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The exact solutions of conformable time-fractional modified nonlinear Schrödinger equation by first integral method and functional variable method

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Using the first integral method (FIM) and the functional variable method (FVM), firstly we find the analytical solutions of conformable time-fractional modified nonlinear Schrödinger equation (CTFMNLSE) and finally, we present numerical results in tables and charts.

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  • Arshed, S., Biswas, A., Alzahrani, A.K., Belic, M.R.: Solitons in nonlinear directional couplers with optical metamaterials by first integral method. Optik 218, 1–16 (2020)

    Article  Google Scholar 

  • Chen, S.-B., Rashid, S., Noor, M.A., Hammouch, Z., Chu, Y.-M.: New fractional approaches for n-polynomial P -convexity with applications in special function theory. Adv. Difference Equ. 1(543), 1–31 (2020)

    MathSciNet  Google Scholar 

  • Chen, S.-B., Rashid, S., Noor, M.A., Ashraf, R., Chu, Y.-M.: A new approach on fractional calculus and probability density function. AIMS Math. 5(6), 1–7 (2020)

    MathSciNet  Google Scholar 

  • Chu, Y.-M., Nazir, U., Sohail, M., Selim, M.M., Lee, J.-R.: Enhancement in thermal energy and solute particles using hybrid nanoparticles by engaging activation energy and chemical reaction over a parabolic surface via finite element approach. Fractal Fract. 5(3), 1–17 (2021)

    Article  Google Scholar 

  • Darvishi, M.T., Najafi, M., Wazwaz, A.M.: Some optical soliton solutions of space-time conformable fractional Schrödinger-type models. Phys. Scr. 96(6), 1–8 (2021)

    Article  Google Scholar 

  • Duran, S.: Solitary wave solutions of the coupled Konno-Oono equation by using the functional variable method and the two variables (G’/G, 1/G)-expansion method. Adıyaman Univ. J. Sci. 10(2), 585–594 (2020)

    Google Scholar 

  • Eslami, M., Rezazadeh, H., Rezazadeh, M., Mosavi, S.S.: Exact solutions to the space-time fractional Schrödinger-Hirota equation and the space-time modified KDV-Zakharov-Kuznetsov equation. Opt. Quant. Electron. 49(8), 1–15 (2017)

    Article  Google Scholar 

  • Ghosh, A., Maitra, S.: The first integral method and some nonlinear models. Comput. Appl. Math. 40(3), 1–16 (2021)

    Article  MathSciNet  Google Scholar 

  • Jahanshahi, H., Orozco-López, O., Munoz-Pacheco, J.M., Alotaibi, N.D., Volos, C., Wang, Z., Chu, Y.M.: Simulation and experimental validation of a non-equilibrium chaotic system. Chaos, Solitons Fractals 143, 1–11 (2021)

    MathSciNet  Google Scholar 

  • Karthikeyan, K., Karthikeyan, P., Baskonus, H.M., Venkatachalam, K., Chu, Y.-M.: Almost sectorial operators on \(\psi \)-Hilfer derivative fractional impulsive integro-differential equations. Math. Methods Appl. Sci. 1, 1–11 (2021)

    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)

  • Rashid, S., Sultana, S., Karaca, Y., Khalid, A., Chu, Y.-M.: Some further extensions considering discrete proportional fractional operators. Fractals 30(1), 1–12 (2022)

    Article  Google Scholar 

  • Rezazadeh, H., Mirhosseini-Alizamini, S.M., Eslami, M., Rezazadeh, M., Mirzazadeh, M., Abbagari, S.: New optical solitons of nonlinear conformable fractional Schrödinger-Hirota equation. Optik 172, 545–553 (2018)

    Article  ADS  Google Scholar 

  • Rezazadeh, H., Vahidi, J., Zafar, A., Bekir, A.: The functional variable method to find new exact solutions of the nonlinear evolution equations with dual-power-law nonlinearity. Int. J. Nonlinear Sci. Numer. Simul. 21(3–4), 249–257 (2020)

    Article  MathSciNet  Google Scholar 

  • Shen, Z.-H., Chu, Y.-M., Khan, M.A., Muhammad, S., AlHartomy, O.A., Higazy, M.: Mathematical modeling and optimal control of the COVID-19 dynamics. Results Phy. 31, 1–9 (2021)

    Google Scholar 

  • Wang, M.-K., Hong, M.-Y., Xu, Y.-F., Shen, Z.-H., Chu, Y.-M.: Inequalities for generalized trigonometric and hyperbolic functions with one parameter. J. Math. Inequal. 1, 1–12 (2020)

    MathSciNet  MATH  Google Scholar 

  • Xu, H.-Z., Qian, W.-M., Chu, Y.-M.: Sharp bounds for the Lemniscatic mean by the one-parameter geometric and quadratic means. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 116(1), 1–15 (2022)

  • Younas, U., Younis, M., Seadawy, A.R., Rizvi, S.T.R., Althobaiti, S., Sayed, S.: Diverse exact solutions for modified nonlinear Schrödinger equation with conformable fractional derivative. Results Phys. 20, 1–10 (2021)

    Article  Google Scholar 

  • Zhou, S.S., Jahanshahi, H., Din, Q., Bekiros, S., Alcaraz, R., Alassafi, M.O., Chu, Y.M.: Discrete-time macroeconomic system: Bifurcation analysis and synchronization using fuzzy-based activation feedback control. Chaos, Solitons & Fractals 142, 1–11 (2021)

    Article  MathSciNet  Google Scholar 

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José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: Cátedras CONACyT para jóvenes investigadores 2014 and SNI-CONACyT.


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Authors and Affiliations



SRA Conceptualization, Methodology, Writing-Original draft, Supervision; RS Conceptualization, Methodology, Writing-Original draft preparation; JV Validation, Formal analysis, Investigation; JFG-A Conceptualization, Methodology, Writing-Original draft preparation, Supervision. All authors read and approved the final manuscript.

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Correspondence to Reza Saadati or J. F. Gómez-Aguilar.

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Aderyani, S.R., Saadati, R., Vahidi, J. et al. The exact solutions of conformable time-fractional modified nonlinear Schrödinger equation by first integral method and functional variable method. Opt Quant Electron 54, 218 (2022).

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