Skip to main content
SpringerLink
Log in

Diverse optical soliton solutions of the fractional coupled (2 + 1)-dimensional nonlinear Schrödinger equations

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

Abstract

Fractional nonlinear models involving the underlying mechanisms of numerous complicated physical phenomena arising in nature of real world have been taken major place of research arena during the couple of years for their significant roles. The study about the nonlinear optical and quantum context connecting to mostly Kerr law media as well as power law, dual-power law, triple-power law, saturable law, logarithm law and polynomial low is increasing at an inconceivable rate. In this exploration, the integrable generalized (2 + 1)-dimensional nonlinear Schrödinger system of equations in the sense of conformable fractional derivative is considered to unravel by means of two innovative schemes namely improved tanh method and rational \(\left( {G^{\prime}/G} \right)\)-expansion method. The advised techniques are employed to seek for appropriate analytic wave solutions after converting the mentioned equation to an ordinary differential equation by introducing a wave variable alteration. The hyperbolic, trigonometric and rational function solutions are successfully gained and put forwarded for graphical representations. The assembled solutions are figured out in 3-D, 2-D and contour formats to illustrate their different views which appeared as kink type, anti-kink type, singular kink type, bell shape, anti-bell shape, singular bell shape, cuspon, peakon, periodic and singular periodic etc. The entire study bears the diversity and novelty of found solutions and applied techniques after making a comparable study with recent work recorded in the literature.

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
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  • Ahmad, H., Khan, T.A., Ahmad, I., Stanimirovic, P.S., Chu, Y.M.: A new analyzing technique for nonlinear time fractional Cauchy reaction-diffusion model equations. Res. Phys. 19, 103462 (2020a)

    Google Scholar 

  • Ahmad, H., Seadawy, A.R., Khan, T.A., Thounthong, P.: Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations. J. Taibah Univ. Sci. 14(1), 346–358 (2020b)

    Article  Google Scholar 

  • Akbar, M.A., Ali, N.H.M., Islam, M.T.: Multiple closed form solutions to some fractional order nonlinear evolution equations in physics and plasma physics. AIMS Math. 4(3), 397–411 (2019)

    Article  MathSciNet  Google Scholar 

  • Akbulut, A., Kaplan, M., Bekir, A.: Auxiliary equation method for fractional differential equations with modified Riemann–Liouville derivative. Int. J. Nonlinear Sci. Numer. Simul. 17(7–8), 413–420 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Akinyemi, L., Senol, M., Rezazadeh, H., Ahmad, H., Wang, H.: Abundant optical soliton solutions for an integrable (2+1)-dimensional nonlinear conformable Schrödinger system. Res. Phys. 25, 104177 (2021)

    Google Scholar 

  • Bulut, H., Baskonus, H.M., Pandir, Y.: The modified trial equation method for fractional wave equation and time fractional generalized Burgers’ equation. Abstr. Appl. Anal. 636802, 1–8 (2013)

    MathSciNet  MATH  Google Scholar 

  • Cheema, N., Younis, M.: New and more general traveling wave solutions for nonlinear Schrödinger equation. Waves Random Complex Media 26(1), 30–41 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Chen, S.J., Ma, W.X., Lu, X.: Backlund transformation, exact solutions and interaction behaviour of the (3+1)-dimensional Hirota–Satsuma–Ito-like equation. Commun. Nonlinear Sci. Numer. Simul. 83, 105135 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, S.J., Lu, X., Tang, X.F.: Novel evolutionary behaviors of the mixed solutions to a generalized Burgers equation with variable coefficients. Commun. Nonlinear Sci. Numer. Simul. 95, 105628 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Chowdhury, M.A., Miah, M.M., Ali, H.M.S., Chu, Y.M., Osman, M.S.: An investigation to the nonlinear (2+1)-dimensional soliton equation for discovering explicit and periodic wave solutions. Res. Phys. 23, 104013 (2021b)

    Google Scholar 

  • Durur, H., Yokus, A.: Exact solutions of (2+1)-Ablowitz-Kaup–Newell–Segur equation. Appl. Math. Non. Sci. 1–6 (2020)

    Article  Google Scholar 

  • Durur, H., Yokuş, A., Kaya, D., Ahmad, H.: Modeling of dark solitons for nonlinear longitudinal wave equation in a magneto-electro-elastic circular rod. Sound Vib. 55(3), 241–251 (2021)

  • Gao, G.H., Sun, Z.Z., Zhang, Y.N.: A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions. J. Comput. Phys. 231(7), 2865–2879 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Gao, W., Ismael, H.F., Husien, A.M., Bulut, H., Baskonus, H.M.: Optical soliton solutions of the cubic-quartic nonlinear Schrödinger and resonant nonlinear Schrödinger equation with the parabolic law. Appl. Sci. 10(1), 219 (2020)

    Article  Google Scholar 

  • Gepreel, K.A.: The homotopy perturbation method applied to nonlinear fractional Kadomtsev–Petviashvili–Piskkunov equations. Appl. Math. Lett. 24(8), 1428–1434 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Guner, O., Eser, D.: Exact solutions of the space time fractional symmetric regularized long wave equation using different methods. Adv. Math. Phys. 2014, 456804 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • He, X.J., Lu, X., Li, M.G.: Backlund transformation, Pfaffian, Wronskian and Grammian solutions to the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation. Anal. Math. Phys. 11, 4 (2021)

    Article  MATH  Google Scholar 

  • Hosseini, K., Mirzazadeh, M., Aguilar, J.F.G.: Soliton solutions of the Sasa–Satsuma equation in the monomode optical fibers including the beta-derivatives. Optik 224, 165425 (2020)

    Article  ADS  Google Scholar 

  • Hosseini, K., Salahshour, S., Mirzazadeh, M., Ahmadian, A., Baleanu, D., Khoshrang, A.: The (2+1)-dimensional Heisenberg ferromagnetic spin chain equation: its solitons and Jacobi elliptic function solutions. Eur. Phys. J. Plus 136, 206 (2021a)

    Article  Google Scholar 

  • Hosseini, K., Sadri, K., Mirzazadeh, M., Chu, Y.M., Ahmadian, A., Pansera, B.A., Salahshour, S.: A high-order nonlinear Schrödinger equation with the weak non-local nonlinearity and its optical solitons. Res. Phys. 23, 104035 (2021)

    Google Scholar 

  • Hosseini, K., Matinfar, M., Mirzazadeh, M.: Soliton solutions of high-order nonlinear Schrödinger equations with different laws of nonlinearities. Reg. Chaotic Dyn. 26(1), 105–112 (2021c)

    Article  ADS  MATH  Google Scholar 

  • Hosseini, K., Sadri, K., Mirzazadeh, M., Salahshour, S.: An integrable (2+1)-dimensional nonlinear Schrödinger system and its optical soliton solutions. Optik 229, 1–6 (2021d)

    Article  Google Scholar 

  • Hu, Y., Luo, Y., Lu, Z.: Analytical solution of the linear fractional differential equation by Adomian decomposition method. J. Comput. Appl. Math. 215(1), 220–229 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Islam, M.T., Akter, M.A.: Exact analytic wave solutions to some nonlinear fractional differential equations for the shallow water wave arise in physics and engineering. J. Res. Eng. Appl. Sci. 6(1), 11–18 (2021a)

    Google Scholar 

  • Islam, M.T., Akter, M.A.: Distinct solutions of nonlinear space-time fractional evolution equations appearing in mathematical physics via a new technique. Partial Differ. Equ. Appl. Math. 3, 100031 (2021)

    Article  Google Scholar 

  • Islam, M.T., Akbar, M.A., Azad, A.K.: A rational -expansion method and its application to the modified KdV-Burgers equation and the (2+1)-dimensional Boussinesq equation. Nonlinear Stud. 6(4), 1–11 (2015)

    MathSciNet  MATH  Google Scholar 

  • Ismael, H.F., Bulut, H., Baskonus, H.M., Gao, W.: Dynamical behaviors to the coupled Schrödinger–Boussinesq system with the beta derivative. AIMS Math. 6(7), 7909–7928 (2021)

    Article  MathSciNet  Google Scholar 

  • Kaplan, M., Unsal, O., Bekir, A.: Exact solutions of nonlinear Schrödinger equation by using symbolic computation. Math. Methods Appl. Sci. 39(8), 2093–2099 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Khalil, R., Horani, M.A., Yousef, A., Sababheh, M.A.M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  • Kumar, S., Kumar, A., Wazwaz, A.M.: New exact solitary wave solutions of the strain wave equation in microstructured solids via the generalized exponential rational function method. Eur. Phys. J. Plus 135(11), 1–17 (2020)

    Article  Google Scholar 

  • Li, C., Guo, Q., Zhao, M.: On the solutions of (2+1)-dimensional time-fractional Schrödinger equation. Appl. Math. Lett. 94, 238–243 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, W., Yu, W., Yang, C., Liu, M., Zhang, Y., Lie, M.: Analytic solutions for the generalized complex Ginzburg–Landau equation in fiber lasers. Nonlinear Dyn. 89(4), 2933–2939 (2017)

    Article  MathSciNet  Google Scholar 

  • Lu, X., Chen, S.J.: Interaction solutions to nonlinear partial differential equations via Hirota bilinear forms: one-lump-multi-stripe and one-lump-multi-soliton types. Nonlinear Dyn. 103(1), 947–977 (2021)

    Article  Google Scholar 

  • Lu, X., Ma, W.X.: Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation. Nonlinear Dyn. 85(2), 1217–1222 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Lu, D., Seadawy, A.R., Arshad, M.: Applications of extended simple equation method on unstable nonlinear Schrödinger equations. Optik 140, 136–144 (2017)

    Article  ADS  Google Scholar 

  • Lu, X., Hua, Y.F., Chen, S.J., Tang, X.F.: Integrability characteristics of a novel (2+1)-dimensional nonlinear model: Painleve analysis, soliton solutions, Backlund transformation, Lax pair and infinitely many conservation laws. Commun. Nonlinear Sci. Numer. Simul. 95, 105612 (2021)

    Article  MATH  Google Scholar 

  • Ma, W.X., Lee, J.H.: A transformed rational function method and exact solutions to the (3+1)-dimensional Jimbo–Miwa equation. Chaos Solitons Fractals 42(3), 1356–1363 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Mahalingam, A., Rajan, M.S.M.: Influence of generalized external potentials on nonlinear tunneling of nonautonomous solitons: soliton management. Opt. Fiber Technol. 25, 44–50 (2015)

    Article  ADS  Google Scholar 

  • Mahalingam, A., Porsezian, K., Rajan, M.S.M., Uthayakumar, A.: Propagation of dispersion-nonlinearity-managed solitons in an inhomogeneous erbium-doped fiber system. J. Phys. A Math. Theor. 42(16), 165101 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Malik, S., Kumar, S., Biswas, A., Ekici, M., Dakova, A., Alzahrani, A.K., Belic, M.R.: Optical solitons and bifurcation analysis in fiber Bragg gratings with Lie symmetry and Kudryashov’s approach. Nonlinear Dyn. 105, 735–751 (2021)

    Article  Google Scholar 

  • Martinez, H.Y., Aguilar, J.F.G., Atangana, A.: First integral method for nonlinear differential equations with conformable derivative. Math. Model. Nat. Phenom. 13(1), 14 (2018)

    Article  MATH  Google Scholar 

  • Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  • Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  • Pandir, Y., Duzgun, H.H.: New exact solutions of the space-time fractional cubic Schrödinger equation using the new type F-expansion method. Waves Random Complex Media 29(3), 425–434 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  • Prakash, S.A., Malathi, V., Rajan, M.S.M., Loomba, S.: Controllable pulse width of bright similaritons in a tapered graded index diffraction decreasing waveguide. Chaos 26(3), 033115 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Radha, R., Lakshmanan, M.: Singularity structure analysis and bilinear form of a (2+1)-dimensional nonlinear Schrödinger (NLS) equation. Inverse Probl. 10(4), 29–32 (1994)

    Article  ADS  MATH  Google Scholar 

  • Rajan, M.S.M.: Dynamics of optical soliton in a tapered erbium-doped fiber under periodic distributed amplification system. Nonlinear Dyn. 85(1), 599–606 (2016)

    Article  Google Scholar 

  • Rajan, M.S.M., Mahalingam, A.: Nonautonomous solitons in modified inhomogeneous Hirota equation: soliton control and soliton interaction. Nonlinear Dyn. 79(4), 2469–2484 (2015)

    Article  MathSciNet  Google Scholar 

  • Rajan, M.S.M., Hakkim, J., Mahalingam, A., Uthayakumar, A.: Dispersion management and cascade compression of femtosecond nonautonomous soliton in birefringent fiber. Eur. Phys. J. D 67(7), 1–8 (2013)

    Google Scholar 

  • Rizvi, S.T.R., Ali, K., Bashir, S., Younis, M., Ashraf, R., Ahmad, M.O.: Exact solution of (2+1)-dimensional fractional Schrödinger equation. Superlattices Microstruct. 107, 234–239 (2017)

    Article  ADS  Google Scholar 

  • Rizvi, S.T.R., Seadawy, A.R., Younis, M., Iqbal, S., Althobaiti, S., El-Shehawi, A.M.: Various optical soliton for weak fractional nonlinear Schrödinger equation with parabolic law. Res. Phys. 23, 103998 (2021)

    Google Scholar 

  • Salam, E.A.-B.A., Yousif, E., El-Aasser, M.: Analytical solution of the space-time fractional nonlinear Schrödinger equation. Rep. Math. Phys. 77(1), 19–34 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R., Cheemaa, N., Biswas, A.: Optical dromions and domain walls in (2+1)-dimensional coupled system. Optik 227, 1–16 (2021)

    Article  Google Scholar 

  • Vijayalekshmi, S., Rajan, M.S.M., Mahalingam, A., Uthayakumar, A.: Hidden possibilities in soliton switching through tunneling in erbium doped birefringence fiber with higher order effects. J. Mod. Opt. 62(4), 278–287 (2015)

    Article  ADS  Google Scholar 

  • Wazwaz, A.M.: Partial Differential Equations: Method and Applications. Taylor and Francis, London (2002)

    MATH  Google Scholar 

  • Wazwaz, A.M., Kaur, L.: Optical solitons for nonlinear Schrödinger (NLS) equation in normal dispersive regimes. Optik Int. J. Light Elect. Opt. 184, 428–435 (2019)

    Article  Google Scholar 

  • Xia, J.W., Zhao, Y.W., Lu, X.: Predictability, fast calculation and simulation for the interaction solutions to the cylindrical Kadomtsev–Petviashvili equation. Commun. Nonlinear Sci. Numer. Simul. 90, 105260 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Xu, T., Chen, Y.: Localised nonlinear waves in the three-component coupled Hirota equations. Z. Naturforschung A 72(11), 1053–1070 (2017)

    Article  ADS  Google Scholar 

  • Xu, H.N., Ruan, W.Y., Zhang, Y., Lu, X.: Multi-exponential wave solutions to two extended Jimbo–Miwa equations and the resonance behavior. Appl. Math. Lett. 99, 105976 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Yin, Y.H., Chen, S.J., Lu, X.: Localized characteristics of lump and interaction solutions to two extended Jimbo–Miwa equations. Chin. Phys. B 29(12), 120502 (2020)

    Article  ADS  Google Scholar 

  • Younis, M., Cheemaa, N., Mehmood, S.A., Rizvi, S.T.R., Bekir, A.: A variety of exact solutions to (2+1)-dimensional Schrödinger equation. Waves Random Complex Media 30(3), 490–499 (2018)

    Article  ADS  Google Scholar 

  • Zayed, E.M.E., Nofal, T.A., Gepreel, K.A., Shohib, R.M.A., Alngar, M.E.M.: Cubic-quartic optical soliton solutions in fiber Bragg gratings with Lakshmanan–Porsezian–Daniel model by two integration schemes. Opt. Quantum Electron. 53(5), 1–17 (2021)

    Article  Google Scholar 

  • Zhang, L.L., Wang, X.M.: Bright-dark soliton dynamics and interaction for the variable coefficient three-coupled nonlinear Schrödinger equations. Mod. Phys. Lett. B 34(05), 2050064 (2020)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hajee Mohammad Danesh Science and Technology University, Dinajpur, Bangladesh

    Md. Tarikul Islam

  2. Department of Applied Mathematics, University of Rajshahi, Rajshahi, Bangladesh

    Md. Ali Akbar

  3. Information Technology Application and Research Center, Istanbul Ticaret University, 34445, Istanbul, Turkey

    Hijaz Ahmad

  4. Department of Mathematics, Faculty of Humanities and Social Sciences, Istanbul Ticaret University, 34445, Istanbul, Turkey

    Hijaz Ahmad

Authors
  1. Md. Tarikul Islam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Md. Ali Akbar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hijaz Ahmad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hijaz Ahmad.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Islam, M.T., Akbar, M.A. & Ahmad, H. Diverse optical soliton solutions of the fractional coupled (2 + 1)-dimensional nonlinear Schrödinger equations. Opt Quant Electron 54, 129 (2022). https://doi.org/10.1007/s11082-021-03472-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-021-03472-z

Keywords

Mathematics Subject Classification

Search

Navigation