Optical soliton wave solutions to the resonant Davey–Stewartson system



We investigate the resonant Davey–Stewartson (DS) system. The resonant DS system is a natural \((2+1)\)-dimensional version of the resonant nonlinear Schrödinger equation. Traveling wave solutions were found. In this paper, we demonstrate the effectiveness of the analytical methods, namely, improved \(\tan (\phi /2)\)-expansion method (ITEM) and generalized (G′/G)-expansion method for seeking more exact solutions via the resonant Davey–Stewartson system. These methods are direct, concise and simple to implement compared to other existing methods. The exact particular solutions containing four types of solutions, i.e., hyperbolic function, trigonometric function, exponential and solutions. We obtained further solutions comparing these methods with other methods. The results demonstrate that the aforementioned methods are more efficient than the multi-linear variable separation method applied by Tang et al. (Chaos Solitons Fractals 42:2707–2712, 2009). Recently the ITEM was developed for searching exact traveling wave solutions of nonlinear partial differential equations. Abundant exact traveling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important role in engineering and physics fields. It is shown that these methods, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving the nonlinear problems.


Resonant Davey–Stewartson system Improved \(\tan (\phi /2)\)-expansion method Generalized (G′/G)-expansion method 


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Conflict of interest

The authors have declared no conflict of interest.


  1. Alam, M.N., Akbar, M.A., Hoque, M.F.: Exact traveling wave solutions of the (3 + 1)-dimensional mKdV-ZK equation and the (1 + 1)-dimensional compound KdVB equation using new approach of the generalized (G′/G)-expansion method. Pramana J. Phys. 83, 317–329 (2014)ADSCrossRefGoogle Scholar
  2. Alam, Md.N: Exact solutions to the foam drainage equation by using the new generalized (G′/G)-expansion method. Results Phys. 5, 168–177 (2015)ADSCrossRefGoogle Scholar
  3. Babaoglu, C.: Some special solutions of a generalized Davey–Stewartson system. Chaos Solitons Fractals 30, 781–790 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. Babaoglu, C.: Long-wave short-wave resonance case for a generalized Davey–Stewartson system. Chaos Solitons Fractals 38, 48–54 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 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
  6. Baskonus, H.M., Bulut, H.: New wave behaviors of the system of equations for the ion sound and Langmuir Waves. Waves Random Complex Media (2016b). doi: 10.1080/17455030.2016.1181811 MathSciNetGoogle Scholar
  7. 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 (2016a)Google Scholar
  8. 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
  9. Bekir, A.: Application of the (G′/G)-expansion method for nonlinear evolution equations. Phys. Lett. A 372, 3400–3406 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. Bekir, A., Güner, Ö.: Topological (dark) soliton solutions for the Camassa–Holm type equations. Ocean Eng. 74, 276–279 (2013)CrossRefGoogle Scholar
  11. Biswas, A.: 1-soliton solution of the generalized Zakharov–Kuznetsov modified equal width equation. Appl. Math. Lett. 22, 1775–1777 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 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 Google Scholar
  13. Chan, W.L., Zixiang, Z.: Line soliton solutions for a generalized Davey–Stewartson equation with variable coefficients. Lett. Math. Phys. 25, 327–334 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. Davey, A., Stewartson, K.: On three-dimensional packets of surfaces waves. Proc. R. Soc. Lond. Ser. A 338, 101–110 (1994)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. Dehghan, M., Manafian, J.: The solution of the variable coefficients fourth-order parabolic partial differential equations by homotopy perturbation method. Zeitschrift für Naturforschung A 64a, 420–430 (2009)ADSGoogle Scholar
  16. 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
  17. 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
  18. 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
  19. 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
  20. Ebadi, G., Biswas, A.: The (G′/G) method and 1-soliton solution of the Davey–Stewartson equation. Math. Comput. Model. 53, 694–698 (2011)MathSciNetCrossRefMATHGoogle Scholar
  21. Eden, A., Erbay, S., Hacinliyan, I.: Reducing a generalized Davey–Stewartson system to a non-local nonlinear Schrödinger equation. Chaos Solitons Fractals 41, 688–697 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. Fan, E.: Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 277, 212–218 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. Fan, E.: Travelling wave solutions for two generalized Hirota–Satsuma KdV systems. Z. Naturforsch. 56A, 312–319 (2001)ADSGoogle Scholar
  24. Fan, E., Zhang, H.: A note on the homogeneous balance method. Phys. Lett. A 246, 403–406 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar
  25. Fan, E., Zhang, J.: Applications of the Jacobi elliptic function method to special-type nonlinear equations. Phys. Lett. A 305, 383–392 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  26. Feng, B., Cai, Y.: Concentration for blow-up solutions of the Davey–Stewartson system in \(\mathbb{R}^3\). Nonlinear Anal. Real World Appl. 26, 330–342 (2015)MathSciNetCrossRefMATHGoogle Scholar
  27. Ganji, Z.Z., Ganji, D.D., Asgari, A.: Finding general and explicit solutions of high nonlinear equations by the Exp-function method. Comput. Math. Appl. 58, 2124–2130 (2009)MathSciNetCrossRefMATHGoogle Scholar
  28. Garagash, T.I., Pogrebkov, A.K.: Inverse scattering transform for the Hamiltonian version of the Davey–Stewartson I equation. Theor. Math. Phys. 99, 583–587 (1994)MathSciNetCrossRefMATHGoogle Scholar
  29. 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 Technol. 24, 252–256 (2013)CrossRefGoogle Scholar
  30. Huang, J., Dai, Z.: Homoclinic solutions for Davey–Stewartson equation. Chaos Solitons Fractals 35, 996–1002 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. Islam, Md.S, Khan, K., Akbar, M.A.: Application of the improved F-expansion method with Riccati equation to find the exact solution of the nonlinear evolution equations. J. Egypt. Math. Soc. 21, 1–6 (2016)Google Scholar
  32. Jahani, M., Manafian, J.: Improvement of the Exp-function method for solving the BBM equation with time-dependent coefficients. Eur. Phys. J. Plus 131, 1–11 (2016)CrossRefGoogle Scholar
  33. Khan, K., Akbar, M.A.: Traveling wave solutions of nonlinear evolution equations via the enhanced (G′/G)-expansion method. J. Egypt. Math. Soc. 22, 220–226 (2013a)MathSciNetCrossRefMATHGoogle Scholar
  34. Khan, K., Akbar, M.A.: Exact and solitary wave solutions for the Tzitzeica–Dodd–Bullough and the modified KdV–Zakharov–Kuznetsov equations using the modified simple equation method. Ain Shams Eng. J. 4, 903–909 (2013b)CrossRefGoogle Scholar
  35. Li, J.H., Lou, S.Y., Chow, K.W.: Doubly periodic patterns of modulated hydrodynamic waves: exact solutions of the Davey–Stewartson system. Acta. Mech. Sin. 27, 620–626 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  36. Liu, D.: Jacobi elliptic function solutions for two variant Boussinesq equations. Chaos Solitons Fractals 24, 1373–1385 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  37. 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
  38. 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
  39. Manafian, J., Lakestani, M.: Optical solitons with Biswas–Milovic equation for Kerr law nonlinearity. Eur. Phys. J. Plus 130, 1–12 (2015a)CrossRefGoogle Scholar
  40. Manafian, J., Lakestani, M.: Solitary wave and periodic wave solutions for Burgers, Fisher, Huxley and combined forms of these equations by the (G′/G)-expansion method. Pramana 130, 31–52 (2015b)ADSCrossRefGoogle Scholar
  41. 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, 1079 (2015c). doi: 10.1155/2015/107978 CrossRefMATHGoogle Scholar
  42. 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
  43. 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
  44. 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
  45. 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. 130, 1–12 (2015)MathSciNetGoogle Scholar
  46. Mirzazadeh, M.: Soliton solutions of Davey–Stewartson equation by trial equation method and ansatz approach. Nonlinear Dyn. (2015). doi: 10.1007/s11071-015-2276-x MathSciNetMATHGoogle Scholar
  47. Nawaz, T., Yildirim, A., Mohyud-Din, S.T.: Analytical solutions Zakharov–Kuznetsov equations. Adv. Powder Technol. 24, 252–256 (2013)CrossRefGoogle Scholar
  48. Pashaev, O.K.: Resonance solitons as black holes in madelung fluid. Mod. Phys. Lett. A 17, 1601 (2002)ADSCrossRefMATHGoogle Scholar
  49. Paul, S.K., Chowdhury, A.R.: On the n-fold backlund transformation for the Davey–Stewartson equation. Chaos Solitons Fractals 9, 1913–1920 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar
  50. 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
  51. Sung, L.Y.: An inverse scattering transform for the Davey–Stewartson equation II equations. J. Math. Anal. Appl. 183, 121–154 (1994)MathSciNetCrossRefMATHGoogle Scholar
  52. Taghizadeh, N., Neirameh, A.: New complex solutions for some special nonlinear partial differential systems. Comput. Math. Appl. 62, 2037–2044 (2011)MathSciNetCrossRefMATHGoogle Scholar
  53. Tang, X.Y., Chow, K.W., Rogers, C.: Propagating wave patterns for the ’resonant’ Davey–Stewartson. Chaos Solitons Fractals 42, 2707–2712 (2009)ADSCrossRefMATHGoogle Scholar
  54. Tascan, F., Bekir, A., Koparan, M.: Travelling wave solutions of nonlinear evolution equations by using the first integral method. Commun. Nonlinear Sci. Num. Simul. 14, 1810–1815 (2009)CrossRefGoogle Scholar
  55. Yusufoglu, E., Bekir, A.: Application of the variational iteration method to the regularized long wave equation. Comput. Math. Appl. 54, 1154–1161 (2007)MathSciNetCrossRefMATHGoogle Scholar
  56. Zayed, E.M.E., Zedan, H.A., Gepreel, K.A.: On the solitary wave solutions for nonlinear Hirota–Sasuma coupled KDV equations. Chaos Solitons Fractals 22, 285–303 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  57. Zedana, H.A., Tantawy, S.S.: Solution of Davey–Stewartson equations by homotopy perturbation method. Comput. Math. Math. Phys. 49, 1382–1388 (2009)MathSciNetCrossRefGoogle Scholar
  58. Zhou, Z.X., Ma, W.X., Zhou, R.G.: Finite-dimensional integrable systems associated with the Davey–Stewartson I equation. Nonlinearity 14, 701–717 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsPayame Noor UniversityTehranIran
  2. 2.Young Researchers and Elite Club, Ilkhchi BranchIslamic Azad UniversityIlkhchiIran

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