New Exact Explicit Solutions of the Generalized Zakharov Equation via the First Integral Method
Affiliation(s)
Institute of Mathematics and Software Science, Sichuan Normal University, Chengdu, China.
Institute of Mathematics and Software Science, Sichuan Normal University, Chengdu, China.
ABSTRACT
The generalized Zakharov equation is a coupled equation which is a classic nonlinear mathematic model in plasma. A series of new exact explicit solutions of the system are obtained, by means of the first integral method, in the form of trigonometric and exponential functions. The results show the first integral method is an efficient way to solve the coupled nonlinear equations and get rich explicit analytical solutions.
The generalized Zakharov equation is a coupled equation which is a classic nonlinear mathematic model in plasma. A series of new exact explicit solutions of the system are obtained, by means of the first integral method, in the form of trigonometric and exponential functions. The results show the first integral method is an efficient way to solve the coupled nonlinear equations and get rich explicit analytical solutions.
Cite this paper
Sun, Y. , Hu, H. and Zhang, J. (2014) New Exact Explicit Solutions of the Generalized Zakharov Equation via the First Integral Method. Open Journal of Applied Sciences, 4, 249-257. doi: 10.4236/ojapps.2014.45025.
Sun, Y. , Hu, H. and Zhang, J. (2014) New Exact Explicit Solutions of the Generalized Zakharov Equation via the First Integral Method. Open Journal of Applied Sciences, 4, 249-257. doi: 10.4236/ojapps.2014.45025.
References
[1] Malomed, B., Anderson, D., Lisak, M., Quiroga-Teixeiro, M.L. and Stenflo, L. (1997) Dynamics of Solitary Waves in the Zakharov Model Equations. Physical Review E, 55, 962-968.
http://dx.doi.org/10.1103/PhysRevE.55.962 [2] Layeni, O.P. (2009) A New Rational Auxiliary Equation Method and Exact Solutions of a Generalized Zakharov System. Applied Mathematics and Computation, 215, 2901-2907.
http://dx.doi.org/10.1016/j.amc.2009.09.034 [3] El-Wakil, S.A., Degheidy, A.R., Abulwafa, E.M., Madkour, M.A., Attia, M.T. and Abdou, M.A. (2009) Exact Travelling Wave Solutions of Generalized Zakharov Equations with Arbitrary Power Nonlinearities. International Journal of Nonlinear Science, 7, 455-461. [4] Li, Y.-Z., Li, K.-M. and Lin, C. (2008) Exp-Function Method for Solving the Generalized-Zakharov Equations. Applied Mathematics and Computation, 205, 197-201.
http://dx.doi.org/10.1016/j.amc.2008.05.138 [5] Borhanifar, A., Kabir, M.M. and Maryam Vahdat, L. (2009) New Periodic and Soliton Wave Solutions for the Generalized Zakharov System and (2 + 1)-Dimensional Nizhnik-Novikov-Veselov System. Chaos, Solitons and Fractals, 42, 1646-1654.
http://dx.doi.org/10.1016/j.chaos.2009.03.064 [6] Hong, B.j., Zhu, W.G. and Lu, D.C. (2012) New Explicit Exact Solutions to the Generalized Zakharov Equations. Journal of Anhui University (Natural Science Edition), 36, 37-42. [7] Javidi, M. and Golbabai, A. (2007) Construction of a Solitary Wave Solution for the Generalized Zakharov Equation by a Variational Iteration Method. Computers and Mathematics with Applications, 54, 1003-1009.
http://dx.doi.org/10.1016/j.camwa.2006.12.044 [8] Guo, B.L., Zhang, J.J. and Pu, X.K. (2010) On the Existence and Uniqueness of Smooth Solution for a Generalized Zakharov Equation. Journal of Mathematical Analysis and Applications, 365, 238-253.
http://dx.doi.org/10.1016/j.jmaa.2009.10.045 [9] Betchewe, G., Thomas, B.B., Victor, K.K. and Crepin, K.T. (2010) Dynamical Survey of a Generalized-Zakharov Equation and Its Exact Travelling Wave Solutions. Applied Mathematics and Computation, 217, 203-211.
http://dx.doi.org/10.1016/j.amc.2010.05.044 [10] Abbasbandy, S., Babolian, E. and Ashtiani, M. (2009) Numerical Solution of the Generalized Zakharov Equation by Homotopy Analysis Method. Communications in Non-linear Science and Numerical Simulation, 14, 4114-4121.
http://dx.doi.org/10.1016/j.amc.2010.05.044 [11] Feng, Z.S. (2002) The First Integral Method to Study the Burgers-Kortewegde Vries Equation. Journal of Physics A: Mathematical and General Physics, 35, 343-349. [12] Lu, B., Zhang, H.Q. and Xie, F.D. (2010) Travelling Wave Solutions of Nonlinear Partial Equations by Using the First Integral Method. Applied Mathematics and Computation, 216, 1329-1336.
http://dx.doi.org/10.1016/j.amc.2010.02.028 [13] Jafari, H., Sooraki, A., Talebi, Y. and Biswas, A. (2012) The First Integral Method and Traveling Wave Solutions to Davey-Stewartson Equation. Nonlinear Analysis: Modelling and Control, 17, 182-193. [14] Ke, Y.-Q. and Yu, J. (2005) The First Integral Method to Study a Class of Reaction-Diffusion Equations. Communications in Theoretical Physics, 43, 597-600. [15] Hosseini, K., Ansari, R. and Gholamin, P. (2012) Exact Solutions of Some Nonlinear Systems of Partial Differential Equations by Using the First Integral Method. Journal of Mathematical Analysis and Applications, 387, 807-814.
http://dx.doi.org/10.1016/j.jmaa.2011.09.044 [16] Kheiri, H., Hajizadeh, R. and Abbasnezhad, N. (2010) The First Integral Method for Solving Some Nonlinear Equations. Armenian Journal of Physics, 3, 82-97. [17] Taghizadeh, N. and Mirzazadeh, M. (2011) The First Integral Method to Some Complex Nonlinear Partial Differential Equations. Journal of Computational and Applied Mathematics, 235, 4871-4877.
http://dx.doi.org/10.1016/j.cam.2011.02.021 [18] El-Sabbagh, M.F. and El-Ganaini, S.I. (2012) The First Integral Method and Its Applications to Nonlinear Equations. Applied Mathematical Sciences, 6, 3893-3906. [19] Lu, B. (2012) The First Integral Method for Some Time Fractional Differential Equations. Journal of Mathematical Analysis and Applications, 395, 684-693.
http://dx.doi.org/10.1016/j.jmaa.2012.05.066 [20] Taghizadeh, N., Mirzazadeh, M. and Tascan, F. (2012) The First-Integral Method Applied to the Eckhaus Equation. Applied Mathematics Letters, 25, 798-802.
http://dx.doi.org/10.1016/j.aml.2011.10.021 [21] Rostamy, D., Zabihi, F., Karimi, K. and Khalehoghli, S. (2011) The First Integral Method for Solving Maccari’s System. Applied Mathematics, 2, 258-263.
http://dx.doi.org/10.4236/am.2011.22030 [22] Deng, X.J. (2008) Exact Peaked Wave Solution of CH-γ Equation by the First-Integral Method. Applied Mathematics and Computation, 206, 806-809.
http://dx.doi.org/10.1016/j.amc.2008.09.039 [23] Tascan, F., Bekir, A. and Koparan, M. (2009) Travelling Wave Solutions of Nonlinear Evolution Equations by Using the First Integral Method. Communications in Nonlinear Science and Numerical Simulation, 14, 1810-1815.
http://dx.doi.org/10.1016/j.cnsns.2008.07.009
[1] Malomed, B., Anderson, D., Lisak, M., Quiroga-Teixeiro, M.L. and Stenflo, L. (1997) Dynamics of Solitary Waves in the Zakharov Model Equations. Physical Review E, 55, 962-968.
http://dx.doi.org/10.1103/PhysRevE.55.962 [2] Layeni, O.P. (2009) A New Rational Auxiliary Equation Method and Exact Solutions of a Generalized Zakharov System. Applied Mathematics and Computation, 215, 2901-2907.
http://dx.doi.org/10.1016/j.amc.2009.09.034 [3] El-Wakil, S.A., Degheidy, A.R., Abulwafa, E.M., Madkour, M.A., Attia, M.T. and Abdou, M.A. (2009) Exact Travelling Wave Solutions of Generalized Zakharov Equations with Arbitrary Power Nonlinearities. International Journal of Nonlinear Science, 7, 455-461. [4] Li, Y.-Z., Li, K.-M. and Lin, C. (2008) Exp-Function Method for Solving the Generalized-Zakharov Equations. Applied Mathematics and Computation, 205, 197-201.
http://dx.doi.org/10.1016/j.amc.2008.05.138 [5] Borhanifar, A., Kabir, M.M. and Maryam Vahdat, L. (2009) New Periodic and Soliton Wave Solutions for the Generalized Zakharov System and (2 + 1)-Dimensional Nizhnik-Novikov-Veselov System. Chaos, Solitons and Fractals, 42, 1646-1654.
http://dx.doi.org/10.1016/j.chaos.2009.03.064 [6] Hong, B.j., Zhu, W.G. and Lu, D.C. (2012) New Explicit Exact Solutions to the Generalized Zakharov Equations. Journal of Anhui University (Natural Science Edition), 36, 37-42. [7] Javidi, M. and Golbabai, A. (2007) Construction of a Solitary Wave Solution for the Generalized Zakharov Equation by a Variational Iteration Method. Computers and Mathematics with Applications, 54, 1003-1009.
http://dx.doi.org/10.1016/j.camwa.2006.12.044 [8] Guo, B.L., Zhang, J.J. and Pu, X.K. (2010) On the Existence and Uniqueness of Smooth Solution for a Generalized Zakharov Equation. Journal of Mathematical Analysis and Applications, 365, 238-253.
http://dx.doi.org/10.1016/j.jmaa.2009.10.045 [9] Betchewe, G., Thomas, B.B., Victor, K.K. and Crepin, K.T. (2010) Dynamical Survey of a Generalized-Zakharov Equation and Its Exact Travelling Wave Solutions. Applied Mathematics and Computation, 217, 203-211.
http://dx.doi.org/10.1016/j.amc.2010.05.044 [10] Abbasbandy, S., Babolian, E. and Ashtiani, M. (2009) Numerical Solution of the Generalized Zakharov Equation by Homotopy Analysis Method. Communications in Non-linear Science and Numerical Simulation, 14, 4114-4121.
http://dx.doi.org/10.1016/j.amc.2010.05.044 [11] Feng, Z.S. (2002) The First Integral Method to Study the Burgers-Kortewegde Vries Equation. Journal of Physics A: Mathematical and General Physics, 35, 343-349. [12] Lu, B., Zhang, H.Q. and Xie, F.D. (2010) Travelling Wave Solutions of Nonlinear Partial Equations by Using the First Integral Method. Applied Mathematics and Computation, 216, 1329-1336.
http://dx.doi.org/10.1016/j.amc.2010.02.028 [13] Jafari, H., Sooraki, A., Talebi, Y. and Biswas, A. (2012) The First Integral Method and Traveling Wave Solutions to Davey-Stewartson Equation. Nonlinear Analysis: Modelling and Control, 17, 182-193. [14] Ke, Y.-Q. and Yu, J. (2005) The First Integral Method to Study a Class of Reaction-Diffusion Equations. Communications in Theoretical Physics, 43, 597-600. [15] Hosseini, K., Ansari, R. and Gholamin, P. (2012) Exact Solutions of Some Nonlinear Systems of Partial Differential Equations by Using the First Integral Method. Journal of Mathematical Analysis and Applications, 387, 807-814.
http://dx.doi.org/10.1016/j.jmaa.2011.09.044 [16] Kheiri, H., Hajizadeh, R. and Abbasnezhad, N. (2010) The First Integral Method for Solving Some Nonlinear Equations. Armenian Journal of Physics, 3, 82-97. [17] Taghizadeh, N. and Mirzazadeh, M. (2011) The First Integral Method to Some Complex Nonlinear Partial Differential Equations. Journal of Computational and Applied Mathematics, 235, 4871-4877.
http://dx.doi.org/10.1016/j.cam.2011.02.021 [18] El-Sabbagh, M.F. and El-Ganaini, S.I. (2012) The First Integral Method and Its Applications to Nonlinear Equations. Applied Mathematical Sciences, 6, 3893-3906. [19] Lu, B. (2012) The First Integral Method for Some Time Fractional Differential Equations. Journal of Mathematical Analysis and Applications, 395, 684-693.
http://dx.doi.org/10.1016/j.jmaa.2012.05.066 [20] Taghizadeh, N., Mirzazadeh, M. and Tascan, F. (2012) The First-Integral Method Applied to the Eckhaus Equation. Applied Mathematics Letters, 25, 798-802.
http://dx.doi.org/10.1016/j.aml.2011.10.021 [21] Rostamy, D., Zabihi, F., Karimi, K. and Khalehoghli, S. (2011) The First Integral Method for Solving Maccari’s System. Applied Mathematics, 2, 258-263.
http://dx.doi.org/10.4236/am.2011.22030 [22] Deng, X.J. (2008) Exact Peaked Wave Solution of CH-γ Equation by the First-Integral Method. Applied Mathematics and Computation, 206, 806-809.
http://dx.doi.org/10.1016/j.amc.2008.09.039 [23] Tascan, F., Bekir, A. and Koparan, M. (2009) Travelling Wave Solutions of Nonlinear Evolution Equations by Using the First Integral Method. Communications in Nonlinear Science and Numerical Simulation, 14, 1810-1815.
http://dx.doi.org/10.1016/j.cnsns.2008.07.009