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On the Solitary Wave Solutions to the (2+1)-Dimensional Davey-Stewartson Equations

  • Hajar F. IsmaelEmail author
  • Hasan Bulut
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1111)

Abstract

In this article, by using the Bernoulli sub-equation, we build the analytical traveling wave solution of the (2+1)-dimensional Davey-Stewartson equation system. First of all, the imaginary (2+1)-dimensional Davey-Stewatson system is transformed into a system of nonlinear differential equations, After getting the resultant equation, the homogeneous method of balance between the highest power and the highest derivative of the ordinary differential equation is authorized and finally the outcomes equations are solved in order to achieve some new analytical solutions. Wolfram Mathematica Package is used for different cases as well as for different values of constants to investigate the solutions of the resulting system of a nonlinear differential equation. The results of this study are shown in 2D and 3D dimensions graphically.

Keywords

Bernoulli sub-equation Davey-Stewatson equations 

References

  1. 1.
    Ilhan, O.A., Esen, A., Bulut, H., Baskonus, H.M.: Singular solitons in the pseudo-parabolic model arising in nonlinear surface waves. Results Phys. (2019).  https://doi.org/10.1016/j.rinp.2019.01.059CrossRefGoogle Scholar
  2. 2.
    Aktürk, T., Gürefe, Y., Bulut, H.: New function method to the (n+1)-dimensional nonlinear problems. Int. J. Optim. Control Theor. Appl. (2017).  https://doi.org/10.11121/ijocta.01.2017.00489CrossRefGoogle Scholar
  3. 3.
    Kocak, Z. F., Bulut, H., Yel, G.: The solution of fractional wave equation by using modified trial equation method and homotopy analysis method. In AIP Conference Proceedings (2014)Google Scholar
  4. 4.
    Nofal, T.A.: An approximation of the analytical solution of the Jeffery-Hamel flow by homotopy analysis method. Appl. Math. Sci. 5(53), 2603–2615 (2011)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Sulaiman, T.A., Bulut, H., Yokus, A., Baskonus, H.M.: On the exact and numerical solutions to the coupled Boussinesq equation arising in ocean engineering. Indian J. Phys. (2019).  https://doi.org/10.1007/s12648-018-1322-1CrossRefGoogle Scholar
  6. 6.
    Yousif, M.A., Mahmood, B.A., Ali, K.K., Ismael, H.F.: Numerical simulation using the homotopy perturbation method for a thin liquid film over an unsteady stretching sheet. Int. J. Pure Appl. Math. 107(2) (2016).  https://doi.org/10.12732/ijpam.v107i2.1
  7. 7.
    Yokus, A., Baskonus, H.M., Sulaiman, T.A., Bulut, H.: Numerical simulation and solutions of the two-component second order KdV evolutionarysystem. Numer. Methods Partial Differ. Equ. (2018).  https://doi.org/10.1002/num.22192zbMATHCrossRefGoogle Scholar
  8. 8.
    Atangana, A., Ahmed, A., Oukouomi Noutchie, S.C.: On the Hamilton-Jacobi-Bellman equation by the homotopy perturbation method. Abstr. Appl. Anal. 2014, 8 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bueno-Orovio, A., Pérez-García, V.M., Fenton, F.H.: Spectral methods for partial differential equations in irregular domains: the spectral smoothed boundary method. SIAM J. Sci. Comput. 28(3), 886–900 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bulut, H., Ergüt, M., Asil, V., Bokor, R.H.: Numerical solution of a viscous incompressible flow problem through an orifice by Adomian decomposition method. Appl. Math. Comput. 153(3), 733–741 (2004)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ismael, H.F., Ali, K.K.: MHD casson flow over an unsteady stretching sheet. Adv. Appl. Fluid Mech. (2017).  https://doi.org/10.17654/FM020040533CrossRefGoogle Scholar
  12. 12.
    Owolabi, K.M., Atangana, A.: On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems. Chaos Interdiscip. J. Nonlinear Sci. 29(2), 23111 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Baskonus, H.M., Bulut, H.: On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method. Open Math. (2015).  https://doi.org/10.1515/math-2015-0052
  14. 14.
    Ismael, H.F.: Carreau-Casson fluids flow and heat transfer over stretching plate with internal heat source/sink and radiation. Int. J. Adv. Appl. Sci. J. 6(2), 81–86 (2017).  https://doi.org/10.1371/journal.pone.0002559CrossRefGoogle Scholar
  15. 15.
    Ali, K.K., Ismael, H.F., Mahmood, B.A., Yousif, M.A.: MHD Casson fluid with heat transfer in a liquid film over unsteady stretching plate. Int. J. Adv. Appl. Sci. 4(1), 55–58 (2017)CrossRefGoogle Scholar
  16. 16.
    Ismael, H.F., Arifin, N.M.: Flow and heat transfer in a Maxwell liquid sheet over a stretching surface with thermal radiation and viscous dissipation. JP J. Heat Mass Transf. 15(4) (2018).  https://doi.org/10.17654/HM015040847CrossRefGoogle Scholar
  17. 17.
    Zeeshan, A., Ismael, H.F., Yousif, M.A., Mahmood, T., Rahman, S.U.: Simultaneous effects of slip and wall stretching/shrinking on radiative flow of magneto nanofluid through porous medium. J. Magn. 23(4), 491–498 (2018).  https://doi.org/10.4283/JMAG.2018.23.4.491CrossRefGoogle Scholar
  18. 18.
    Baskonus, H.M., Bulut, H., Sulaiman, T.A.: New complex hyperbolic structures to the Lonngren-wave equation by using sine-Gordon expansion method. Appl. Math. Nonlinear Sci. 4(1), 141–150 (2019)MathSciNetGoogle Scholar
  19. 19.
    Eskitaşçıoğlu, Eİ., Aktaş, M.B., Baskonus, H.M.: New complex and hyperbolic forms for Ablowitz-Kaup-Newell-Segur wave equation with fourth order. Appl. Math. Nonlinear Sci. 4(1), 105–112 (2019)MathSciNetGoogle Scholar
  20. 20.
    Vakhnenko, V.O., Parkes, E.J., Morrison, A.J.: A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation. Chaos Solitons Fractals (2003).  https://doi.org/10.1016/S0960-0779(02)00483-6zbMATHCrossRefGoogle Scholar
  21. 21.
    Hammouch, Z., Mekkaoui, T.: Traveling-wave solutions of the generalized Zakharov equation with time-space fractional derivatives. J. MESA 5(4), 489–498 (2014)zbMATHGoogle Scholar
  22. 22.
    Baskonus, H.M., Bulut, H.: An effective schema for solving some nonlinear partial differential equation arising in nonlinear physics. Open Phys. (2015).  https://doi.org/10.1515/phys-2015-0035
  23. 23.
    Baskonus, H.M., Bulut, H.: Exponential prototype structures for (2+1)-dimensional Boiti-Leon-Pempinelli systems in mathematical physics. Waves Random Complex Media (2016).  https://doi.org/10.1080/17455030.2015.1132860MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Wei, G., Ismael, H.F., Bulut, H., Baskonus, H.M.: Instability modulation for the (2+1)-dimension paraxial wave equation and its new optical soliton solutions in Kerr media. Phys. Scr. (2019). http://iopscience.iop.org/10.1088/1402-4896/ab4a50
  25. 25.
    Ilhan, O.A., Bulut, H., Sulaiman, T.A., Baskonus, H.M.: Dynamic of solitary wave solutions in some nonlinear pseudoparabolic models and Dodd–Bullough–Mikhailov equation. Indian J. Phys. (2018).  https://doi.org/10.1007/s12648-018-1187-3CrossRefGoogle Scholar
  26. 26.
    Cattani, C., Sulaiman, T.A., Baskonus, H.M., Bulut, H.: Solitons in an inhomogeneous Murnaghan’s rod. Eur. Phys. J. Plus (2018).  https://doi.org/10.1140/epjp/i2018-12085-y
  27. 27.
    Houwe, A., Hammouch, Z., Bienvenue, D., Nestor, S., Betchewe, G.: Nonlinear Schrödingers equations with cubic nonlinearity: M-derivative soliton solutions by \(\exp (-\varPhi (\xi )) \)-expansion method (2019)Google Scholar
  28. 28.
    Manafian, J., Aghdaei, M.F.: Abundant soliton solutions for the coupled Schrödinger-Boussinesq system via an analytical method. Eur. Phys. J. Plus (2016).  https://doi.org/10.1140/epjp/i2016-16097-3CrossRefGoogle Scholar
  29. 29.
    Hammouch, Z., Mekkaoui, T., Agarwal, P.: Optical solitons for the Calogero-Bogoyavlenskii-Schiff equation in (2 + 1) dimensions with time-fractional conformable derivative. Eur. Phys. J. Plus (2018).  https://doi.org/10.1140/epjp/i2018-12096-8CrossRefGoogle Scholar
  30. 30.
    Khalique, C.M., Mhlanga, I.E.: Travelling waves and conservation laws of a (2+1)-dimensional coupling system with Korteweg-de Vries equation. Appl. Math. Nonlinear Sci. (2018).  https://doi.org/10.21042/amns.2018.1.00018MathSciNetCrossRefGoogle Scholar
  31. 31.
    Aghdaei, M.F., Manafian, J.: Optical soliton wave solutions to the resonant davey-stewartson system. Opt. Quantum Electron. (2016).  https://doi.org/10.1007/s11082-016-0681-0CrossRefGoogle Scholar
  32. 32.
    Yang, X., Yang, Y., Cattani, C., Zhu, C.M.: A new technique for solving the 1-D Burgers equation. Therm. Sci. (2017).  https://doi.org/10.2298/TSCI17S1129YCrossRefGoogle Scholar
  33. 33.
    Bulut, H., Sulaiman, T.A., Baskonus, H.M.: Dark, bright optical and other solitons with conformable space-time fractional second-order spatiotemporal dispersion. Optik (Stuttg). (2018).  https://doi.org/10.1016/j.ijleo.2018.02.086CrossRefGoogle Scholar
  34. 34.
    Cattani, C., Sulaiman, T.A., Baskonus, H.M., Bulut, H.: On the soliton solutions to the Nizhnik-Novikov-Veselov and the Drinfel’d-Sokolov systems. Opt. Quantum Electron. (2018).  https://doi.org/10.1007/s11082-018-1406-3
  35. 35.
    Osman, M.S., Ghanbari, B.: New optical solitary wave solutions of Fokas-Lenells equation in presence of perturbation terms by a novel approach. Optik (Stuttg). (2018).  https://doi.org/10.1016/j.ijleo.2018.08.007CrossRefGoogle Scholar
  36. 36.
    Ghanbari, B., Kuo, C.-K.: New exact wave solutions of the variable-coefficient (1 + 1)-dimensional Benjamin-Bona-Mahony and (2 + 1)-dimensional asymmetric Nizhnik-Novikov-Veselov equations via the generalized exponential rational function method. Eur. Phys. J. Plus 134(7), 334 (2019)CrossRefGoogle Scholar
  37. 37.
    Ebadi, G., Biswas, A.: The \(G^{\prime }/G\) method and 1-soliton solution of the Davey-Stewartson equation. Math. Comput. Model. 53(5–6), 694–698 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Zedan, H.A., Al Saedi, A.: Periodic and solitary wave solutions of the Davey-Stewartson equation. Appl. Math. Inf. Sci. 4(2), 253–260 (2010)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Besse, C., Mauser, N.J., Stimming, H.P.: Numerical study of the Davey-Stewartson system. ESAIM Math. Model. Numer. Anal. 38(6), 1035–1054 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Ye, X.: On the fully discrete Davey-Stewartson system with self-consistent sources. Pacific J. Appl. Math. 7(3), 163 (2015)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Li, Z.-F., Ruan, H.-Y.: (2+1)-dimensional Davey-Stewartson II equation for a two-dimensional nonlinear monatomic lattice. Zeitschrift für Naturforsch. A 61(1–2), 45–52 (2006)CrossRefGoogle Scholar
  42. 42.
    Baskonus, H.M.: New acoustic wave behaviors to the Davey-Stewartson equation with power-law nonlinearity arising in fluid dynamics. Nonlinear Dyn. (2016).  https://doi.org/10.1007/s11071-016-2880-4MathSciNetCrossRefGoogle Scholar
  43. 43.
    Abdelaziz, M.A.M., Moussa, A.E., Alrahal, D.M.: Exact solutions for the nonlinear (2+1)-dimensional Davey-Stewartson equation using the generalized \(({G^\prime }/{G})\)-expansion method. J. Math. Res. 6(2) (2014)Google Scholar
  44. 44.
    Gurefe, Y., Misirli, E., Pandir, Y., Sonmezoglu, A., Ekici, M.: New exact solutions of the Davey-Stewartson equation with power-law nonlinearity. Bull. Malaysian Math. Sci. Soc. 38(3), 1223–1234 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Cevikel, A.C., Bekir, A.: New solitons and periodic solutions for (2+1)-dimensional Davey-Stewartson equations. Chin. J. Phys. 51(1), 1–13 (2013)MathSciNetzbMATHGoogle Scholar
  46. 46.
    El-Kalaawy, O.H., Ibrahim, R.S.: Solitary wave solution of the two-dimensional regularized long-wave and Davey-Stewartson equations in fluids and plasmas. Appl. Math. 3(08), 833 (2012)CrossRefGoogle Scholar
  47. 47.
    Baskonus, H.M., Bulut, H.: On the complex structures of Kundu-Eckhaus equation via improved Bernoulli sub-equation function method. Waves Random Complex Media (2015).  https://doi.org/10.1080/17455030.2015.1080392MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Baskonus, H.M., Bulut, H.: An effective schema for solving some nonlinear partial differential equation arising in nonlinear physics. Open Phys. (2015). https://doi.org/10.1515/phys-2015-0035
  49. 49.
    Anker, D., Freeman, N.C.: On the soliton solutions of the Davey-Stewartson equation for long waves. Proc. R. Soc. London Ser. A (1978).  https://doi.org/10.1098/rspa.1978.0083MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Mirzazadeh, M.: Soliton solutions of Davey-Stewartson equation by trial equation method and ansatz approach. Nonlinear Dyn. 82(4), 1775–1780 (2015)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ZakhoZakhoIraq
  2. 2.Department of MathematicsUniversity of FiratElazigTurkey

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