Skip to main content
SpringerLink
Log in

New methods to solve the resonant nonlinear Schrödinger’s equation with time-dependent coefficients

  • Published:
Optical and Quantum Electronics Aims and scope Submit manuscript

Abstract

In this study, the generalized \(\tan (\phi /2)\)-expansion method and He’s semi-inverse variational method (HSIVM) are applied to seek the exact solitary wave solutions for the resonant nonlinear Schrödinger equation with time-dependent coefficients. Using these methods, we investigate exact solutions for the nonlinear resonant Schrödinger equation with time-dependent coefficients two forms of nonlinearity, including power and dual-power law nonlinearity. Moreover, many new analytical exact solutions are obtained which are expressed by hyperbolic solutions, trigonometric solutions, and rational solutions. In addition, we obtained the bright soliton by HSIVM. These methods are powerful, efficient and those can be used as an alternative to establishing new solutions of different types of differential equations in mathematical physics and engineering.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  • Aghdaei, M.F.: Kerr-law nonlinearity of the resonant nonlinear Schrodinger’s equation with time-dependent coefficients. Opt. Quantum Electron. 49(245), 1–22 (2017)

    Google Scholar 

  • Aghdaei, M.F., Manafianheris, J.: Exact solutions of the couple Boiti-Leon-Pempinelli system by the generalized \((\frac{{\text{ G }}^{\prime }}{{\text{ G }}})\)-expansion method. J. Math. Ext. 5, 91–104 (2011)

    MathSciNet  MATH  Google Scholar 

  • Ahmed, M.T., Khan, K., Akbar, M.A.: Study of nonlinear evolution equations to construct traveling wave solutions via modified simple equation method. Phys. Rev. Res. Int. 3(4), 490–503 (2013)

    Google Scholar 

  • Baskonus, H.M.: New acoustic wave behaviors to the Davey-Stewartson equation with power-law nonlinearity arising in fluid dynamics. Nonlinear Dyn. 86, 177–183 (2016)

    Article  MathSciNet  Google Scholar 

  • Baskonus, H.M.: New complex and hyperbolic function solutions to the generalized double combined Sinh-Cosh-Gordon equation. AIP Conf. Proc. 1798, 020018 (2017). doi:10.1063/1.4972610

    Article  Google Scholar 

  • 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)

    MathSciNet  Google Scholar 

  • 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, 613–625 (2016b). doi:10.1080/17455030.2016.1181811

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • 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

    Article  ADS  Google Scholar 

  • Bhrawy, A.H., Alshaery, A.A., Hilal, E.M., Khan, K.R., Mahmood, M.F., Biswas, A.: Optical soliton perturbation with spatio-temporal dispersion in parabolic and dual-power law media by semi-inverse variational principle. Optik 125, 4945–4950 (2014a)

    Article  ADS  Google Scholar 

  • Bhrawy, A.H., Abdelkawy, M.A., Biswas, A.: Optical solitons in \((1+1)\) and \((2+1)\) dimensions. Optik 125, 1537–1549 (2014b)

    Article  ADS  Google Scholar 

  • Biswas, A.: Soliton solutions of the perturbed resonant nonlinear Schrodinger’s equation with full nonlinearity by semi-inverse variational principle. Quantum Phys. Lett. 1, 79–84 (2012)

    Google Scholar 

  • Biswas, A., Alqahtani, R.T.: Chirp-free bright optical solitons for perturbed Gerdjikov-Ivanov equation by semi-inverse variational principle. Optik (2017a). doi:10.1016/j.ijleo.2017.08.019

  • Biswas, A., Alqahtani, R.T.: Optical soliton perturbation with complex Ginzburg-Landau equation by semi-inverse variational principle. Optik (2017b). doi:10.1016/j.ijleo.2017.08.018

  • Biswas, A., Milovic, D.: Bright and dark solitons of the generalized nonlinear Schrödinger’s equation. Commun. Nonlinear Sci. Numer. Simul. 15, 1473–1484 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Biswas, A., Milovic, D.: Chiral solitons with Bohm potential by He’s variational principle. Phys. Atomic Nuclei 74, 781–783 (2011)

    Article  Google Scholar 

  • Biswas, A., Kara, A.H., Zerrad, E.: Dynamics and conservation laws of the generalized chiral solitons. Open Nuclear Particle Phys. J. 4, 21–24 (2011)

    Article  ADS  Google Scholar 

  • Biswas, A., Milovic, D., Kohl, R.: Optical soliton perturbation in a log-law medium with full nonlinearity by He’s semi-inverse variational principle. Inverse Probl. Sci. Eng. 20, 227–232 (2012a)

    Article  MATH  Google Scholar 

  • Biswas, A., Johnson, S., Fessak, M., Siercke, B., Zerrad, E., Konar, S.: Dispersive optical solitons by semi-inverse variational principle. J. Mod. Opt. 59, 213–217 (2012b)

    Article  ADS  Google Scholar 

  • Biswas, A., Fessak, M., Johnson, S., Beatrice, S., Milovic, D., Jovanoski, Z., et al.: Optical soliton perturbation in non-Kerr law media: traveling wave solution. Opt. Laser Technol. 44, 263–268 (2012c)

    Article  ADS  Google Scholar 

  • Biswas, A., Alqahtani, R.T., Abdelkawy, M.A.: Optical soliton perturbation with parabolic and dual-power law nonlinearities by semi-inverse variational principle. Optik (2017a). doi:10.1016/j.ijleo.2017.08.022

    Google Scholar 

  • Biswas, A., Ullah, M.Z., Zhou, Q., Moshokoa, S.P., Triki, H., Belic, M.: Resonant optical solitons with quadratic-cubic nonlinearity by semi-inverse variational principle. Optik 145, 18–21 (2017b)

    Article  Google Scholar 

  • Biswas, A., Zhou, Q., Ullah, M.Z., Triki, H., Moshokoa, S.P., Belic, M.: Optical soliton perturbation with anti-cubic nonlinearity by semi-inverse variational principle. Optik 143, 131–134 (2017c)

    Article  ADS  Google Scholar 

  • Biswas, A., Ullah, M.Z., Asma, M., Zhou, Q., Moshokoa, S.P., Belic, M.: Optical solitons with quadratic-cubic nonlinearity by semi-inverse variational principle. Optik 139, 16–19 (2017d)

    Article  ADS  Google Scholar 

  • Biswas, A., Zhou, Q., Moshokoa, S.P., Triki, H., Belic, M., Alqahtani, R.T.: Resonant 1-soliton solution in anti-cubic nonlinear medium with perturbations. Optik 145, 14–17 (2017e)

    Article  Google Scholar 

  • 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

    MathSciNet  Google Scholar 

  • 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 Fract. 24, 745–757 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Dehghan, M., Manafian, J.: The solution of the variable coefficients fourth-order parabolic partial differential equations by homotopy perturbation method. Z. Naturforsch. 64a, 420–430 (2009)

    ADS  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • 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 (2011)

    Article  MathSciNet  Google Scholar 

  • Ekici, M., Zhou, Q., Sonmezoglu, A., Manafian, J., Mirzazadeh, M.: The analytical study of solitons to the nonlinear Schrödinger equation with resonant nonlinearity. Opt. Int. J. Electron Opt. 130, 378–382 (2017a)

    Article  Google Scholar 

  • Ekici, M., Mirzazadeh, M., Sonmezoglu, A., Zhou, Q., Triki, H., Zaka Ullah, M., Moshokoa, S.P., Biswas, A.: Optical solitons in birefringent fibers with Kerr nonlinearity by exp-function method. Opt. Int. J. Light Electron Opt. 131, 964–976 (2017b)

    Article  Google Scholar 

  • Eslami, M., Mirzazadeh, M., Vajargah, B.F., Biswas, A.: Optical solitons for the resonant nonlinear Schrödinger’s equation with time-dependent coefficients by the first integral method. Opt. Int. J. Light Electron Opt. 125, 3107–3116 (2014)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Hasegawa, A., Kodama, Y.: Solitons in Optical Communications. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  • Jahani, M., Manafian, J.: Improvement of the Exp-function method for solving the BBM equation with time-dependent coefficients. Eur. Phys. J. Plus 131(54), 1–12 (2016)

    Google Scholar 

  • Jawad, A.J.M., Mirzazadeh, M., Zhou, Q., Biswas, A.: Optical solitons with anti-cubic nonlinearity using three integration schemes. Superlattices Microstruct. 105, 1–10 (2017)

    Article  ADS  Google Scholar 

  • 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(4), 903–909 (2013)

    Article  Google Scholar 

  • Khan, K., Akbar, M.A., Rashidi, M.M., Zamanpour, I.: Exact traveling wave solutions of an autonomous system via the enhanced (G′/G)-expansion method. Waves Random Complex Media 25, 1–12 (2015)

    Article  MathSciNet  Google Scholar 

  • Kohl, R., Biswas, A., Milovic, D., Zerrad, E.: Optical soliton perturbation in a non-Kerr law media. Opt. Laser Technol. 40, 647–662 (2008)

    Article  ADS  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Manafian, J.: Optical soliton solutions for Schrödinger type nonlinear evolution equations by the \(\tan(\phi /2)\)-expansion method. Opt. Int. J. Electron Opt. 127, 4222–4245 (2016a)

    Article  Google Scholar 

  • Manafian, J.: Optical soliton solutions for Schrödinger type nonlinear evolution equations by the \(\tan(\phi /2)\)-expansion method. Optik 127, 4222–4245 (2016b)

    Article  ADS  Google Scholar 

  • Manafian, J., Lakestani, M.: Optical solitons with Biswas-Milovic equation for Kerr law nonlinearity. Eur. Phys. J. Plus 130, 1–12 (2015)

    Article  Google Scholar 

  • Manafian, J., Lakestani, M.: Application of \(\tan(\phi /2)\)-expansion method for solving the Biswas-Milovic equation for Kerr law nonlinearity. Opt. Int. J. Electron Opt. 127, 2040–2054 (2016a)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • Mirzazadeh, M.: Topological and non-topological soliton solutions of Hamiltonian amplitude equation by He’s semi-inverse method and ansatz approach. J. Egypt. Math. Soc. 23, 292–296 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Mirzazadeh, M., Eslami, M.: Exact multisoliton solutions of nonlinear Klein-Gordon equation in \(1+2\) dimensions. Eur. Phys. J. Plus 128, 1–9 (2015)

    Google Scholar 

  • Mirzazadeh, M., Eslami, M., Milovic, D., Biswas, A.: Topological solitons of resonant nonlinear Schödinger’s equation with dual-power law nonlinearity by G′/G-expansion technique. Opt. Int. J. Light Electron Opt. 125, 5480–5489 (2014)

    Article  Google Scholar 

  • Mirzazadeh, M., Eslami, M., Arnous, A.H.: Dark optical solitons of Biswas-Milovic equation with dual-power law nonlinearity. Eur. Phys. J. Plus 130, 1–7 (2015)

    Article  Google Scholar 

  • Mirzazadeh, M., Ekici, M., Sonmezoglu, A., Ortakaya, S., Eslami, M., Biswas, A.: Soliton solutions to a few fractional nonlinear evolution equations in shallow water wave dynamics. Eur. Phys. J. Plus 131(166), 1–11 (2016)

    Google Scholar 

  • Nishino, A., Umeno, Y., Wadati, M.: Chiral nonlinear Schrödinger equation. Chaos Solitons Fract. 9, 1063–1069 (1998)

    Article  ADS  MATH  Google Scholar 

  • Pashaev, O.K., Lee, J.H.: Resonance solitons as black holes in Madelung fluid. Mod. Phys. Lett. A 17, 1601–1619 (2002)

    Article  ADS  MATH  Google Scholar 

  • Rogers, C., Yip, L.P., Chow, K.W.: A resonant Davey-Stewartson capillary model system. Int. J. Nonlinear Sci. Numer. Simul. 10, 397–405 (2009)

    Article  Google Scholar 

  • Sindi, C.T., Manafian, J.: Wave solutions for variants of the KdV-Burger and the K(n, n)-Burger equations by the generalized \(G^{\prime }/G\)-expansion method. Math. Methods Appl. Sci. 87, 1–14 (2016)

    MATH  Google Scholar 

  • Taghizadeh, N., Zhou, Q., Ekici, M., Mirzazadeh, M.: Soliton solutions for Davydov solitons in \(\alpha\)-helix proteins. Superlattices Microstruct. 102, 323–341 (2017)

    Article  ADS  Google Scholar 

  • Tang, X.Y., Chow, K.W., Rogers, C.: Propagating wave patterns for the ’resonant’ Davey-Stewartson system. Chaos Solitons Fract. 42, 2707–2712 (2009)

    Article  ADS  MATH  Google Scholar 

  • Triki, H., Hayat, T., Aldossary, O.M., Biswas, A.: Bright and dark solitons for the resonant nonlinear Schrödinger’s equation with time-dependent coefficients. Opt. Laser Technol. 44, 2223–2231 (2012)

    Article  ADS  Google Scholar 

  • Wazwaz, A.M.: Reliable analysis for nonlinear Schrödinger equations with a cubic nonlinearity and a power law nonlinearity. Math. Comput. Model. 43, 178–184 (2006)

    Article  MATH  Google Scholar 

  • Xu, Y., Vega-Guzman, J., Milovic, D., Mirzazadeh, M., Eslami, M., Mahmood, M.F., Biswas, A., Belic, M.: Bright and exotic solitons in optical metamaterials by semi-inverse variational principle. J. Nonlinear Opt. Phys. Mater. 24, 1550042 (2015). doi:10.1142/S0218863515500423

    Article  ADS  Google Scholar 

  • Zhao, X., Wang, L., Sun, W.: The repeated homogeneous balance method and its applications to nonlinear partial differential equations. Chaos Solitons Fract. 28, 448–453 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Zhao, X., Wang, L., Sun, W.: Exact solutions for nonlinear fractional differential equations using exponential rational function method. Opt. Quantum Electron. 49(64), 1–12 (2017)

    Google Scholar 

  • Zhou, Q., Ekici, M., Sonmezoglu, A., Manafian, J., Khaleghizadeh, S., Mirzazadeh, M.: Exact solitary wave solutions to the generalized Fisher equation. Optik 127, 12085–12092 (2016)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for their valuable suggestions and comments.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Avenue, Tehran, 15914, Iran

    Mehdi Fazli Aghdaei & Hojatollah Adibi

Authors
  1. Mehdi Fazli Aghdaei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hojatollah Adibi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hojatollah Adibi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aghdaei, M.F., Adibi, H. New methods to solve the resonant nonlinear Schrödinger’s equation with time-dependent coefficients. Opt Quant Electron 49, 316 (2017). https://doi.org/10.1007/s11082-017-1152-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-017-1152-y

Keywords

Mathematics Subject Classification

Search

Navigation