Journal of Applied Mathematics

New Exact Solutions of Ion-Acoustic Wave Equations by (G/G)-Expansion Method

Wafaa M. Taha, M. S. M. Noorani, and I. Hashim

Full-text: Open access

Abstract

The (G/G)-expansion method is used to study ion-acoustic waves equations in plasma physic for the first time. Many new exact traveling wave solutions of the Schamel equation, Schamel-KdV (S-KdV), and the two-dimensional modified KP (Kadomtsev-Petviashvili) equation with square root nonlinearity are constructed. The traveling wave solutions obtained via this method are expressed by hyperbolic functions, the trigonometric functions, and the rational functions. In addition to solitary waves solutions, a variety of special solutions like kink shaped, antikink shaped, and bell type solitary solutions are obtained when the choice of parameters is taken at special values. Two- and three-dimensional plots are drawn to illustrate the nature of solutions. Moreover, the solution obtained via this method is in good agreement with previously obtained solutions of other researchers.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 810729, 11 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808186

Digital Object Identifier
doi:10.1155/2013/810729

Mathematical Reviews number (MathSciNet)
MR3124606

Zentralblatt MATH identifier
06950889

Citation

Taha, Wafaa M.; Noorani, M. S. M.; Hashim, I. New Exact Solutions of Ion-Acoustic Wave Equations by ( ${G}^{\prime }/G$ )-Expansion Method. J. Appl. Math. 2013 (2013), Article ID 810729, 11 pages. doi:10.1155/2013/810729. https://projecteuclid.org/euclid.jam/1394808186


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References

  • G. B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1974.
  • R. C. Davidson, Methods in Nonlinear Plasma Theory, Academic Press, New York, NY, USA, 1972.
  • H. Schamel, “A modified Korteweg de Vries equation for ion acoustic waves due to resonant electrons,” Journal of Plasma Physics, vol. 9, pp. 377–387, 1973.
  • J. Lee and R. Sakthivel, “Exact travelling wave solutions of the Schamel–Korteweg–de Vries equation,” Reports on Mathematical Physics, vol. 68, no. 2, pp. 153–161, 2011.
  • M. M. Hassan, “New exact solutions of two nonlinear physical models,” Communications in Theoretical Physics, vol. 53, no. 4, pp. 596–604, 2010.
  • A. H. Khater, M. M. Hassan, and D. K. Callebaut, “Travelling wave solutions to some important equations of mathematical physics,” Reports on Mathematical Physics, vol. 66, no. 1, pp. 1–19, 2010.
  • D. Chakraborty and K. P. Das, “Stability of ion-acoustic solitons in a multispecies plasma consisting of non-isothermal electrons,” Journal of Plasma Physics, vol. 60, no. 1, pp. 151–158, 1998.
  • B. B. Kadomtsev and V. I. Petviashvili, “On the stability of solitary in weakly dispersive media,” Soviet Physics Doklady, vol. 15, pp. 539–541, 1970.
  • M. Krisnangkura, S. Chinviriyasit, and W. Chinviriyasit, “Analytic study of the generalized Burger's-Huxley equation by hyperbolic tangent method,” Applied Mathematics and Computation, vol. 218, no. 22, pp. 10843–10847, 2012.
  • A.-M. Wazwaz, “The tanh method for traveling wave solutions of nonlinear equations,” Applied Mathematics and Computation, vol. 154, no. 3, pp. 713–723, 2004.
  • Q. Shi, Q. Xiao, and X. Liu, “Extended wave solutions for a nonlinear Klein-Gordon-Zakharov system,” Applied Mathematics and Computation, vol. 218, no. 19, pp. 9922–9929, 2012.
  • A.-M. Wazwaz, “The tanh and the sine-cosine methods for a reliable treatment of the modified equal width equation and its variants,” Communications in Nonlinear Science and Numerical Simulation, vol. 11, no. 2, pp. 148–160, 2006.
  • A. Jabbari and H. Kheiri, “New exact traveling wave solutions for the Kawahara and modified Kawahara equations by using modified tanh-coth method,” Acta Universitatis Apulensis, no. 23, pp. 21–38, 2010.
  • K. Parand and J. A. Rad, “Exp-function method for some nonlinear PDE's and a nonlinear ODE's,” Journal of King Saud University-Science, vol. 24, no. 1, pp. 1–10, 2012.
  • M. K. Elboree, “New soliton solutions for a Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation,” Applied Mathematics and Computation, vol. 218, no. 10, pp. 5966–5973, 2012.
  • A. S. Abdel Rady, E. S. Osman, and M. Khalfallah, “On soliton solutions of the $(2+1)$ dimensional Boussinesq equation,” Applied Mathematics and Computation, vol. 219, no. 8, pp. 3414–3419, 2012.
  • B. Hong and D. Lu, “New Jacobi elliptic function-like solutions for the general KdV equation with variable coefficients,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1594–1600, 2012.
  • J. Lee, R. Sakthivel, and L. Wazzan, “Exact traveling wave solutions of a higher-dimensional nonlinear evolution equation,” Modern Physics Letters B, vol. 24, no. 10, pp. 1011–1021, 2010.
  • S. Abbasbandy and A. Shirzadi, “The first integral method for modified Benjamin-Bona-Mahony equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 7, pp. 1759–1764, 2010.
  • N. Taghizadeh, M. Mirzazadeh, and F. Tascan, “The first-integral method applied to the Eckhaus equation,” Applied Mathematics Letters, vol. 25, no. 5, pp. 798–802, 2012.
  • W. Malfliet, “Solitary wave solutions of nonlinear wave equations,” American Journal of Physics, vol. 60, no. 7, pp. 650–654, 1992.
  • M. Wang, X. Li, and J. Zhang, “The (${G}^{\prime }/G$)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008.
  • W. M. Taha and M. S. M. Noorani, “Exact solutions of equation generated by the Jaulent-Miodek hierarchy by (${G}^{\prime }/G$)-expansion method,” Mathematical Problems in Engineering, vol. 2013, Article ID 392830, 7 pages, 2013.
  • W. M. Taha, M. S. M. Noorani, and I. Hashim, “New application of the (${G}^{\prime }/G$)-expansion method for thin film equations,” Abstract and Applied Analysis, vol. 2013, Article ID 535138, 6 pages, 2013.
  • B. Ayhan and A. Bekir, “The (${G}^{\prime }/G$)-expansion method for the nonlinear lattice equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 9, pp. 3490–3498, 2012.
  • J. Feng, W. Li, and Q. Wan, “Using (${G}^{\prime }/G$)-expansion method to seek the traveling wave solution of Kolmogorov-Petrovskii-Piskunov equation,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5860–5865, 2011.
  • M. M. Kabir, A. Borhanifar, and R. Abazari, “Application of (${G}^{\prime }/G$)-expansion method to regularized long wave (RLW) equation,” Computers & Mathematics with Applications, vol. 61, no. 8, pp. 2044–2047, 2011.
  • A. Malik, F. Chand, H. Kumar, and S. C. Mishra, “Exact solutions of the Bogoyavlenskii equation using the multiple (${G}^{\prime }/G$)-expansion method,” Computers & Mathematics with Applications, vol. 64, no. 9, pp. 2850–2859, 2012.
  • A. Malik, F. Chand, and S. C. Mishra, “Exact travelling wave solutions of some nonlinear equations by (${G}^{\prime }/G$)-expansion method,” Applied Mathematics and Computation, vol. 216, no. 9, pp. 2596–2612, 2010.
  • A. Jabbari, H. Kheiri, and A. Bekir, “Exact solutions of the coupled Higgs equation and the Maccari system using He's semi-inverse method and (${G}^{\prime }/G$)-expansion method,” Computers & Mathematics with Applications, vol. 62, no. 5, pp. 2177–2186, 2011.
  • H. Naher and F. A. Abdullah, “Some new traveling wave solutions of the nonlinear reaction diffusion equation by using the improved (${G}^{\prime }/G$)-expansion method,” Mathematical Problems in Engineering, vol. 2012, Article ID 871724, 17 pages, 2012.
  • E. M. E. Zayed and M. A. M. Abdelaziz, “The two-variable $({G}^{\prime }/G,1/G)$-expansion method for solving the nonlinear KdV-mKdV equation,” Mathematical Problems in Engineering, vol. 2012, Article ID 725061, 14 pages, 2012.
  • H. Naher and F. A. Abdullah, “The improved (${G}^{\prime }/G$)-expansion method for the $(2+1)$-dimensional modified Zakharov-Kuznetsov equation,” Journal of Applied Mathematics, vol. 2012, Article ID 438928, 20 pages, 2012.
  • H. Naher and F. A. Abdullah, “New traveling wave solutions by the extended generalized Riccati equation mapping method of the $(2+1)$-dimensional evolution equation,” Journal of Applied Mathematics, vol. 2012, Article ID 486458, 18 pages, 2012.
  • A. H. Khater and M. M. Hassan, “Exact solutions expressible in hyperbolic and jacobi elliptic functions of some important equations of ion-acoustic waves,” in Acoustic Waves–-From Microdevices to Helioseismology, 2011. \endinput