Skip to main content
SpringerLink
Log in

Abundant soliton solutions of the resonant nonlinear Schrödinger equation with time-dependent coefficients by ITEM and He’s semi-inverse method

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

Abstract

The improved \(\tan (\phi /2)\)-expansion method (ITEM) and He’s semi-inverse variational method (HSIVM) are the efficient methods for obtaining exact solutions of nonlinear differential equations. In this paper, the ITEM and HSIVM are applied to construct exact solutions of the resonant nonlinear Schrödinger equation (RNLSE) with time-dependent coefficients for parabolic law nonlinearity. The resonant nonlinear Schrödinger equation plays a very important role in mathematical physics and nonlinear optics. We compare analytical findings with the results of the other analytical schemes describing the ansatz method approach and expansion method are used to carry out the integration. Description of the ITEM is given and the obtained results reveal that the ITEM is a new significant method for exploring nonlinear partial differential models. Moreover, by help of the HSIVM we obtained the bright and dark soliton wave solutions. Finally, by using Matlab, some graphical simulations were drawn to see the behavior of these solutions.

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
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Aghdaei, M.F., Manafian, J.: Optical soliton wave solutions to the resonant Davey–Stewartson system. Opt. Quantum Electron. 48, 1–33 (2016)

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Arnous, A.H., Mahmood, S.A., Younis, M.: Dynamics of optical solitons in dual-core fibers via two integration schemes. Superlattices Microstruct. 106, 156–162 (2017)

    Article  ADS  Google Scholar 

  • 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. At. Nucl. 74, 781–783 (2011)

    Article  Google Scholar 

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

    Article  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 (2012a)

    Article  ADS  Google Scholar 

  • Biswas, A., Fessak, M., Johnson, S., Beatrice, S., Milovic, D., Jovanoski, Z., et al.: Optical soliton perturbation in a non-Kerr law media: traveling wave solution. Opt. Laser Technol. 44(1), 1775–1780 (2012b)

    Article  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(3), 213–217 (2012c)

    Article  ADS  Google Scholar 

  • Biswas, A., Milovic, D., Savescu, M., Mahmood, M.F., Khan, K.R.: Optical soliton perturbation in nanofibers with improved nonlinear Schrödinger equation by semi-inverse variational principle. J. Nonlinear Opt. Phys. Mater. 21(4), 1250054 (2012d)

    Article  ADS  Google Scholar 

  • Cheemaa, N., Mehmood, S.A., Rizvi, S.T.R., Younis, M.: Single and combined optical solitons with third order dispersion in Kerr media. Optik 127, 8203–8207 (2017)

    Article  ADS  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 Fractals 24, 745–757 (2005)

    Article  ADS  MathSciNet  MATH  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 Schödinger equation with resonant nonlinearity. Opt. Int. J. Light Electron Opt. 130, 378–382 (2017)

    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 

  • Gagnon, L.: Exact traveling wave solutions for optical models based on the nonlinear cubic–quintic Schrödinger equation. J. Opt. Soc. Am. A 6, 1477–1483 (1989)

    Article  ADS  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 

  • He, J.H.: Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B. 20, 1141–1199 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Islam, W., Younis, M., Rizvi, S.T.R.: Optical solitons with time fractional nonlinear Schrödinger equation and competing weakly nonlocal nonlinearity. Optik 130, 562–567 (2017)

    Article  ADS  Google Scholar 

  • Kohl, R., Milovic, D., Zerrad, E., Biswas, A.: Optical solitons by He’s variational principle in a non-Kerr law media. J. Infrared Millim. Terahertz Waves 30(5), 526–537 (2009)

    Article  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 evolutionequations by the \(tan(\phi /2)\)-expansion method. Opt. Int. J. Electron Opt. 127, 4222–4245 (2016)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • 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, 1–35 (2015b)

    Article  MATH  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. Quant. Electron. 48, 1–32 (2016b)

    Article  Google Scholar 

  • Manafian, J., Lakestani, M.: Abundant soliton solutions for the Kundu–Eckhaus equation via \(tan(\phi /2)\)-expansion method. Opt. Int. J. Elecron. Opt. 127, 5543–5551 (2016c)

    Article  Google Scholar 

  • Manafian, J., Lakestani, M.: Optical soliton solutions for the Gerdjikov–Ivanov model via \(tan(\phi /2)\)-expansion method. Opt. Int. J. Electron Opt. 127, 9603–9620 (2016d)

    Article  Google Scholar 

  • Manafian, J., Aghdaei, M.F., Zadahmad, M.: Analytic study of sixth-order thin-film equation by \(tan(\phi /2)\)-expansion method. Opt. Quant. Electron 48, 1–16 (2016)

    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., 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 

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

    Article  ADS  MathSciNet  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 

  • Rizva, S.T.R., Salim, S., Ali, K., Younis, M.: New Thirring optical solitons with vector-coupled Schrödinger equations in birefringent fibers. Waves Random Complex Media 27, 359–366 (2017)

    Article  MathSciNet  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 

  • Sassaman, R., Heidari, A., Biswas, A.: Topological and nontopological solitons of nonlinear Klein–Gordon equations by He’s semi-inverse variational principle. J. Franklin Inst. 347, 1148–1157 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Tang, X.Y., Chow, K.W., Rogers, C.: Propagating wave patterns for the ’resonant’ Davey–Stewartson system. Chaos Solitons Fractals 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 

  • Younis, M.: Optical solitons in \((n+1)\) dimensions with Kerr and power law nonlinearities. Mod. Phys. Lett. B 31, 1750186 (2017). doi:10.1142/S021798491750186X

    Article  ADS  MathSciNet  Google Scholar 

  • Younis, M., ur Rehman, H., Rizvi, S.T.R., Mahmood, S.A.: Dark and singular optical solitons perturbation with fractional temporal evolution. Superlattices Microstruct. 104, 525–531 (2017a)

    Article  ADS  Google Scholar 

  • Younis, M., Younas, U., ur Rehman, S., Bilal, M., Waheed, A.: Optical bright-dark and Gaussian soliton with third order dispersion. Optik 134, 233–238 (2017b)

    Article  ADS  Google Scholar 

  • Zhang, J.: Variational approach to solitary wave solution of the generalized Zakharov equation. Comput. Math. Appl. 54, 1043–1046 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  ADS  MathSciNet  MATH  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

Author information

Authors and Affiliations

  1. Young Researchers and Elite Club, Ilkhchi Branch, Islamic Azad University, Ilkhchi, Iran

    Jalil Manafian

  2. Department of Applied Mathematics, Science and Technology of Iran, Tehran, Iran

    Peyman Bolghar

  3. Department of Engineering, Tonkabon Branch, Islamic Azad University, Tonkabon, Iran

    Atefeh Mohammadalian

Authors
  1. Jalil Manafian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peyman Bolghar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Atefeh Mohammadalian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jalil Manafian.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Manafian, J., Bolghar, P. & Mohammadalian, A. Abundant soliton solutions of the resonant nonlinear Schrödinger equation with time-dependent coefficients by ITEM and He’s semi-inverse method. Opt Quant Electron 49, 322 (2017). https://doi.org/10.1007/s11082-017-1156-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-017-1156-7

Keywords

Mathematics Subject Classification

Search

Navigation