Text
Feng, Z: On explicit exact solutions to the compound Burgers-KdV equation. Phys. Lett. A. 293, 57–66 (2002). Publisher Full Text
Feng, Z: Exact solution to an approximate sine-Gordon equation in -dimensional space. Phys. Lett. A. 302, 64–76 (2002). Publisher Full Text
Feng, Z, Wang, X: The first integral method to the two-dimensional Burgers-Korteweg-de Vries equation. Phys. Lett. A. 308, 173–178 (2003). Publisher Full Text
Feng, Z: Traveling wave behavior for a generalized Fisher equation. Chaos Solitons Fractals. 38, 481–488 (2008). Publisher Full Text
Tascan, F, Bekir, A: Travelling wave solutions of the Cahn-Allen equation by using first integral method. Appl. Math. Comput.. 207, 279–282 (2009). Publisher Full Text
Raslan, KR: The first integral method for solving some important nonlinear partial differential equations. Nonlinear Dyn.. 53(4), 281–286 (2008). Publisher Full Text
Taghizadeh, N, Mirzazadeh, M, Farahrooz, F: Exact solutions of the nonlinear Schrodinger equation by the first integral method. J. Math. Anal. Appl.. 374, 549–553 (2011). Publisher Full Text
Zhang, TS: Exp-function method for solving Maccari’s system. Phys. Lett. A. 371(1-2), 65–71 (2007). Publisher Full Text
Jiang, Z: The construction of the scattering data for a class of multidimensional scattering operators. Inverse Probl.. 5, 349–374 (1989). Publisher Full Text
Hereman, W, Malfliet, W: The tanh method: a tool to solve nonlinear partial differential equations with symbolic software. Phys. Scr.. 54, 563–568 (1996). Publisher Full Text