Abstract
A improvement of the expansion methods namely the improved \(\tan \left( \varPhi (\xi )/2\right)\)-expansion method for solving the Tzitzéica type nonlinear evolution equations is proposed. In this work, the dispersive optical solitons that are governed by the Tzitzéica type nonlinear evolution equations. As a result, many new and more general exact travelling wave solutions are obtained including periodic function solutions, soliton-like solutions and trigonometric function solutions. The exact particular solutions containing four types hyperbolic function solution, trigonometric function solution, exponential solution and rational solution. We obtained the further solutions comparing with other methods. Recently this method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important role in engineering fields. It is shown that this method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving the nonlinear problems.
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References
Abazari, R.: The (G’/G)-expansion method for Tzitzéica type nonlinear evolution equations. Math. Comput. Model. 52, 1834–1845 (2010)
Abdou, M.A., Soliman, A.A., Basyony, S.T.: New application of exp-function method for improved Boussinesq equation. Phys. Lett. A 369, 469–475 (2007)
Abdou, M.A., Soliman, A.A.: Modified extended tanh-function method and its application on nonlinear physical equations. Phys. Lett. A 353, 487–492 (2006)
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, 020202 (2014)
Borhanifar, A., Moghanlu, A.Z.: Application of the (G’/G )-expansion method for the Zhiber–Shabat equation and other related equations. Math. Comput. Model. 54, 2109–2116 (2011)
Chen, Y., Wang, Q.: Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic functions solutions to (1+1)-dimensional dispersive long wave equation. Chaos Solitons Fractals 24, 745–757 (2005)
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 (2010)
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 (2010)
Dehghan, M., Manafian Heris, 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 (2011)
Dehghan, M., Manafian, J.: The solution of the variable coefficients fourth–order parabolic partial differential equations by homotopy perturbation method. Z. Naturfor. 64a, 420–430 (2009)
Ebrahimi Ghogdia, S., Ghomanjani, F., Saberi-Nadjafi, J.: Expansion of the Exp-function method for solving systems of two-dimensional Navier–Stokes equations. J. Taibah Univ. Sci. 9, 121–125 (2015)
El-Wakil, S.A., Abdou, M.A.: New exact travelling wave solutions using modified extended tanh-function method. Chaos Solitons Fractals 31, 840–852 (2007)
Gray, P., Scott, S.K.: Chemical Oscillation and Instabilities-Nonlinear Chemical Kinetics. Oxford Science Publications, Clarendon, Oxford (1990)
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)
He, J.H.: Variational iteration method a kind of non-linear analytical technique: some examples. Int. J. Nonlinear Mech. 34, 699–708 (1999)
Huber, A.: A note on a class of solitary-like solutions of the Tzitzéica equation generated by a similarity reduction. Phys. D. 237, 1079–1087 (2008)
Jafari, H., Kadem, A., Baleanu, D.: Variational iteration method for a fractional-order Brusselator system. Abstract Appl. Anal. 2014, 1–6 (2014)
Jawad, A.J.M., Petkovic, M.D., Biswas, A.: Modified simple equation method for nonlinear evolution equations. Appl. Math. Comput. 217, 869–877 (2010)
Kabir, M.M., Khajeh, A.: New explicit solutions for the Vakhnenko and a generalized form of the nonlinear heat conduction equations via exp-function method. Int. J. Nonlinear Sci. Numer. Simul. 10, 1307–1318 (2009)
Ma, W.X., You, Y.: Rational solutions of the Toda lattice equation in Casoratian form. Chaos Solitons Fractals 22, 395–406 (2004)
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 J. Phys. 4, 1–22 (2015a)
Manafian, J., Lakestani, M.: Optical solitons with Biswas–Milovic equation for Kerr law nonlinearity. Eur. Phys. J. Plus 130, 1–12 (2015b)
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
Manafian, J., Lakestani, M.: Application of \(tan(\phi /2)\)-expansion method for solving the Biswas–Milovic equation for Kerr law nonlinearity. Optik-Int. J. Light Electron Opt. 127, 2040–2054 (2016)
Manafian Heris, J., Lakestani, M.: Solitary wave and periodic wave solutions for variants of the KdV-Burger and the K(n, n)-Burger equations by the generalized tanh-coth method. Commun. Numer. Anal. 2013, 1–18 (2013)
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. 1, 1–26 (2015). doi:10.1007/s40819-015-0058-2
Mikhailov, A.V.: The reduction problem and the inverse scattering method. Physica 3D(2), 73–117 (1910)
Mohyud-Din, S.T., Noor, M.A., Noor, K.I.: Some relatively new techniques for nonlinear problems. Math. Problems Eng. 2009, 1–26 (2009). doi:10.1155/2009/234849. Article ID 234849
Mohyud-Din, S.T., Noor, M.A., Asif, W.: Exp-function method for generalized traveling solutions of Calogero-Degasperis-Fokas equation. Z. Naturfor. A 65a, 78–84 (2010)
Mohyud-Din, S.T., Yildirim, A., Sezer, S.A.: Numerical soliton solutions of the improved Boussinesq equation. Int. J. Numer. Methods Heat Fluid Flow 21, 822–827 (2011)
Mohyud-Din, S.T., Yildirim, A., Sariaydin, S.: Numerical soliton solution of the Kaup-Kupershmidt equation. Int. J. Numer. Methods Heat Fluid Flow 21, 272–281 (2011)
Mohyud-Din, S.T., Khan, Y., Naeem, F., Yildirim, A.: Exp-function method for solitary and periodic solutions of Fitzhugh Nagumo equations. Int. J. Numer. Methods Heat Fluid Flow 22, 335–341 (2012)
Naher, H., Abdullah, F.A., Mohyud-Din, S.T.: Extended generalized Riccati equation mapping method for the fifth-order Sawada–Kotera equation. AIP Adv. 3, 052104 (2013). doi:10.1063/1.4804433
Naher, H., Abdullah, F.A.: New approach of (G’/G)-expansion method and new approach of generalized (G’/G)-expansion method for nonlinear evolution equation. AIP Adv. 3, 032116 (2013). doi:10.1063/1.4794947
Noor, M.A., Mohyud-Din, S.T., Asif, W.: Exp-function method for generalized traveling solutions of master partial differential equations. Acta Appl. Math. 104, 131–137 (2008)
Noor, M.A., Mohyud-Din, S.T., Asif, W., Eisa, A.A.S.: Exp-function method for traveling wave solutions of nonlinear evolution equations. Appl. Math. Comput. 216, 477–483 (2010)
Roshid, O.R., Rahman, M.A.: The exp(-\(\Phi (\xi )\))-expansion method with application in the (1+1)-dimensional classical Boussinesq equations. Results Phys. 4, 150–155 (2014)
Saba, F., Jabeen, S., Akbar, H., Tauseef Mohyud-Din, S.: Modified alternative (G’/G)-expansion method to general Sawada–Kotera equation of fifth-order. J. Egypt. Math. Soc. 23, 416–423 (2015)
Tzitzéica, G.: Géometric infinitésimale-sur une nouvelle classe de surface. C. R. Math. Acad. Sci. Paris 150, 227–232 (1910)
Wazwaz, A.M.: The tanh method: solitons and periodic solutions for the Dodd–Bullough–Mikhailov and the Tzitzéica–Dodd–Bullough equations. Chaos Solitons Fractals 25, 55–63 (2005)
Wazwaz, A.M.: Travelling wave solutions for combined and double combined sine–cosine–Gordon equations by the variable separated ODE method. Appl. Math. Comput. 177, 755–760 (2006)
Yildirim, A., Pinar, Z.: Application of the exp-function method for solving nonlinear reaction-diffusion equations arising in mathematical biology. Comput. Math. Appl. 60, 1873–1880 (2010)
Zhang, X., Zhao, J., Liu, J., Tang, B.: Homotopy perturbation method for two dimensional time-fractional wave equation. Appl. Math. Model. 38, 5545–5552 (2014)
Zhao, X., Wang, L., Sun, W.: The repeated homogeneous balance method and its applications to nonlinear partial differential equations. Chaos Solitons Fractals 28, 448–453 (2006)
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Manafian, J., Lakestani, M. Dispersive dark optical soliton with Tzitzéica type nonlinear evolution equations arising in nonlinear optics. Opt Quant Electron 48, 116 (2016). https://doi.org/10.1007/s11082-016-0371-y
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DOI: https://doi.org/10.1007/s11082-016-0371-y