Application of the ITEM for the modified dispersive water-wave system



The aim of this paper is to introduce a novel study of obtaining an analytical solutions to the modified dispersive water-wave system. An analytical technique based on the improved \(\tan (\phi /2)\)-expansion method (ITEM) is extended to handle such a system. Description of the method is given and the obtained results reveal that the ITEM is a new significant method for exploring nonlinear partial differential models. By using this method, exact solutions including the hyperbolic function solution, traveling wave solution, soliton solution, rational function solution, and periodic wave solution of this system of equations have been obtained. Moreover, by using Matlab, some graphical simulations were done to see the behavior of these solutions.


Improved tan(\(\phi\)/2)-expansion method Modified dispersive water-wave system Analytical solutions Soliton wave solution 

Mathematics Subject Classification

65D19 65H10 35A20 35A24 35C08 35G50 



This Paper is Published as Part of a Research Project Supported by the University of Tabriz Research Affairs Office.


  1. Alam, M.N., Akbar, M.A., Mohyud-Din, S.T.: A novel (G′/G)-expansion method and its application to the Boussinesq equation. Chin. Phys. B 23(2), 020203 (2014). doi: 10.1088/1674-1056/23/2/020203 CrossRefGoogle Scholar
  2. Ali, S., Rizvi, S.T.R., Younis, M.: Traveling wave solutions for nonlinear dispersive water-wave systems with time-dependent coefficients. Nonlinear Dyn. 82, 1755–1762 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. Baskonus, H.M., Bulut, H.: New wave behaviors of the system of equations for the ion sound and Langmuir Waves. Waves Random Complex Media 26(4), 1–14 (2016c). doi: 10.1080/17455030.2016.1181811
  4. Baskonus, H.M., Koç, D.A., Bulut, H.: New travelling wave prototypes to the nonlinear Zakharov–Kuznetsov equation with power law nonlinearity. Nonlinear Sci. Lett. A 7, 67–76 (2016)Google Scholar
  5. Baskonus, H.M., Bulut, H., Atangana, A.: On the complex and hyperbolic structures of longitudinal wave equation in a magneto-electro-elastic circular rod. Smart Mater. Struct. 25, 035022 (2016b). doi: 10.1088/0964-1726/25/3/035022 ADSCrossRefGoogle Scholar
  6. Baskonus, H.M., Bulut, H.: Exponential prototype structures for (2+1)-dimensional Boiti-Leon-Pempinelli systems in mathematical physics. Waves Random Complex Media 26, 201–208 (2016a)MathSciNetGoogle Scholar
  7. Biswas, A.: 1-soliton solution of the generalized Zakharov–Kuznetsov modified equal width equation. Appl. Math. Letters 22, 1775–1777 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. Bulut, H., Baskonus, H.M.: New complex hyperbolic function solutions for the (2+1)-dimensional dispersive long water-wave system. Math. Comput. Appl. 21, 6 (2016). doi: 10.3390/mca21020006 MathSciNetGoogle Scholar
  9. Dehghan, M., Manafian, J., Saadatmandi, A.: Application of semi-analytic methods for the Fitzhugh–Nagumo equation, which models the transmission of nerve impulses. Math. Methods Appl. Sci. 33, 1384–1398 (2010b)MathSciNetMATHGoogle Scholar
  10. Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Methods Partial Differ. Equ. J. 26, 448–479 (2010a)MathSciNetMATHGoogle Scholar
  11. Dehghan, M., Manafian, J., Saadatmandi, A.: Application of the Exp-function method for solving a partial differential equation arising in biology and population genetics. Int. J. Numer. Methods Heat Fluid Flow 21, 736–753 (2011a)MathSciNetCrossRefGoogle Scholar
  12. Dehghan, M., Manafian, J., Saadatmandi, A.: Analytical treatment of some partial differential equations arising in mathematical physics by using the Exp-function method. Int. J. Mod. Phys. B 25, 2965–2981 (2011b)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. Dehghan, M., Manafian, J.: The solution of the variable coefficients fourth-order parabolic partial differential equations by homotopy perturbation method. Z. Naturforschung A 64a, 420–430 (2009)ADSGoogle Scholar
  14. Demiray, H.: Exact solution of perturbed KdV equation with variable dissipation coefficient. Appl. Comput. Math. 16, 12–16 (2017)Google Scholar
  15. Fan, E.: Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 277, 212–218 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. Fan, E.: Travelling wave solutions for two generalized Hirota-Satsuma KdV systems. Z. Naturforsch. 56A, 312–319 (2001)ADSGoogle Scholar
  17. Fan, E., Zhang, H.: A note on the homogeneous balance method. Phys. Lett. A 246, 403–406 (1998)ADSCrossRefMATHGoogle Scholar
  18. Hasseine, A., Barhoum, Z., Attarakih, M., Bart, H.J.: Analytical solutions of the particle breakage equation by the Adomian decomposition and the variational iteration methods. Adv. Powder Tech. 24, 252–256 (2013)CrossRefGoogle Scholar
  19. Huang, W.H.: Periodic folded waves for a \((2+1)\)-dimensional modified dispersive water wave equation. Chin. Phys. B 18, 3163–3168 (2009)ADSCrossRefGoogle Scholar
  20. Liu, Q., Zhou, Y., Zhang, W.: Bifurcation of travelling wave solutions for the modified dispersive water wave equation. Nonlinear Anal. 69, 151–166 (2008)MathSciNetCrossRefMATHGoogle Scholar
  21. Li, D.S., Zhang, H.Q.: New families of non-travelling wave solutions to the \((2+1)\)-dimensional modified dispersive water-wave system. Chin. Phys. 13, 1377–1381 (2004)ADSCrossRefGoogle Scholar
  22. Lou, S.Y., Hu, X.B.: Infinitely many Lax pairs and symmetry constraints of the KP equation. J. Math. Phys. 38, 6401–6427 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. Ma, W.X.: Complexiton solutions to the Korteweg–de Vries equation. Phys. Lett. A 301, 35–44 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  24. Ma, W.X., Wu, H., He, J.: Partial differential equations possessing Frobenius integrable decompositions. Phys. Lett. A 364, 29–32 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  25. Ma, W.X., Fuchssteiner, Y.: Explicit and exact solutionns to a Kolmogorov–Petrovskii–Piskunov equation. Int. J. Nonlinear Mech. 31, 329–338 (1996)CrossRefMATHGoogle Scholar
  26. Ma, W.X., Maruno, K.-I.: Complexiton solutions of the Toda lattice equation. Phys. A 343, 219–237 (2004)MathSciNetCrossRefGoogle Scholar
  27. Manafian, J., Lakestani, M.: New improvement of the expansion methods for solving the generalized Fitzhugh–Nagumo equation with time-dependent coefficients. Int. J. Eng. Math. 2015, 107978 (2015c). doi: 10.1155/2015/107978
  28. Manafian, J.: On the complex structures of the Biswas–Milovic equation for power, parabolic and dual parabolic law nonlinearities. Eur. Phys. J. Plus 130, 1–20 (2015)CrossRefGoogle Scholar
  29. Manafian, J., Lakestani, M., Bekir, A.: Study of the analytical treatment of the (2+1)-dimensional Zoomeron, the Duffing and the SRLW equations via a new analytical approach. Int. J. Appl. Comput. Math. 2, 243–268 (2016)MathSciNetCrossRefGoogle Scholar
  30. Manafian, J.: Optical soliton solutions for Schrödinger type nonlinear evolution equations by the tan(\(\phi /2\))-expansion method. Optik 127, 4222–4245 (2016)ADSCrossRefGoogle Scholar
  31. Manafian, J.: Application of the ITEM for the system of equations for the ion sound and Langmuir waves. Opt. Quantum Electron. 49(17), 1–26 (2017)Google Scholar
  32. Manafian, J., Lakestani, M.: Solitary wave and periodic wave solutions for Burgers, Fisher, Huxley and combined forms of these equations by the \((G^{\prime }/G)\)-expansion method. Pramana J. Phys. 130, 31–52 (2015b)ADSCrossRefGoogle Scholar
  33. Manafian, J., Lakestani, M.: Optical solitons with Biswas–Milovic equation for Kerr law nonlinearity. Eur. Phys. J. Plus 130, 1–12 (2015a)CrossRefGoogle Scholar
  34. Manafian, J., Lakestani, M.: Application of \(tan(\phi /2)\)-expansion method for solving the Biswas–Milovic equation for Kerr law nonlinearity. Optik 127, 2040–2054 (2016a)ADSCrossRefGoogle Scholar
  35. Manafian, J., Lakestani, M.: Dispersive dark optical soliton with Tzitzéica type nonlinear evolution equations arising in nonlinear optics. Opt. Quantum Electron. 48, 1–32 (2016b)CrossRefGoogle Scholar
  36. Manafian, J., Lakestani, M.: Abundant soliton solutions for the Kundu–Eckhaus equation via tan(\(\phi /2\))-expansion method. Optik 127, 5543–5551 (2016c)ADSCrossRefGoogle Scholar
  37. Ma, W.X., You, Y.: Solving the Kortewegde Vries equation by its bilinear form: Wronskian solutions. Trans. Am. Math. Soc 357, 1753–1778 (2004a)MathSciNetCrossRefMATHGoogle Scholar
  38. Ma, W.X., You, Y.: Rational solutions of the Toda lattice equation in Casoratian form. Chaos Solitons Fractals 22, 395–406 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  39. Ma, W.X., Zhou, D.T.: Explicit exact solution of a generalized KdV equation. Acta Math. Sci. 17, 168–174 (1997)Google Scholar
  40. Meng, D.X., Gao, Y.T., Wang, L., Xu, P.B.: Elastic and inelastic interactions of solitons for a variable-coefficient generalized dispersive water-wave system. Nonlinear Dyn. 69, 391–398 (2012)MathSciNetCrossRefMATHGoogle Scholar
  41. Mohyud-Din, S.T., Noor, M.A., Noor, K.I.: Some relatively new techniques for nonlinear problems. Math. Probl. Eng. 1–25 (2009b). doi: 10.1155/2009/234849
  42. Mohyud-Din, S.T., Noor, M.A., Noor, K.I.: Traveling wave solutions of seventh-order generalized KdV equations using He’s polynomials. Int. J. Nonlinear Sci. Numer. Simul. 10, 223–229 (2009a)CrossRefMATHGoogle Scholar
  43. Mohyud-Din, S.T., Noor, M.A., Noor, K.I., Hosseini, M.M.: Variational iteration method for re-formulated partial differential equations. Int. J. Nonlinear Sci. Numer. Simul. 11(2), 87–92 (2010a)MathSciNetGoogle Scholar
  44. Mohyud-Din, S.T., Noor, M.A., Waheed, A.: Exp-function method for generalized traveling solutions of Calogero–Degasperis–Fokas equation. Z. Naturforschung A 65a, 78–84 (2010)ADSGoogle Scholar
  45. Mohyud-Din, S.T., Yildirim, A., Sariaydin, S.: Numerical soliton solution of the Kaup–Kupershmidt equation. Int. J. Numer. Methods Heat Fluid Flow 21(3), 272–281 (2011a)MathSciNetCrossRefMATHGoogle Scholar
  46. Mohyud-Din, S.T., Yildirim, A., Sezer, S.A.: Numerical soliton solution of the Kaup–Kupershmidt equation. Int. J. Numer. Methods Heat Fluid Flow 21(7), 822–827 (2011b)MathSciNetCrossRefGoogle Scholar
  47. Mohyud-Din, S.T., Negahdary, E., Usman, M.: A meshless numerical solution of the family of generalized fifth-order Korteweg-de Vries equations. Int. J. Numer. Methods Heat Fluid Flow 22, 641–658 (2012a)MathSciNetCrossRefMATHGoogle Scholar
  48. Mohyud-Din, S.T., Khan, Y., Faraz, N., Yildirim, A.: Exp-function method for solitary and periodic solutions of Fitzhugh–Nagumo equations. Int. J. Numer. Methods Heat Fluid Flow 22(3), 335–341 (2012b)MathSciNetCrossRefMATHGoogle Scholar
  49. Nawaz, T., Yildirim, A., Mohyud-Din, S.T.: Analytical solutions Zakharov–Kuznetsov equations. Adv. Powder Technol. 24, 252–256 (2013)CrossRefGoogle Scholar
  50. Noor, M.A., Mohyud-Din, S.T., Waheed, A.: Exp-function method for generalized traveling solutions of master partial differential equations. Acta Appl. Math. 104(2), 131–137 (2008). doi: 10.1007/s10440-008-9245-z MathSciNetCrossRefMATHGoogle Scholar
  51. Noor, M.A., Mohyud-Din, S.T., Waheed, A., Al-Said, E.A.: Exp-function method for traveling wave solutions of nonlinear evolution equations. Appl. Math. Comput. 216, 477–483 (2010b)MathSciNetMATHGoogle Scholar
  52. Rashidi, M.M., Hayat, T., Keimanesh, T., Yousefian, H.: A study on heat transfer in a second-grade fluid through a porous medium with the modified differential transform method. Heat Transf. Asian Res. 42, 31–45 (2013)CrossRefGoogle Scholar
  53. Sabattia, M., Fabbrini, F., Harfouche, A., et al.: Evaluation of biomass production potential and heating value ofhybrid poplar genotypes in a short-rotation culture in Italy. Ind. Crops Prod. 61, 62–73 (2014)CrossRefGoogle Scholar
  54. Tang, X.Y., Lou, S.Y., Zhang, Y.: Localized exicitations in \((2+1)\)-dimensional systems. Phys. Rev. E 66, 046601 (2002)ADSMathSciNetCrossRefGoogle Scholar
  55. Tang, X.Y., Lou, S.Y.: Extended multilinear variable separation approach and multivalued localized excitations for some \((2+1)\)-dimensional integrable systems. J. Math. Phys. 44, 4000–4025 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  56. Wen, X.Y., Xu, X.G.: Multiple soliton solutions and fusion interaction phenomena for the (2+1)-dimensional modified dispersive water-wave system. Appl. Math. Comput. 219, 7730–7740 (2013)MathSciNetMATHGoogle Scholar
  57. Zheng, C.L., Fang, J.P., Chen, L.Q.: Localized excitations with and without propagating properties in \((2+1)\)-dimensions obtained by a mapping approach. Chin. Phys. 14, 676–682 (2005)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Mathematical ScienceUniversity of TabrizTabrizIran
  2. 2.Young Researchers and Elite Club, Ilkhchi BranchIslamic Azad UniversityIlkhchiIran

Personalised recommendations