Abstract
Recently, the \(\exp (-\phi (\xi ))\)-expansion method has attracted many authors’ interest. In this article, by making use of a certain Riccati equation, we obtain its equivalent form. Compared with the original \(\exp (-\phi (\xi ))\)-expansion method, the equivalent form is simpler, more direct and facile for application. The nonlinear Gerdjikov–Ivanov equation serves as an example to show its advantages.
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Rahman, N., Akter, S., Roshid, H.O., Alam, M.N.: Traveling wave solutions of the (1+1)-dimensional compound KdVB equation by \(\exp (-\phi (\eta ))\)-expansion method. Glob. J. Sci. Front. Res. 13, 7–13 (2013)
Islam, R., Alam, M.N., Hossain, A.K.M.K.S., Roshid, H.O., Akbar, M.A.: Traveling wave solutions of nonlinear evolution equations via \(\exp (-\phi (\eta ))\)-expansion method. Glob. J. Sci. Front. Res. 13, 63–71 (2013)
Khan, K., Akbar, M.A.: Application of \(\exp (-\phi (\xi ))\)-expansion method to find the exact solutions of modified Benjamin–Bona–Mahony equation. World Appl. Sci. J. 24, 1373–1377 (2013)
Rahman, N., Alam, M.N., Roshid, H.O., Akter, S., Akbar, M.A.: Application of \(\exp (-\phi (\xi ))\)-expansion method to find the exact solutions of Shorma–Tasso–Olver equation. Afr. J. Math. Comput. Sci. Res. 7, 1–6 (2014)
Khan, K., Akbar, M.A.: The \(\exp (-\phi (\xi ))\)-expansion method for finding travelling wave solutions of Vakhnenko–Parkes equation. Int. J. Dyn. Syst. Differ. Equ. 5, 72–83 (2014)
Akbar, M.A., Ali, N.H.M.: Solitary wave solutions of the fourth order Boussinesq equation through the \(\exp (-\phi (\eta ))\)-expansion method. SpringerPlus 3, 344 (2014)
Roshid, H.O., Rahman, M.A.: The \(\exp (-\phi (\eta ))\)-expansion method with application in the (1+1)-dimensional classical Boussinesq equations. Results Phys. 4, 150–155 (2014)
Hafez, M.G., Alam, M.N., Akbar, M.A.: Traveling wave solutions for some important coupled nonlinear physical models via the coupled Higgs equation and the Maccari system. J. King Saud Univ. Sci. 27, 105–112 (2015)
Roshid, H.O., Alam, M.N., Akbar, M.A.: Traveling wave solutions for fifth order (1+1)-dimensional Kaup–Kupershmidt equation with the help of \(\exp (-\phi (\eta ))\)-expansion method. Walailak J. Sci. Technol. 12, 1063–1073 (2015)
Zahran, E.H.M.: Exact traveling wave solutions of nano-ionic solitons and nano-ionic current of MTs using the \(\exp (-\phi (\xi ))\)-expansion method. Adv. Nanopart. 4, 25–36 (2015)
Alam, M.N., Hafez, M.G., Akbar, M.A., Roshid, H.O.: Exact solutions to the (2+1)-dimensional Boussinesq equation via \(\exp (-\phi (\eta ))\)-expansion method. J. Sci. Res. 7, 1–10 (2015)
Baskonus, H.M., Bulut, H., Atangana, A.: On the complex and hyperbolic structures of the longitudinal wave equation in a magneto-electro-elastic circular rod. Smart Mater. Struct. 25, 035022 (2016)
Hafez, M.G.: Exact solutions to the (3+1)-dimensional coupled Klein–Gordon–Zakharov equation using \(\exp (-\phi (\xi ))\)-expansion method. Alex. Eng. J. 55, 1635–1645 (2016)
Ali, A., Iqbal, M.A., Mohyud-Din, S.T.: Traveling wave solutions of generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony and simplified modified form of Camassa–Holm equation \(\exp (-\phi (\eta ))\)-expansion method. Egypt. J. Basic Appl. Sci. 3, 134–140 (2016)
Bulut, H., Baskonus, H.M.: New complex hyperbolic function solutions for the (2+1)-dimensional dispersive long waterwave system. Math. Comput. Appl. 21, 6 (2016)
Alam, M.N., Tunc, C.: An analytical method for solving exact solutions of the nonlinear Bogoyavlenskii equation and the nonlinear diffusive predator-prey system. Alex. Eng. J. 55, 1855–1865 (2016)
Khater, M.M.A.: Exact traveling wave solutions for the generalized Hirota–Satsuma couple KdV system using the \(\exp (-\phi (\xi ))\)-expansion method. Cogent Math. 3, 1172397 (2016)
Kaplan, M., Bekir, A.: A novel analytical method for time-fractional differential equations. Optik 127, 8209–8214 (2016)
Ali, A., Iqbal, M.A., Mohyud-Din, S.T.: New analytical solutions for nonlinear physical models of the coupled Higgs equation and the Maccari system via rational \(\exp (-\phi (\eta ))\)-expansion method. Pramana J. Phys. 87, 79 (2016)
Baskonus, H.M., Bulut, H., Belgacem, F.B.M.: Analytical solutions for nonlinear longshort wave interaction systems with highly complex structure. J. Comput. Appl. Math. 312, 257–266 (2017)
Islam, M.R., Roshid, H.O.: Application of \(\exp (-\phi (\xi ))\)-expansion method for Tzitzeica type nonlinear evolution equations. J. Found. Appl. Phys. 4, 8–18 (2017)
Alam, M.N., Alam, M.M.: An analytical method for solving exact solutions of a nonlinear evolution equation describing the dynamics of ionic currents along microtubules. J. Taibah Univ. Sci. 11, 939–948 (2017)
Mirzazadeh, M., Ekici, M., Zhou, Q., Sonmezoglu, A.: Analytical study of solitons in the fiber waveguide with power law nonlinearity. Superlattices Microstruct. 101, 493–506 (2017)
Akbulut, A., Kaplan, M., Tascan, F.: The investigation of exact solutions of nonlinear partial differential equations by using \(\exp (-\phi (\xi ))\) method. Optik 132, 382–387 (2017)
Kadkhoda, N., Jafari, H.: Analytical solutions of the GerdjikovIvanov equation by using \(\exp (-\phi (\xi ))\)-expansion method. Optik 139, 72–76 (2017)
Ni, W.G., Dai, C.Q.: Note on same result of different anstz based on extended tanh-function method for nonlinear models. Appl. Math. Comput. 270, 434–440 (2015)
Rogers, C., Chow, K.W.: Localized pulses for the quintic derivative nonlinear Schödinger equation on a continuous-wave background. Phys. Rev. E 86, 037601 (2012)
Triki, H., Alqahtani, R.T., Zhou, Q., Biswas, A.: New envelope solitons for Gerdjikov–Ivanov model in nonlinear fiber optics. Superlattices Microstruct. 111, 326–334 (2017)
Biswas, A., Ekici, M., Sonmezoglu, A., Triki, H., Alshomrani, A.S., Zhou, Q., Moshokoa, S.P., Belic, M.: Optical solitons for GerdjikovIvanov model by extended trial equation scheme. Optik 157, 1241–1248 (2018)
Lü, X., Ma, W.X., Yu, J., Lin, F., Khalique, C.M.: Envelope bright- and dark-soliton solutions for the Gerdjikov–Ivanov model. Nonlinear Dyn. 82, 1211–1220 (2015)
Acknowledgements
Many thanks are due to the helpful comments and suggestions from the anonymous referee and support from the Scientific Research Fund of Zhejiang Provincial Education Department (Grant number Y201432746).
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Liu, HZ. An equivalent form for the \(\exp (-\phi (\xi ))\)-expansion method. Japan J. Indust. Appl. Math. 35, 1153–1161 (2018). https://doi.org/10.1007/s13160-018-0324-x
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DOI: https://doi.org/10.1007/s13160-018-0324-x