Symbolic Computation and New Exact Travelling Solutions for the (2+1)-Dimensional Zoomeron Equation ()

Hua Gao

Department of Applied Mathematics, Yuncheng University, Yuncheng, China.

**DOI: **10.4236/ijmnta.2014.32004
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Department of Applied Mathematics, Yuncheng University, Yuncheng, China.

In this paper, we present Yan’s sine-cosine method and Wazwaz’s sine-cosine method to solve the (2+1)-dimensional Zoomeron equation. New exact travelling wave solutions are explicitly obtained with the aid of symbolic computation. The study confirms the power of the two schemes.

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Gao, H. (2014) Symbolic Computation and New Exact Travelling Solutions for the (2+1)-Dimensional Zoomeron Equation. *International Journal of Modern Nonlinear Theory and Application*, **3**, 23-28. doi: 10.4236/ijmnta.2014.32004.

Sine-Cosine Method, (2+1)-Dimensional Zoomeron Equation, Nonlinear Evolution Equations

1. Introduction

In recent years, nonlinear evolution equations (NLEES) are widely used to describe complex phenomena in various fields of sciences, such as physics, biology, chemistry, etc. Therefore, seeking exact solutions of nonlinear evolution equations (NLEES) plays an important role in mathematical physics. In the past decades, many effective methods have been presented, such as the inverse scattering method [1] , Hirota bilinear method [2] , the tanh-function method [3] [4] , homogeneous balance method [5] [6] , Jacobi elliptic function method [7] [8] , the first-integral method [9] [10] , the Exp-function method [11] -[13] , the -expansion method [14] -[16] and so on.

Recently, Yan [17] directly obtained a simple transformation from the famous sine-Gordon equation. The simple transformation was used to get more solutions of a wide class of nonlinear wave equations [17] -[19] . The simple transformation which named sine-cosine method is based on the assumptions that the travelling wave solutions can be expressed by a trigonometric polynomial as follows:

(1)

The degree of the polynomial can be determined by considering the homogeneous balance between the highest order derivative and nonlinear terms appearing in the given NLEE. The coefficients of the polynomial can be obtained by solving a set of algebraic equations resulted from the process of using the method. More recently, a new sine-cosine method was proposed by Wazwaz [20] . The new sine-cosine algorithm admits the use of the ansatzes

(2)

(3)

where, and are parameters that will be determined later. By using Wazwaz’s sine-cosine method, many nonlinear equations [20] -[28] have been successfully solved.

In the present paper, we will extend the two sine-cosine methods to the following (2+1)-dimensional Zoomeron equation:

(4)

where is the amplitude of the relevant wave mode; see [29] . To the best of our knowledge, there are a few articles about this equation. By applying the -expansion method, Abazari [30] obtained some periodic and soliton solutions to the Zoomeron equation. Recently, Alquran and Al-Khaled [31] studied the Zoomeron equation using the extended tanh, the exp-function and the methods. In the subsequent section, we will illustrate the two sine-cosine methods in detail with the (2+1)-dimensional Zoomeron equation.

2. Yan’s Sine-Cosine Method for the (2+1)-Dimensional Zoomeron Equation

In this section, we start out our study for Equation (4) by Yan’s sine-cosine method. Firstly, making the following wave variable

(5)

where c, and and are constants to be determined later. Substitute Equation (5) into Equation (4) and integrating twice with respect to, by setting the second constant of integration to zero, we obtain the following ODE:

(6)

where is integration constant. According to Yan’s sine-cosine method, we make an ansatz (1) for the solution of Equation (6). Balancing the terms and in Equation (6) yields the leading order (from). Therefore, we can write the solution of Equation (6) in the form

(7)

and

(8)

where, and are unknown constants.

Substituting (7) into (6), collecting the coefficients of and set it to zero we obtain the following system of algebraic equations for:, , , , , and:

Solving the above system by Matlab gives

Case 1.

(9)

where, and are arbitrary constants.

Case 2.

(10)

where, and are arbitrary constants.

Case 3.

(11)

where, , and are arbitrary constants.

Now, we consider Equation (8). By using the separation of variables method the solutions of Equation (8) are easily written in the following form

(12)

or

(13)

where is the integration constant.

Finally, combining (5), (7), (12), (13) along with cases 1 - 3, we find the following three types of travelling wave solutions for Equation (4):

Type 1.

(14)

where, and are arbitrary constants.

Type 2.

(15)

where, and are arbitrary constants.

Type 3.

(16)

where, , and are arbitrary constants.

Then if we take and in the new form of (14), it is easy that our results can reduce to Abazari’s [30] result (21a). When setting, our solution (15) will be same as Alquran’s [31] result (15). It is worth to note that our solution (16) is not derived in [30] [31] .

3. Wazwaz’s Sine-Cosine Method for the (2+1)-Dimensional Zoomeron Equation

Now, we use Wazwaz’s sine-cosine method to handle Equation (4). Substituting (2) into (6) gives

(17)

The equation is satisfied only if the following system of algebraic equations hold

(18)

Solving the system (18) leads to the following sets of solutions:

(19)

where, and are any arbitrary constant. Therefore, the solution of Equation (4) is

(20)

Now, if we use the ansatze (3) instead of (2), then we get the same system (18) and therefore, the solution is

(21)

To the best of our knowledge, solutions (20) and (21) have not been reported in the literature.

4. Conclusion

The two sine-cosine methods have been successfully applied here to seek exact solutions of the (2+1)-dimensional Zoomeron equation. As a result, a series of new exact solutions are obtained and some solutions given in [30] [31] are only our special cases. The solution procedure is very simple, and the obtained solution is very concise. It is shown that the sine-cosine method provides a very effective and powerful mathematical tool for solving nonlinear equations in mathematical physics.

Acknowledgements

This work was supported by the research project of Yuncheng University (No. YQ-2011013).

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | Ablowitz, M.J. and Clarkson, P.A. (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, New York. http://dx.doi.org/10.1017/CBO9780511623998 |

[2] |
Hirota, R. (2004) The Direct Method in Soliton Theory. Cambridge University Press, New York. http://dx.doi.org/10.1017/CBO9780511543043 |

[3] |
Parkes, E.J. and Duffy, B.R. (1996) An Automated tanh-Function Method for Finding Solitary Wave Solutions to Nonlinear Evolution Equations. Computer Physics Communications, 98, 288-300. http://dx.doi.org/10.1016/0010-4655(96)00104-X |

[4] | Fan, E.G. (2000) Extended tanh-Function Method and Its Applications to Nonlinear Equations. Physics Letters A, 277, 212-218. http://dx.doi.org/10.1016/S0375-9601(00)00725-8 |

[5] |
Wang, M.L. (1995) Solitary Wave Solutions for Variant Boussinesq Equations. Physics Letters A, 199, 169-172. http://dx.doi.org/10.1016/0375-9601(95)00092-H |

[6] |
Wang, M.L., Zhou, Y.B. and Li, Z.B. (1996) Applications of a Homogeneous Balance Method to Exact Solutions of Nonlinear Equations in Mathematical Physics. Physics Letters A, 216, 67-75. http://dx.doi.org/10.1016/0375-9601(96)00283-6 |

[7] | Liu, S.K., Fu, Z.T. and Liu, S.D. (2001) Jacobi Elliptic Function Expansion Method and Periodic Wave Solutions of Nonlinear Wave Equations. Physics Letters A, 289, 69-74. http://dx.doi.org/10.1016/S0375-9601(01)00580-1 |

[8] | Fu, Z.T., Liu, S.K. and Liu, S.D. (2001) New Jacobi Elliptic Function Expansion and New Periodic Wave Solutions of Nonlinear Wave Equations. Physics Letters A, 290, 72-76. http://dx.doi.org/10.1016/S0375-9601(01)00644-2 |

[9] |
Feng, Z.S. (2002) On Explicit Exact Solutions to the Compound Burgers-KdV Equation. Physics Letters A, 293, 57-66. http://dx.doi.org/10.1016/S0375-9601(01)00825-8 |

[10] | Feng, Z.S. (2002) Exact Solution to an Approximate Sine-Gordon Equation in (n+1)-Dimensional Space. Physics Letters A, 302, 64-76. http://dx.doi.org/10.1016/S0375-9601(02)01114-3 |

[11] | He, J.H. and Wu, X.H. (2006) Exp-Function Method for Nonlinear Wave Equations. Chaos Solitons Fractals, 30, 700-708. http://dx.doi.org/10.1016/j.chaos.2006.03.020 |

[12] | He, J.H. and Abdou, M.A. (2007) New Periodic Solutions for Nonlinear Evolutions Using Exp-Function Method. Chaos Solitons Fractals, 34, 1421-1429. http://dx.doi.org/10.1016/j.chaos.2006.05.072 |

[13] | Gao, H. and Zhao, R.X. (2010) New Exact Solutions to the Generalized Burgers-Huxley Equation. Applied Mathematics and Computation, 217, 1598-1603. http://dx.doi.org/10.1016/j.amc.2009.07.020 |

[14] |
Wang, M.L., Li, X.Z. and Zhang, J.L. (2008) The (G’/G)-Expansion Method and Travelling Wave Solutions of Nonlinear Evolution Equations in Mathematical Physics. Physics Letters A, 372, 417-423. http://dx.doi.org/10.1016/j.physleta.2007.07.051 |

[15] |
Wang, M.L., Zhang, J.L. and Li, X.Z. (2008) Application of the (G’/G)-Expansion to Travelling Wave Solutions of the Broer-Kaup and the Approximate Long Water Wave Equations. Applied Mathematics and Computation, 206, 321-326. http://dx.doi.org/10.1016/j.amc.2008.08.045 |

[16] |
Gao, H. and Zhao, R.X. (2009) New Application of the (G’/G)-Expansion Method to Higher-Order Nonliear Equations. Applied Mathematics and Computation, 215, 2781-2786. http://dx.doi.org/10.1016/j.amc.2009.08.041 |

[17] |
Yan, C.T. (1996) A Simple Transformation for Nonlinear Waves. Physics Letters A, 224, 77-84. http://dx.doi.org/10.1016/S0375-9601(96)00770-0 |

[18] |
Yan, Z.Y. and Zhang, H.Q. (1999) New Explicit and Exact Travelling Wave Solutions for a System of Variant Boussinesq Equations in Mathematical Physics. Physics Letters A, 252, 291-296. http://dx.doi.org/10.1016/S0375-9601(98)00956-6 |

[19] | Yan, Z.Y. and Zhang, H.Q. (2000) On a New Algorithm of Constructing Solitary Wave Solutions for Systems of Nonlinear Evolution Equations in Mathematical Physics. Applied Mathematics and Computation, 21, 382-388. |

[20] | Wazwaz, A.M. (2004) A Sine-Cosine Method for Handling Nonlinear Wave Equations. Mathematical and Computer Modelling, 40, 499-508. http://dx.doi.org/10.1016/j.mcm.2003.12.010 |

[21] | Wazwaz, A.M. (2004) Distinct Variants of the KdV Equation with Compact and Noncompact Structures. Applied Mathematics and Computation, 150, 365-377. http://dx.doi.org/10.1016/S0096-3003(03)00238-8 |

[22] | Wazwaz, A.M. (2004) Variants of the Generalized KdV Equation with Compact and Noncompact Structures. Computers & Mathematics with Applications, 47, 583-591. http://dx.doi.org/10.1016/S0898-1221(04)90047-8 |

[23] | Wazwaz, A.M. (2006) Solitons and Periodic Solutions for the Fifth-Order KdV Equation. Applied Mathematics Letters, 19, 162-167. http://dx.doi.org/10.1016/j.aml.2005.07.014 |

[24] | Tang, S., Xiao, Y. and Wang, Z. (2009) Travelling Wave Solutions for a Class of Nonlinear Fourth Order Variant of a Generalized Camassa-Holm Equation. Applied Mathematics Letters, 210, 39-47. |

[25] | Alquran, M. (2012) Solitons and Periodic Solutions to Nonlinear Partial Differential Equations by the Sine-Cosine Method. Applied Mathematics & Information Sciences, 6, 85-88. |

[26] | Alquran, M., Ali, M. and Al-Khaled, K. (2012) Solitary Wave Solutions to Shallow Water Waves Arising in Fluid Dynamics. Journal of Nonlinear Studies, 19, 555-562. |

[27] | Alquran, M. and Qawasmeh, A. (2013) Classifications of Solutions to Some Generalized Nonlinear Evolution Equations and Systems by the Sine-Cosine Method. Journal of Nonlinear Studies, 20, 263-272. |

[28] |
Alquran, M. and Al-Khaled, K. (2011) The tanh and Sine-Cosine Methods for Higher Order Equation of Kortewe-de Vrie Type. Physica Scripta, 84, 25010-25013.
http://dx.doi.org/10.1088/0031-8949/84/02/025010 |

[29] | Calogero, F. and Degasperis, A. (1976) Nonlinear Evolution Equations Solvable by the Inverse Spectral Transform I. Nuovo Cimento B, 32, 201-242. http://dx.doi.org/10.1007/BF02727634 |

[30] | Abazari, R. (2011) The Solitary Wave Solutions of Zoomeron Equation. Applied Mathematical Sciences, 5, 2943-2949. |

[31] | Alquran, M. and Al-Khaled, K. (2012) Mathematical Methods for a Reliable Treatment of the (2+1)-Dimensional Zoomeron Equation. Mathematical Sciences, 6, 1-12. http://dx.doi.org/10.1186/2251-7456-6-11 |

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