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On new traveling wave solutions of the Hirota-Satsuma coupled KdV equation

Year 2017, Volume: 5 Issue: 3, 262 - 272, 01.07.2017

Abstract

The improved (G′ /G)-expansion method is applied to reach the different type soliton solitions of the Hirota-Satsuma Coupled KdV (HSCKdV) equation. It is obtained hyperbolic, triangular, periodic wave and kink soliton solitions of this equation. The method is an effective one to reach the different types of solutions of nonlinear partial differential equations and systems. Finally, the numerical simulations add to these obtained solutions.

References

  • Jawad, A. J. M.,Johnson S, Yildirim, A., Kumar, S., Biswas, A., 2013. Soliton solutions to coupled nonlinear wave equations in (2 + 1)-dimensions, Indian J Phys 87: 281.
  • Wang, M. L., Li, X. Z., Zhang, J. L., 2008. The (G^I/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A 372 , 417–423.
  • Bhrawy, A. H., Abdelkawy, M. A.,Biswas, A., 2013. Topological solitons and cnoidal waves to a few nonlinear wave equations in theoretical physics, Indian J. Phys. 87, 1125-1131.
  • Biswas, A.., Krishnan, E.V.,2011. Exact solutions for Ostrovsky equation, Indian J Phys 85, 1513-1521.
  • Ray, S. S.,Shaoo, S.,2015. A Novel Analytical Method with Fractional Complex Transform for New Exact Solutions of Time-Fractional Fifth-Order Sawada-Kotera Equation, Rep. Math. Phys.75,63-72.
  • Wu, X. H.,He., J. H.,2007. Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method, Comput. Math. Appl. 54 966-986.
  • Liu, X.,Zhang, W., Li, Z., 2012. Application of Improved (G’/G)–Expansion Method to Traveling Wave Solutions of Two Nonlinear Evolution Equations, Adv. App. math. Mech. 4, 122-130.
  • Hashemi, M. S., Abbasbandy, S., Alhuthali, M.S. and Alsulami, H.H., 2015. Conservation laws and symmetries of mKdV-KP equation, Romanian Journal of Physics 60 904–917.
  • Hashemi, M. S. and Wang, G.W., 2017. Lie symmetry analysis and soliton solutions of time-fractional K(m,n) equation, Pramana – J. Phys. 88: 7.
  • Hashemi, M. S., 2016. On Black-Scholes equation; method of Heir-equations, nonlinear self-adjointness and conservation laws." Bulletin of the Iranian Mathematical Society 42 903–921.
  • Wu, Y.,Geng, X., Hu, X.,Zhu, S., 1999. A generalized Hirota–Satsuma coupled Korteweg–de Vries equation and Miura transformations, Phys. Lett. A 255 , 259–264.
  • Hirota, R., Satsuma, J.,1981. Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett. A 85, 407-408.
  • Guo-Zhong, Z.,Xi-Jun,Y.,Yun, X.,Jiang, Z.,Di, W.,2010. Approximate analytic solutions for a generalized Hirota–Satsuma coupled KdV equation and a coupled mKdV equation, Chinese Phys. B 19.
  • Kangalgil, F., Ayaz, F., 2010, Solitary Wave Solutions for HSCKdV Equation and Coupled mKdV Equation by DTM, Arabian J. Sci. Eng., 35(2D), 203-213.,
  • Liu, J., Li, H., 2013. Approximate Analytic Solutions of Time-Fractional Hirota-Satsuma Coupled KdV Equation and Coupled MKdV Equation, Abstract and Applied Analysis 11.
  • Raslan, K. R., 2004. The decomposition method for a Hirota–Satsuma coupled KdV equation and a coupled MKdV equation, Appl. Math. Comput. 81, 1497–1505.
  • Abbasbandy, S., 2007. The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation, Physics Letters A, 361, 6, 478-483.
  • Lu, D., Hong, B., Tian, L., 2007, New Explicit Exact Solutions for The Generalized Coupled Hirota-Satsuma KdV System, Comp. Math. Applic., 53, 1181-1190.
  • Wen, C., Zheng, B., 2013. A New Fractional Sub-equation Method For Fractional Partial Differential Equations, WSEAS Transactions on Math.12,564.
  • Yong, C.,Yan, Z.,Li, B., Zhang, H., 2003.New explicit exact solutions for a generalized Hirota-Satsuma coupled KdV system and a coupled mKdV Equation, Chin. Phys.,12,1-10.
  • Feng, D., Kezan, L., 2011, Exact traveling wave solutions for a generalized Hirota–Satsuma coupled KdV equation by Fan sub-equation method, Phys. Lett., A 375, 2201-2210.
  • Feng, D., Zhang, H., 2008, A new auxiliary function method for solving the generalized coupled Hirota–Satsuma KdV system, Appl. Math. Comput., 200, 283-288.
  • Yan, Z., 2003, The extended Jacobian elliptic function expansion method and its application in the generalized Hirota–Satsuma coupled KdV system, Chaos, Solitons and Fractals, 15, 575–583.
Year 2017, Volume: 5 Issue: 3, 262 - 272, 01.07.2017

Abstract

References

  • Jawad, A. J. M.,Johnson S, Yildirim, A., Kumar, S., Biswas, A., 2013. Soliton solutions to coupled nonlinear wave equations in (2 + 1)-dimensions, Indian J Phys 87: 281.
  • Wang, M. L., Li, X. Z., Zhang, J. L., 2008. The (G^I/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A 372 , 417–423.
  • Bhrawy, A. H., Abdelkawy, M. A.,Biswas, A., 2013. Topological solitons and cnoidal waves to a few nonlinear wave equations in theoretical physics, Indian J. Phys. 87, 1125-1131.
  • Biswas, A.., Krishnan, E.V.,2011. Exact solutions for Ostrovsky equation, Indian J Phys 85, 1513-1521.
  • Ray, S. S.,Shaoo, S.,2015. A Novel Analytical Method with Fractional Complex Transform for New Exact Solutions of Time-Fractional Fifth-Order Sawada-Kotera Equation, Rep. Math. Phys.75,63-72.
  • Wu, X. H.,He., J. H.,2007. Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method, Comput. Math. Appl. 54 966-986.
  • Liu, X.,Zhang, W., Li, Z., 2012. Application of Improved (G’/G)–Expansion Method to Traveling Wave Solutions of Two Nonlinear Evolution Equations, Adv. App. math. Mech. 4, 122-130.
  • Hashemi, M. S., Abbasbandy, S., Alhuthali, M.S. and Alsulami, H.H., 2015. Conservation laws and symmetries of mKdV-KP equation, Romanian Journal of Physics 60 904–917.
  • Hashemi, M. S. and Wang, G.W., 2017. Lie symmetry analysis and soliton solutions of time-fractional K(m,n) equation, Pramana – J. Phys. 88: 7.
  • Hashemi, M. S., 2016. On Black-Scholes equation; method of Heir-equations, nonlinear self-adjointness and conservation laws." Bulletin of the Iranian Mathematical Society 42 903–921.
  • Wu, Y.,Geng, X., Hu, X.,Zhu, S., 1999. A generalized Hirota–Satsuma coupled Korteweg–de Vries equation and Miura transformations, Phys. Lett. A 255 , 259–264.
  • Hirota, R., Satsuma, J.,1981. Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett. A 85, 407-408.
  • Guo-Zhong, Z.,Xi-Jun,Y.,Yun, X.,Jiang, Z.,Di, W.,2010. Approximate analytic solutions for a generalized Hirota–Satsuma coupled KdV equation and a coupled mKdV equation, Chinese Phys. B 19.
  • Kangalgil, F., Ayaz, F., 2010, Solitary Wave Solutions for HSCKdV Equation and Coupled mKdV Equation by DTM, Arabian J. Sci. Eng., 35(2D), 203-213.,
  • Liu, J., Li, H., 2013. Approximate Analytic Solutions of Time-Fractional Hirota-Satsuma Coupled KdV Equation and Coupled MKdV Equation, Abstract and Applied Analysis 11.
  • Raslan, K. R., 2004. The decomposition method for a Hirota–Satsuma coupled KdV equation and a coupled MKdV equation, Appl. Math. Comput. 81, 1497–1505.
  • Abbasbandy, S., 2007. The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation, Physics Letters A, 361, 6, 478-483.
  • Lu, D., Hong, B., Tian, L., 2007, New Explicit Exact Solutions for The Generalized Coupled Hirota-Satsuma KdV System, Comp. Math. Applic., 53, 1181-1190.
  • Wen, C., Zheng, B., 2013. A New Fractional Sub-equation Method For Fractional Partial Differential Equations, WSEAS Transactions on Math.12,564.
  • Yong, C.,Yan, Z.,Li, B., Zhang, H., 2003.New explicit exact solutions for a generalized Hirota-Satsuma coupled KdV system and a coupled mKdV Equation, Chin. Phys.,12,1-10.
  • Feng, D., Kezan, L., 2011, Exact traveling wave solutions for a generalized Hirota–Satsuma coupled KdV equation by Fan sub-equation method, Phys. Lett., A 375, 2201-2210.
  • Feng, D., Zhang, H., 2008, A new auxiliary function method for solving the generalized coupled Hirota–Satsuma KdV system, Appl. Math. Comput., 200, 283-288.
  • Yan, Z., 2003, The extended Jacobian elliptic function expansion method and its application in the generalized Hirota–Satsuma coupled KdV system, Chaos, Solitons and Fractals, 15, 575–583.
There are 23 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

M. Tuncay Gencoglu This is me

Ali Akgul This is me

Mustafa Inc This is me

Publication Date July 1, 2017
Published in Issue Year 2017 Volume: 5 Issue: 3

Cite

APA Gencoglu, M. T., Akgul, A., & Inc, M. (2017). On new traveling wave solutions of the Hirota-Satsuma coupled KdV equation. New Trends in Mathematical Sciences, 5(3), 262-272.
AMA Gencoglu MT, Akgul A, Inc M. On new traveling wave solutions of the Hirota-Satsuma coupled KdV equation. New Trends in Mathematical Sciences. July 2017;5(3):262-272.
Chicago Gencoglu, M. Tuncay, Ali Akgul, and Mustafa Inc. “On New Traveling Wave Solutions of the Hirota-Satsuma Coupled KdV Equation”. New Trends in Mathematical Sciences 5, no. 3 (July 2017): 262-72.
EndNote Gencoglu MT, Akgul A, Inc M (July 1, 2017) On new traveling wave solutions of the Hirota-Satsuma coupled KdV equation. New Trends in Mathematical Sciences 5 3 262–272.
IEEE M. T. Gencoglu, A. Akgul, and M. Inc, “On new traveling wave solutions of the Hirota-Satsuma coupled KdV equation”, New Trends in Mathematical Sciences, vol. 5, no. 3, pp. 262–272, 2017.
ISNAD Gencoglu, M. Tuncay et al. “On New Traveling Wave Solutions of the Hirota-Satsuma Coupled KdV Equation”. New Trends in Mathematical Sciences 5/3 (July 2017), 262-272.
JAMA Gencoglu MT, Akgul A, Inc M. On new traveling wave solutions of the Hirota-Satsuma coupled KdV equation. New Trends in Mathematical Sciences. 2017;5:262–272.
MLA Gencoglu, M. Tuncay et al. “On New Traveling Wave Solutions of the Hirota-Satsuma Coupled KdV Equation”. New Trends in Mathematical Sciences, vol. 5, no. 3, 2017, pp. 262-7.
Vancouver Gencoglu MT, Akgul A, Inc M. On new traveling wave solutions of the Hirota-Satsuma coupled KdV equation. New Trends in Mathematical Sciences. 2017;5(3):262-7.