# Exact solutions of two nonlinear partial differential equations by using the first integral method

- Hossein Jafari
^{1, 2}Email author, - Rahmat Soltani
^{1}, - Chaudry Masood Khalique
^{2}and - Dumitru Baleanu
^{3, 4, 5}

**2013**:117

**DOI: **10.1186/1687-2770-2013-117

© Jafari et al.; licensee Springer. 2013

**Received: **22 August 2012

**Accepted: **12 April 2013

**Published: **7 May 2013

## Abstract

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## 1 Introduction

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 [12], 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

*c*is constant. Then we have

If we can find the integrals of (6) under the same conditions, the qualitative theory of differential equations [24] 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 [25])

*Let* $P(x,y)$ *and* $Q(x,y)$ *be polynomials of two variables* *x* *and* *y* *in* $\mathbf{C}[x,y]$, *and let* $P(x,y)$ *be irreducible in* $\mathbf{C}[x,y]$. *If* $Q(x,y)$ *vanishes at all zero points of* $P(x,y)$, *then there exists a polynomial* $G(x,y)$ *in* $\mathbf{C}[x,y]$ *such that* $Q(x,y)=P(x,y)G(x,y)$.

The division theorem follows immediately from the Hilbert-Nullstellensatz theorem [26].

**Theorem 2.2** (Hilbert-Nullstellensatz theorem)

*Let*

*K*

*be a field and*

*L*

*be an algebraic closure of*

*K*.

*Then*:

- (i)
*Every ideal**γ**of*$K[{X}_{1},\dots ,{X}_{n}]$*not containing*1*admits at least one zero in*${L}^{n}$. - (ii)
*Let*$x=({x}_{1},\dots ,{x}_{n})$, $y=({y}_{1},\dots ,{y}_{n})$*be two elements of*${L}^{n}$;*for the set of polynomials of*$K[{X}_{1},\dots ,{X}_{n}]$*zero at**x**to be identical with the set of polynomials of*$K[{X}_{1},\dots ,{X}_{n}]$*zero at**y*,*it is necessary and sufficient that there exists a K*-*automorphism**S**of**L**such that*${y}_{i}=S({x}_{i})$*for*$1\le i\le n$. - (iii)
*For an ideal**α**of*$K[{X}_{1},\dots ,{X}_{n}]$*to be maximal*,*it is necessary and sufficient that there exists an**x**in*${L}^{n}$*such that**α**is the set of polynomials of*$K[{X}_{1},\dots ,{X}_{n}]$*zero at**x*. - (iv)
*For a polynomial**Q**of*$K[{X}_{1},\dots ,{X}_{n}]$*to be zero on the set of zeros in*${L}^{n}$*of an ideal**γ**of*$K[{X}_{1},\dots ,{X}_{n}]$,*it is necessary and sufficient that there exists an integer*$m>0$*such that*${Q}^{m}\in \gamma $.

## 3 Applications

### 3.1 Exact solutions to the double sine-Gordon equation

where $a=\frac{1}{2c}$.

*X*and ${a}_{m}(X)\ne 0$. Equation (14) is called the first integral to Eq. (13). Applying the division theorem, one sees that there exists a polynomial $H(X,Y)=h(X)+g(X)Y$ in the complex domain $\mathbf{C}[X,Y]$ such that

*X*, from (16) we conclude that ${a}_{1}(X)$ is a constant and $g(X)=1$. For simplicity, we take ${a}_{1}(X)=1$. Then Eq. (17) indicates that $degh(X)\le deg{a}_{0}(X)$. Thus, from Eq. (18) we conclude that $degh(X)=deg{a}_{0}(X)=2$. Now suppose that

*X*to be zero, we obtain a system of nonlinear algebraic equations, and by solving it, we obtain

where ${c}_{1}$ is an arbitrary constant and $a=\frac{1}{2c}$.

where ${c}_{1}$ is an arbitrary constant.

### 3.2 Exact solutions to the Burgers equation

is one of the most famous nonlinear diffusion equations. The positive parameter *a* refers to a dissipative effect.

*X*and ${a}_{m}(X)\ne 0$. Equation (31) is called the first integral to Eq. (30). Due to the division theorem, there exists a polynomial $H(X,Y)=h(X)+g(X)Y$ in $\mathbf{C}[X,Y]$ such that

*X*, from (33) we conclude that ${a}_{1}(X)$ is a constant and $g(X)=0$. For simplicity, we take ${a}_{1}(X)=1$, and balancing the degrees of $h(X)$ and ${a}_{0}(X)$, we conclude that $degh(X)=0$ or 1. If $degh(X)=0$, suppose that $h(X)=A$, then from (34), we find

where *B* is an arbitrary integration constant.

*X*to be zero, we obtain a system of nonlinear algebraic equations, and by solving it, we obtain

where ${\xi}_{0}$ is an arbitrary constant.

where ${\xi}_{0}$ is an arbitrary constant and $a>0$.

*X*, from (41) we conclude that ${a}_{2}(X)$ is a constant and $g(X)=0$. For simplicity, we take ${a}_{2}(X)=1$, and balancing the degrees of $h(X)$, ${a}_{0}(X)$ and ${a}_{1}(X)$, we conclude that $degh(X)=0$ or 1. If $degh(X)=0$, suppose that $h(X)=A$. Then from (42), we find

*B*is an arbitrary constant of integration. From (43) we have

*D*is an arbitrary constant of integration. Substituting ${a}_{0}(X)$ and $h(X)$ 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

where ${\xi}_{0}$ is an arbitrary constant and $a>0$.

## 4 Conclusions

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.

## Declarations

## Authors’ Affiliations

## References

- Wazwaz AM:New solitary wave and periodic wave solutions to the $(2+1)$-dimensional Nizhnik-Novikov-Veselov system.
*Appl. Math. Comput.*2007, 187: 1584-1591. 10.1016/j.amc.2006.09.069MATHMathSciNetView ArticleGoogle Scholar - Abdou MA: The extended tanh method and its applications for solving nonlinear physical models.
*Appl. Math. Comput.*2007, 190: 988-996. 10.1016/j.amc.2007.01.070MATHMathSciNetView ArticleGoogle Scholar - Shukri S, Al-Khaled K: The extended tanh method for solving systems of nonlinear wave equations.
*Appl. Math. Comput.*2010, 217(5):1997-2006. 10.1016/j.amc.2010.06.058MATHMathSciNetView ArticleGoogle Scholar - Malfliet W: The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations.
*J. Comput. Appl. Math.*2004, 164: 529-541.MathSciNetView ArticleGoogle Scholar - Malfliet W, Hereman W: The tanh method: I. Exact solutions of nonlinear evolution and wave equations.
*Phys. Scr.*1996, 54: 563-568. 10.1088/0031-8949/54/6/003MATHMathSciNetView ArticleGoogle Scholar - Malfliet W, Hereman W: The tanh method: II. Perturbation technique for conservative systems.
*Phys. Scr.*1996, 54: 569-575. 10.1088/0031-8949/54/6/004MATHMathSciNetView ArticleGoogle Scholar - Soliman AA: The modified extended tanh-function method for solving Burgers-type equations.
*Physica A*2006, 361: 394-404. 10.1016/j.physa.2005.07.008MathSciNetView ArticleGoogle Scholar - Abdou MA, Soliman AA: Modified extended tanh-function method and its application on nonlinear physical equations.
*Phys. Lett. A*2006, 353: 487-492. 10.1016/j.physleta.2006.01.013View ArticleGoogle Scholar - Wazwaz AM:Single and multiple-soliton and solutions for the $(2+1)$-dimensional KdV equation.
*Appl. Math. Comput.*2008, 204: 20-26. 10.1016/j.amc.2008.05.126MATHMathSciNetView ArticleGoogle Scholar - Wazwaz AM, Mehanna MS:A variety of exact travelling wave solutions for the $(2+1)$-dimensional Boiti-Leon-Pempinelli equation.
*Appl. Math. Comput.*2010, 217(4):1484-1490. 10.1016/j.amc.2009.06.024MATHMathSciNetView ArticleGoogle Scholar - Zhang S, Zhang HQ: An exp-function method for new
*N*-soliton solutions with arbitrary functions of a $(2+1)$ -dimensional vcBK system.*Comput. Math. Appl.*2011, 61(8):1923-1930. 10.1016/j.camwa.2010.07.042MATHMathSciNetView ArticleGoogle Scholar - Deng X, Cao J, Li X:Travelling wave solutions for the nonlinear dispersion Drinfeld-Sokolov $(D(m,n))$ system.
*Commun. Nonlinear Sci. Numer. Simul.*2010, 15: 281-290. 10.1016/j.cnsns.2009.03.023MATHMathSciNetView ArticleGoogle Scholar - Khan Y, Austin F: Application of the Laplace decomposition method to nonlinear. homogeneous and non-homogenous advection equations.
*Z. Naturforsch.*2010, 65a: 849-853.Google Scholar - Baleanu D, Diethelm K, Scalas E, Trujillo JJ Series on Complexity, Nonlinearity and Chaos. In
*Fractional Calculus Models and Numerical Methods*. World Scientific, Singapore; 2012.Google Scholar - Lu B, Zhang HQ, Xie FD: Travelling wave solutions of nonlinear partial equations by using the first integral method.
*Appl. Math. Comput.*2010, 216: 1329-1336. 10.1016/j.amc.2010.02.028MATHMathSciNetView ArticleGoogle Scholar - Feng Z: On explicit exact solutions to the compound Burgers-KdV equation.
*Phys. Lett. A*2002, 293: 57-66. 10.1016/S0375-9601(01)00825-8MATHMathSciNetView ArticleGoogle Scholar - Feng Z:Exact solution to an approximate sine-Gordon equation in $(n+1)$-dimensional space.
*Phys. Lett. A*2002, 302: 64-76. 10.1016/S0375-9601(02)01114-3MATHMathSciNetView ArticleGoogle Scholar - Feng Z, Wang X: The first integral method to the two-dimensional Burgers-Korteweg-de Vries equation.
*Phys. Lett. A*2003, 308: 173-178. 10.1016/S0375-9601(03)00016-1MATHMathSciNetView ArticleGoogle Scholar - Feng Z: Traveling wave behavior for a generalized Fisher equation.
*Chaos Solitons Fractals*2008, 38: 481-488. 10.1016/j.chaos.2006.11.031MATHMathSciNetView ArticleGoogle Scholar - Tascan F, Bekir A: Travelling wave solutions of the Cahn-Allen equation by using first integral method.
*Appl. Math. Comput.*2009, 207: 279-282. 10.1016/j.amc.2008.10.031MATHMathSciNetView ArticleGoogle Scholar - Raslan KR: The first integral method for solving some important nonlinear partial differential equations.
*Nonlinear Dyn.*2008, 53(4):281-286. 10.1007/s11071-007-9262-xMATHMathSciNetView ArticleGoogle Scholar - Taghizadeh N, Mirzazadeh M, Farahrooz F: Exact solutions of the nonlinear Schrodinger equation by the first integral method.
*J. Math. Anal. Appl.*2011, 374: 549-553. 10.1016/j.jmaa.2010.08.050MATHMathSciNetView ArticleGoogle Scholar - Jafari H, Sooraki A, Talebi Y, Biswas A: The first integral method and traveling wave solutions to Davey-Stewartson equation.
*Nonlinear Anal., Model. Control*2012, 17(2):182-193.MATHMathSciNetGoogle Scholar - Ding TR, Li CZ:
*Ordinary Differential Equations*. Peking University Press, Peking; 1996.Google Scholar - Zhang TS: Exp-function method for solving Maccari’s system.
*Phys. Lett. A*2007, 371(1-2):65-71. 10.1016/j.physleta.2007.05.091MATHView ArticleGoogle Scholar - Bourbaki N:
*Commutative Algebra*. Addison-Wesley, Paris; 1972.MATHGoogle Scholar - Fan E, Hon YC: Generalized tanh method extended to special types of nonlinear equations.
*Z. Naturforsch.*2002, 57a: 692-700.Google Scholar - Jiang Z: The construction of the scattering data for a class of multidimensional scattering operators.
*Inverse Probl.*1989, 5: 349-374. 10.1088/0266-5611/5/3/009MATHView ArticleGoogle Scholar - Hereman W, Malfliet W: The tanh method: a tool to solve nonlinear partial differential equations with symbolic software.
*Phys. Scr.*1996, 54: 563-568. 10.1088/0031-8949/54/6/003MATHMathSciNetView ArticleGoogle Scholar

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