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Generalized Jacobi Elliptic Function Method for Traveling Wave Solutions of (2+1)-Dimensional Breaking Soliton Equation

Year 2010, Volume: 7 Issue: 1, - , 01.02.2010

Abstract

In this study, we implemented the generalized Jacobi elliptic function method
with symbolic computation to construct periodic and multiple soliton solutions for the
(2+1)-dimensional breaking soliton equation.

References

  • [1] L. Debtnath, Nonlinear Partial Differential Equations for Scientist and Engineers, Birkhauser, Boston, MA, 1997.
  • [2] A.M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema, Rotterdam, 2002.
  • [3] I. Aslan, T. Ozi¸s, Analytic study on two nonlinear evolution equations by using the ( G'/G )- expansion method, Applied Mathematics and Computation 209 (2009), 425–429.
  • [4] I. Aslan, T. Ozi¸s, On the validity and reliability of the ( G'/G )- expansion method by using higher-order nonlinear equations, Applied Mathematics and Computation 211 (2009), 531–536.
  • [5] A. Bekir, Application of the ( G'/G )- expansion method for nonlinear evolution equations, Physics Letters A 372 (2008), 3400–3406.
  • [6] M. A. Abdou, S. Zhang, New periodic wave solutions via extended mapping method, Communications in Nonlinear Science and Numerical Simulation 14 (2009), 2–11.
  • [7] X. Zhao, H. Zhi and H. Zhang, Improved Jacobi-function method with symbolic computation to construct new double-periodic solutions for the generalized Ito system, Chaos, Solitons & Fractals 28 (2006), 112–126.
  • [8] Q. Wang, Y. Chen, Z. Hongqing, A new Jacobi elliptic function rational expansion method and its application to (1 + 1)-dimensional dispersive long wave equation, Chaos, Solitons & Fractals 23 (2005), 477–483.
  • [9] Y. Yu, Q. Wang, H. Zhang , The extended Jacobi elliptic function method to solve a generalized Hirota-Satsuma coupled KdV equations, Chaos, Solitons & Fractals 26 (2005), 1415–1421.
  • [10] A. H. Khater, O. H. El-Kalaawy, M.A. Helal, Two new classes of exact solutions for the KdV equation via B¨acklund transformations, Chaos, Solitons & Fractals 12 (1997), 1901–1909.
  • [11] M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Physics Letters A 213 (1996), 279–287.
  • [12] M. L. Wang, Y. Zhou, Z. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Physics Letters A 216 (1996), 67–75.
  • [13] M. A. Helal, M. S. Mehanna, The tanh method and Adomian decomposition method for solving the foam drainage equation, Applied Mathematics and Computation 190 (2007), 599–607.
  • [14] A. M. Wazwaz, The tanh and the sine-cosine methods for the complex modified KdV and the generalized KdV equations, Computers & Mathematics with Applications 49 (2005), 1101–1112.
  • [15] B. R. Duffy, E. J. Parkes, Travelling solitary wave solutions to a seventh-order generalized KdV equation, Physics Letters A 214 (1996), 271–272.
  • [16] E. J. Parkes, B. R. Duffy, Travelling solitary wave solutions to a compound KdV-Burgers equation, Physics Letters A 229 (1997), 217–220.
  • [17] A. Borhanifar, M. M. Kabir, New periodic and soliton solutions by application of Exp-function method for nonlinear evolution equations, Journal of Computational and Applied Mathematics 229 (2009), 158–167.
  • [18] A. M. Wazwaz, Multiple kink solutions and multiple singular kink solutions for two systems of coupled Burgers-type equations, Communications in Nonlinear Science and Numerical Simulation 14 (2009), 2962–2970.
  • [19] E. Demetriou, N. M. Ivanova, C. Sophocleous, Group analysis of (2+1)- and (3+1)- dimensional diffusion-convection equations, Journal of Mathematical Analysis and Applications 348 (2008), 55–65.
  • [20] Y. Yu, Q. Wang, H. Zhang, The extension of the Jacobi elliptic function rational expansion method, Communications in Nonlinear Science and Numerical Simulation 12 (2007), 702–713.
  • [21] T. Ozi¸s, A. Yıldırım, Reliable analysis for obtaining exact soliton solutions of nonlinear Schrödinger (NLS) equation, Chaos, Solitons & Fractals 38 (2008), 209–212.
  • [22] A. Yıldırım, Application of He’s homotopy perturbation method for solving the Cauchy reaction-diffusion problem, Computers & Mathematics with Applications 57 (2009), 612–618.
  • [23] A. Yıldırım, Variational iteration method for modified Camassa-Holm and Degasperis-Procesi equations, Communications in Numerical Methods in Engineering (2008), (in press).
  • [24] T. Ozi¸s, A.Yıldırım, Comparison between Adomian’s method and He’s homotopy perturbation method, Computers & Mathematics with Applications 56 (2008), 1216–1224.
  • [25] A.Yıldırım, An Algorithm for Solving the Fractional Nonlinear Schr¨odinger Equation by Means of the Homotopy Perturbation Method, International Journal of Nonlinear Sciences and Numerical Simulation 10 (2009), 445–451.
  • [26] H. Zhao, H. J. Niu , A new method applied to obtain complex Jacobi elliptic function solutions of general nonlinear equations, Chaos, Solitons & Fractals 41 (2009), 224–232.
  • [27] W. Hereman, M. Takaoka, Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA, Journal of Physics A: Mathematical and General 23 (1990), 4805–4822.
  • [28] C. Huai-Tang, Z. Hong-Qing, New double periodic and multiple soliton solutions of the generalized (2 + 1)-dimensional Boussinesq equation, Chaos, Solitons & Fractals 20 (2004), 765–769.
  • [29] Y. Chen, Q. Wang, Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (1 + 1)-dimensional dispersive long wave equation, Chaos, Solitons & Fractals 24 (2005), 745–757.
  • [30] E. Fan, J. Zhang, Applications of the Jacobi elliptic function method to special-type nonlinear equations, Physics Letters A, Volume 305 (2002), 383–392.
  • [31] E. J. Parkes, B. R. Duffy, An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Computer Physics Communications 98 (1996), 288–300.
  • [32] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Physics Letters A 277 (2000), 212–218.
  • [33] S. A. Elwakil, S. K. El-labany, M. A. Zahran and R. Sabry, Modified extended tanh-function method for solving nonlinear partial differential equations, Physics Letters A 299 (2002), 179–188.
  • [34] X. Zheng, Y. Chen and H. Zhang, Generalized extended tanh-function method and its application to (1+1)-dimensional dispersive long wave equation, Physics Letters A 311 (2003), 145–157.
  • [35] E. Yomba, Construction of new soliton-like solutions of the (2+1) dimensional dispersive long wave equation, Chaos, Solitons & Fractals 20 (2004), 1135–1139.
  • [36] H. Chen, H. Zhang, New multiple soliton solutions to the general Burgers-Fisher equation and the Kuramoto-Sivashinsky equation, Chaos, Solitons & Fractals 19 (2004), 71–76.
Year 2010, Volume: 7 Issue: 1, - , 01.02.2010

Abstract

References

  • [1] L. Debtnath, Nonlinear Partial Differential Equations for Scientist and Engineers, Birkhauser, Boston, MA, 1997.
  • [2] A.M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema, Rotterdam, 2002.
  • [3] I. Aslan, T. Ozi¸s, Analytic study on two nonlinear evolution equations by using the ( G'/G )- expansion method, Applied Mathematics and Computation 209 (2009), 425–429.
  • [4] I. Aslan, T. Ozi¸s, On the validity and reliability of the ( G'/G )- expansion method by using higher-order nonlinear equations, Applied Mathematics and Computation 211 (2009), 531–536.
  • [5] A. Bekir, Application of the ( G'/G )- expansion method for nonlinear evolution equations, Physics Letters A 372 (2008), 3400–3406.
  • [6] M. A. Abdou, S. Zhang, New periodic wave solutions via extended mapping method, Communications in Nonlinear Science and Numerical Simulation 14 (2009), 2–11.
  • [7] X. Zhao, H. Zhi and H. Zhang, Improved Jacobi-function method with symbolic computation to construct new double-periodic solutions for the generalized Ito system, Chaos, Solitons & Fractals 28 (2006), 112–126.
  • [8] Q. Wang, Y. Chen, Z. Hongqing, A new Jacobi elliptic function rational expansion method and its application to (1 + 1)-dimensional dispersive long wave equation, Chaos, Solitons & Fractals 23 (2005), 477–483.
  • [9] Y. Yu, Q. Wang, H. Zhang , The extended Jacobi elliptic function method to solve a generalized Hirota-Satsuma coupled KdV equations, Chaos, Solitons & Fractals 26 (2005), 1415–1421.
  • [10] A. H. Khater, O. H. El-Kalaawy, M.A. Helal, Two new classes of exact solutions for the KdV equation via B¨acklund transformations, Chaos, Solitons & Fractals 12 (1997), 1901–1909.
  • [11] M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Physics Letters A 213 (1996), 279–287.
  • [12] M. L. Wang, Y. Zhou, Z. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Physics Letters A 216 (1996), 67–75.
  • [13] M. A. Helal, M. S. Mehanna, The tanh method and Adomian decomposition method for solving the foam drainage equation, Applied Mathematics and Computation 190 (2007), 599–607.
  • [14] A. M. Wazwaz, The tanh and the sine-cosine methods for the complex modified KdV and the generalized KdV equations, Computers & Mathematics with Applications 49 (2005), 1101–1112.
  • [15] B. R. Duffy, E. J. Parkes, Travelling solitary wave solutions to a seventh-order generalized KdV equation, Physics Letters A 214 (1996), 271–272.
  • [16] E. J. Parkes, B. R. Duffy, Travelling solitary wave solutions to a compound KdV-Burgers equation, Physics Letters A 229 (1997), 217–220.
  • [17] A. Borhanifar, M. M. Kabir, New periodic and soliton solutions by application of Exp-function method for nonlinear evolution equations, Journal of Computational and Applied Mathematics 229 (2009), 158–167.
  • [18] A. M. Wazwaz, Multiple kink solutions and multiple singular kink solutions for two systems of coupled Burgers-type equations, Communications in Nonlinear Science and Numerical Simulation 14 (2009), 2962–2970.
  • [19] E. Demetriou, N. M. Ivanova, C. Sophocleous, Group analysis of (2+1)- and (3+1)- dimensional diffusion-convection equations, Journal of Mathematical Analysis and Applications 348 (2008), 55–65.
  • [20] Y. Yu, Q. Wang, H. Zhang, The extension of the Jacobi elliptic function rational expansion method, Communications in Nonlinear Science and Numerical Simulation 12 (2007), 702–713.
  • [21] T. Ozi¸s, A. Yıldırım, Reliable analysis for obtaining exact soliton solutions of nonlinear Schrödinger (NLS) equation, Chaos, Solitons & Fractals 38 (2008), 209–212.
  • [22] A. Yıldırım, Application of He’s homotopy perturbation method for solving the Cauchy reaction-diffusion problem, Computers & Mathematics with Applications 57 (2009), 612–618.
  • [23] A. Yıldırım, Variational iteration method for modified Camassa-Holm and Degasperis-Procesi equations, Communications in Numerical Methods in Engineering (2008), (in press).
  • [24] T. Ozi¸s, A.Yıldırım, Comparison between Adomian’s method and He’s homotopy perturbation method, Computers & Mathematics with Applications 56 (2008), 1216–1224.
  • [25] A.Yıldırım, An Algorithm for Solving the Fractional Nonlinear Schr¨odinger Equation by Means of the Homotopy Perturbation Method, International Journal of Nonlinear Sciences and Numerical Simulation 10 (2009), 445–451.
  • [26] H. Zhao, H. J. Niu , A new method applied to obtain complex Jacobi elliptic function solutions of general nonlinear equations, Chaos, Solitons & Fractals 41 (2009), 224–232.
  • [27] W. Hereman, M. Takaoka, Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA, Journal of Physics A: Mathematical and General 23 (1990), 4805–4822.
  • [28] C. Huai-Tang, Z. Hong-Qing, New double periodic and multiple soliton solutions of the generalized (2 + 1)-dimensional Boussinesq equation, Chaos, Solitons & Fractals 20 (2004), 765–769.
  • [29] Y. Chen, Q. Wang, Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (1 + 1)-dimensional dispersive long wave equation, Chaos, Solitons & Fractals 24 (2005), 745–757.
  • [30] E. Fan, J. Zhang, Applications of the Jacobi elliptic function method to special-type nonlinear equations, Physics Letters A, Volume 305 (2002), 383–392.
  • [31] E. J. Parkes, B. R. Duffy, An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Computer Physics Communications 98 (1996), 288–300.
  • [32] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Physics Letters A 277 (2000), 212–218.
  • [33] S. A. Elwakil, S. K. El-labany, M. A. Zahran and R. Sabry, Modified extended tanh-function method for solving nonlinear partial differential equations, Physics Letters A 299 (2002), 179–188.
  • [34] X. Zheng, Y. Chen and H. Zhang, Generalized extended tanh-function method and its application to (1+1)-dimensional dispersive long wave equation, Physics Letters A 311 (2003), 145–157.
  • [35] E. Yomba, Construction of new soliton-like solutions of the (2+1) dimensional dispersive long wave equation, Chaos, Solitons & Fractals 20 (2004), 1135–1139.
  • [36] H. Chen, H. Zhang, New multiple soliton solutions to the general Burgers-Fisher equation and the Kuramoto-Sivashinsky equation, Chaos, Solitons & Fractals 19 (2004), 71–76.
There are 36 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

İbrahim Enam İnan

Publication Date February 1, 2010
Published in Issue Year 2010 Volume: 7 Issue: 1

Cite

APA İnan, İ. E. (2010). Generalized Jacobi Elliptic Function Method for Traveling Wave Solutions of (2+1)-Dimensional Breaking Soliton Equation. Cankaya University Journal of Science and Engineering, 7(1).
AMA İnan İE. Generalized Jacobi Elliptic Function Method for Traveling Wave Solutions of (2+1)-Dimensional Breaking Soliton Equation. CUJSE. February 2010;7(1).
Chicago İnan, İbrahim Enam. “Generalized Jacobi Elliptic Function Method for Traveling Wave Solutions of (2+1)-Dimensional Breaking Soliton Equation”. Cankaya University Journal of Science and Engineering 7, no. 1 (February 2010).
EndNote İnan İE (February 1, 2010) Generalized Jacobi Elliptic Function Method for Traveling Wave Solutions of (2+1)-Dimensional Breaking Soliton Equation. Cankaya University Journal of Science and Engineering 7 1
IEEE İ. E. İnan, “Generalized Jacobi Elliptic Function Method for Traveling Wave Solutions of (2+1)-Dimensional Breaking Soliton Equation”, CUJSE, vol. 7, no. 1, 2010.
ISNAD İnan, İbrahim Enam. “Generalized Jacobi Elliptic Function Method for Traveling Wave Solutions of (2+1)-Dimensional Breaking Soliton Equation”. Cankaya University Journal of Science and Engineering 7/1 (February 2010).
JAMA İnan İE. Generalized Jacobi Elliptic Function Method for Traveling Wave Solutions of (2+1)-Dimensional Breaking Soliton Equation. CUJSE. 2010;7.
MLA İnan, İbrahim Enam. “Generalized Jacobi Elliptic Function Method for Traveling Wave Solutions of (2+1)-Dimensional Breaking Soliton Equation”. Cankaya University Journal of Science and Engineering, vol. 7, no. 1, 2010.
Vancouver İnan İE. Generalized Jacobi Elliptic Function Method for Traveling Wave Solutions of (2+1)-Dimensional Breaking Soliton Equation. CUJSE. 2010;7(1).