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Optical soliton solutions to the time-fractional Kundu–Eckhaus equation through the \((G^{\prime}/G,1/G)\)-expansion technique

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Abstract

The present research focuses on fractional nonlinear evolution equations and their optical soliton solutions, which have become the inquisitive context to study their significant attributes in understanding natural kernels ascending in the field of science and technology. This article has been dedicated to searching out the analytical soliton solution of an important fractional nonlinear evolution equation, named the time-fractional Kundu–Eckhaus equation in the sense of beta fractional derivative through the (\(G^{\prime}/G, 1/G\))-expansion approach. This equation was originated to search out the transmission of data through the optical fiber. By exerting the stated method, abundant novel soliton solutions, like kink soliton, compacton, periodic soliton, singular periodic, singular bell-shaped soliton, and others have been established. In accordance with the trail solutions generated in this method, the solutions contain arbitrary parameters and hyperbolic, rational, and trigonometric functions. Soliton solutions are extracted from analytical solutions for apposite values of the parameters. Contour, three- and two-dimensional graphs are plotted to demonstrate the physical structure and characteristics of the attained solitons. The obtained results imply that the concerned method can be used to attain diverse, improved, useful, and compatible solutions for other significant fractional nonlinear evolution equations.

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References

  • Acan, O., Baleanu, D.: A new numerical technique for solving fractional partial differential equations. ar**v Preprint ar**v:1704.02575 (2017)

  • Ahmed, H.M., Rabie, W.B., Ragusa, M.A.: Optical solitons and other solutions to Kaup–Newell equation with Jacobi elliptic function expansion method. Anal. Math. Phys. 11(1), 1–16 (2021)

    MathSciNet  MATH  Google Scholar 

  • Ahmed, M.S., Zaghrout, A.A.S., Ahmed, H.M.: Travelling wave solutions for the doubly dispersive equation using improved modified tanh-function method. Alex. Eng. J. 61(10), 7987–7994 (2022a)

    Google Scholar 

  • Ahmed, H., Khan, T.A., Stanimirovic, P.S., Shatanawi, W., Botmart, T.: New approach on conventional solutions to nonlinear partial differential equations describing physical phenomena. Results Phys. 41, 1–13 (2022b)

    Google Scholar 

  • Akbar, M.A., Ali, N.H., Zayed, E.: A generalized and improved (G′⁄G,1/G)-expansion method for nonlinear evolution equations. Math. Probl. Eng. 2012, 1–22 (2012)

    MathSciNet  MATH  Google Scholar 

  • Akbulut, A., Kaplan, M.: Auxiliary equation method for time-fractional differential equations with conformable derivative. Comput. Math. Appl. 75(3), 876–882 (2018)

    MathSciNet  MATH  Google Scholar 

  • Akram, G., Sadaf, M., Khan, M.A.U.: Abundant optical solitons for Lakshmanan–Porsezian–Daniel model by the modified auxiliary equation method. Optik 252, 1–10 (2022)

    Google Scholar 

  • Alam, L.M.B., Jiang, X., Maun, A.A.: Exact and explicit travelling wave solution to the time-fractional phi-four and (2+1) dimensional CBS equation using the modified extended tanh-function method in mathematical physics. Partial Differ. Equ. Appl. Math. 4, 1–11 (2021)

    Google Scholar 

  • Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 44, 460–481 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  • Alquran, M.: Solitons and periodic solutions to nonlinear partial differential equations by the sine–cosine method. Appl. Math. Inf. Sci. 6(1), 85–88 (2012)

    MathSciNet  MATH  Google Scholar 

  • Aniqa, A., Ahmad, J.: Soliton solution of fractional Sharma–Tasso–Olever equation via an efficient (GG’)-expansion method. Ain Shams Eng. J. 13, 1–23 (2022)

    Google Scholar 

  • Arshed, S., Biswas, A., Majid, F.B., Zhou, Q., Moshokoa, S.P., Belic, M.: Optical solitons in birefringent fibers for Lakshmanan–Porsezian–Daniel model using exp (-ϕ(ξ))-expansion method. Optik 170, 555–560 (2018)

    ADS  Google Scholar 

  • Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763–769 (2016)

    Google Scholar 

  • Baskonus, H.M., Bulut, H., Sulaiman, T.S.: New complex hyperbolic structure to the Lonngren-wave equation by using sine-Gordon expansion method. Appl. Math. Nonlinear Sci. 4(1), 129–138 (2019)

    MathSciNet  Google Scholar 

  • Bekir, A., Zahran, E.H.M.: Bright and dark soliton solutions for the complex Kundu–Eckhaus equation. Optik 223, 1–18 (2020)

    Google Scholar 

  • Bian, C., Pang, J., **, L., Ying, X.: Solving two fifth order strong nonlinear evolution equations by using the (GG’)-expansion method. Commun. Nonlinear Sci. Numer. Simul. 15(9), 2337–2343 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  • Bilal, M., Ahmad, J.: A variety of exact optical soliton solutions to the generalized (2+1)-dimensional dynamical conformable fractional Schrödinger model. Results Phys. 33, 1–12 (2022)

    Google Scholar 

  • Bilal, M., Rehaman, S.U., Ahmad, J.: Dispersive solitary wave solutions for the dynamical soliton model by three versatile analytical mathematical methods. Eur. Phys. J. plus 137, 674 (2022)

    Google Scholar 

  • Biswas, A., Yidirim, Y., Yasar, E., Zhou, Q., Moshokoa, S.P., Belic, M.: Optical solitons for Lakshmanan–Porsezian–Daniel model by modified simple equation method. Optik 160, 24–32 (2018)

    ADS  Google Scholar 

  • Biswas, A., Ekici, M., Kara, A.S.A.H.: Optical solitons and conservation law in birefringent fibers with Kundu–Eckhaus equation by extended trial function method. Optik 179, 471–478 (2019)

    ADS  Google Scholar 

  • Demiray, S.T., Bayrakci, U.: Soliton solutions for the space-time fractional Heisenberg ferromagnetic spin chain equation by generalized Kudryashov method and modified exp (-Ω(η))-expansion function method. Rev. Mex. Fis. 67(3), 393–402 (2020)

    Google Scholar 

  • Duran, S.: Extractions of travelling wave solutions of (2+1)-dimensional Boiti–Leon–Pempinelli system via (G’/G,1/G)-expansion method. Opt. Quant. Electron. 53(6), 1–12 (2021)

    ADS  Google Scholar 

  • Eckhaus, W.: The long-time behaviour for perturbed wave-equations and related problems. Trends Appl. Pure Math. Mech. 249, 168–194 (1986)

    ADS  MathSciNet  MATH  Google Scholar 

  • Ekici, M., Mirzazadeh, M., Sonmezoglu, A., Zhou, Q., Moshokoa, S.P., Biswas, A., Belic, M.: Dark and singular optical solitons with Kundu–Eckhaus equation by extended trial equation method and extended G’/G-expansion scheme. Optik 127(22), 10490–10497 (2016)

    ADS  Google Scholar 

  • Elboree, M.K.: Soliton molecules and exp (-ϕ(ζ))-expansion method for the new (3+1)-dimensional Kadomtsev–Petviashvii (KP) equation. Chin. J. Phys. 71, 623–633 (2021)

    MathSciNet  Google Scholar 

  • Elwakil, S.A., El-Labany, S.K., Zahran, M.A., Sabry, R.: Modified extended tanh-function method for solving nonlinear partial differential equations. Phys. Lett. A 299(2–3), 189–188 (2002)

    ADS  MathSciNet  MATH  Google Scholar 

  • Fan, E.: Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 277(4–5), 212–218 (2000)

    ADS  MathSciNet  MATH  Google Scholar 

  • Ghanbari, B., Inc, M.: A new generalized exponential rational function method to find exact special solutions for the resonance nonlinear Schrödinger equation. Eur. Phys. J. Plus 133(4), 1–18 (2018)

    Google Scholar 

  • Gou, S., Zhou, Y.: The extended (G’/G)-expansion method and its application to the Whitham–Broer–Kaup-like equation and coupled Hirota–Satsuma KdV equations. Appl. Math. Comput. 215(9), 3214–3221 (2010)

    MathSciNet  MATH  Google Scholar 

  • Jawad, A.J.M., Petkovic, M.D., Biswas, A.: Modified simple equation method for nonlinear evolution equations. Appl. Math. Comput. 217(2), 869–877 (2010)

    MathSciNet  MATH  Google Scholar 

  • Kadkhoda, N., Jafari, H.: An analytical approach to obtain exact solutions of some space–time conformable fractional differential equations. Adv. Differ. Equ. 428, 1–10 (2019)

    MathSciNet  MATH  Google Scholar 

  • Kaplan, M.: Optical solutions of the Kundu–Eckhaus equation via two different methods. Adlyaman Univ. J. Sci. 11(1), 126–135 (2021)

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Khater, M., Akbar, M.A., Akinyemi, L., Jhangeer, A., Rezazadeh, H.: Bifurcation of new optical solitary wave solutions for the nonlinear long-short wave interaction system via two improved models of (G’/G)-expansion method. Opt. Quant. Electron. 53(9), 1–16 (2021)

    Google Scholar 

  • Kumar, V.S., Razazadeh, H., Eslami, M., Izadi, F.: Jacobi elliptic function expansion method for solving KdV equation with conformable derivative and dual-power law nonlinearity. Int. J. Appl. Comput. Math. 5(5), 1–10 (2019)

    MathSciNet  Google Scholar 

  • Kundu, A.: Landau–Lifshitz and higher order nonlinear systems gauge generated from nonlinear Schrödinger-type equations. J. Math. Phys. 25(12), 3433–3438 (1984)

    ADS  MathSciNet  Google Scholar 

  • Kundu, P.R., Fahim, M.R.A., Islam, M.E., Akbar, M.A.: The sine-Gordon expansion method for higher-dimensional NLEEs and parametric analysis. Heliyon 7(3), e06459 (2021)

    Google Scholar 

  • Li, L.X., Li, E.Q., Wang, M.L.: The (G’/G,1/G)-expansion method and its applications to travelling wave solutions of the Zakharov equations. Appl. Math. 25(4), 454–462 (2010)

    MathSciNet  MATH  Google Scholar 

  • Liang, X., Cai, Z., Wang, M., Zhao, X., Chen, H., Li, C.: Chaotic oppositional sine–cosine method for solving global optimization problems. Eng. Comput. 38, 1223–12139 (2022)

    Google Scholar 

  • Liu, J.G., Zeng, Z.F.: Solving (3+1)-dimensional generalized BKP equations by the improved (G’/G)-expansion method. Indian J. Pure Appl. Phys. (IJPAP) 53(11), 713–717 (2015)

    Google Scholar 

  • Liu, S., Fu, Z., Liu, S., Zhao, Q.: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 289(1–2), 69–74 (2001)

    ADS  MathSciNet  MATH  Google Scholar 

  • Mahak, N., Akram, G.: The modified auxiliary equation method to investigate solutions of the perturbed nonlinear Schrodinger equation with Kerr law nonlinearity. Optik 207, 1–17 (2020)

    Google Scholar 

  • Mamun, A.A., Ananna, S.N., An, T., Asaduzzaman, M., Rana, M.S.: Sine-Gordon expansion method to construct the solitary wave solutions of a family of 3D fractional WBBM equations. Results Phys. 40, 1–9 (2022)

    Google Scholar 

  • Manafian, J., Lakestani, M.: Abundant soliton solutions for the Kundu–Eckhaus equation via tanϕξ-expansion method. Optik 127(14), 5543–5551 (2016)

    ADS  Google Scholar 

  • Miah, M.M., Ali, H.M.S., Akbar, M.A., Wazwaz, A.M.: Some applications of the (G’/G,1/G)-expansion method to find new exact solutions of NLEEs. Eur. Phys. J. plus 132(6), 1–15 (2017)

    Google Scholar 

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

    MATH  Google Scholar 

  • Ni, W.G., Dai, C.Q.: Note on some result of differential ansätz based on extended tanh-function method for nonlinear models. Appl. Math. Comput. 270, 434–440 (2015)

    MathSciNet  MATH  Google Scholar 

  • Nofal, T.A.: Simple equation method for nonlinear partial differential equations and its applications. J. Egypt. Math. Soc. 24(2), 204–209 (2016)

    MathSciNet  MATH  Google Scholar 

  • Radhakrishnan, R., Kundu, A., Lakshmana, M.: Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: integrability and soliton interaction in non-kerr media. Phys. Rev. E 60(3), 3314–3323 (1999)

    ADS  Google Scholar 

  • Rahman, M.A.: The exp (-ϕ(η))-expansion method with application in the (1+1)-dimensional classical Boussinesq equations. Results Phys. 4, 150–155 (2014)

    Google Scholar 

  • Rezazadeh, H., Korkmaz, A., Eslami, M., Alizamini, S.M.M.: A large family of optical solutions to Kundu–Eckhaus model by a new auxiliary equation method. Opt. Quant. Electron. 51(3), 1–12 (2019)

    Google Scholar 

  • Smadi, M.A., Arqub, O.A., Hadid, S.: Approximate solutions of nonlinear fractional Kundu–Eckhaus and coupled fractional massive Thirring equations emerging in quantum field theory using conformable residual power series method. Phys. Scr. 95(10), 1–18 (2020)

    Google Scholar 

  • Unal, A.G.: Exact solutions of some complex partial differential equations of fractional order. J. Fract. Calc. Appl. 5, 209–214 (2014)

    MathSciNet  MATH  Google Scholar 

  • Wang, M., Li, X., Zhang, J.: The (G’/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372(4), 417–423 (2008)

    ADS  MathSciNet  MATH  Google Scholar 

  • Wazwaz, A.M.: A sine–cosine method for handling nonlinear wave equations. Math. Comput. Model. 40(5–6), 499–508 (2004)

    MathSciNet  MATH  Google Scholar 

  • Yang, Z., Hon, B.Y.C.: An improved modified tanh-function method. Zeitschrift Für Naturforschung A 61(3–4), 103–115 (2006)

    ADS  MATH  Google Scholar 

  • Zahran, E.H.M., Khater, M.M.A.: Modified tanh-function method and its applications to the Bogoyavlenskii equation. Appl. Math. Model. 40(3), 1769–1775 (2016)

    MathSciNet  MATH  Google Scholar 

  • Zhang, J., Wei, X., Lu, Y.: A generalized (G’/G)-expansion method and its applications. Phys. Lett. A 372(20), 3653–3658 (2008)

    ADS  MathSciNet  MATH  Google Scholar 

  • Zhang, J., Jiang, F., Zhao, X.: An improved (G’/G)-expansion method for solving nonlinear evolution equations. Int. J. Comput. Math. 87(8), 1716–1725 (2010)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors appreciate the insightful comments and suggestions of the anonymous referees to enhance the quality of the article.

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Authors and Affiliations

  1. School of Mathematical Sciences, Universiti Sains Malaysia, George Town, Penang, Malaysia

    M. Ali Akbar & Farah Aini Abdullah

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

    M. Ali Akbar & Mst. Munny Khatun

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  1. M. Ali Akbar
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  2. Farah Aini Abdullah
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  3. Mst. Munny Khatun
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Contributions

MAA: Conceptualization, Methodology, Resources, Writing-original draft, Data curation, Visualization. FAA: Supervision, Software, Project administration, Funding acquisition, Writing-review editing. MMK: Software, Investigation, Formal analysis, Validation, Writing-review editing.

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Correspondence to M. Ali Akbar.

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Akbar, M.A., Abdullah, F.A. & Khatun, M.M. Optical soliton solutions to the time-fractional Kundu–Eckhaus equation through the \((G^{\prime}/G,1/G)\)-expansion technique. Opt Quant Electron 55, 291 (2023). https://doi.org/10.1007/s11082-022-04530-w

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