The Solutions of the Space-Time Fractional Cubic Nonlinear Schrödinger Equation by Using the Unified Method

Year 2024, , 78 - 89, 30.06.2024

Tuğba Aydemir

https://doi.org/10.47000/tjmcs.1356268

Abstract

Representing physical processes by introducing fractional derivatives in partial differential equations
provides more realistic and flexible mathematical models. The solutions of nonlinear partial differential equations (NPDEs) can be derived from the solutions of the fractional nonlinear partial differential equations (FNPDEs) when the fractional derivatives go to 1 because FNPDEs are a generalization of NPDEs. Most of the exact solution methods for NPDEs based on the ansatz method can be extended easily to solve FNPDEs. In this study, we employ the unified method to obtain exact solutions in a more general form for the space-time fractional cubic nonlinear Schr¨odinger equation (stFCSE). Compared to other methods, this method not only gives more general solution forms with free parameters for the stFCSE, but also provides many novel solutions including hyperbolic, trigonometric, and rational function solutions. The solutions of the stFCSE approach the solutions of the cubic nonlinear Schr¨odinger equation when the fractional orders go to 1 for time and space. Moreover, three-dimensional graphs of some selected solutions with specific values of the parameters are presented to visualize the behavior and physical structures of the stFCSE.

Keywords

The unified method, the space-time fractional cubic nonlinear Schrödinger equation, fractional differential equation, conformable derivative

Ethical Statement

2.Lanre Akinyemi, USA, Hampton University, [e-mail]

Supporting Institution

3.Hasan Bulut, Türkiye, Fırat Üniversitesi, [e-mail]

Project Number

1.Hadi Rezazadeh, İran, Amol University of Special Modern Technologies ,[e-mail]

Thanks

4.Hacı Mehmet Başkonuş, Türkiye, Harran Üniversitesi, [e-mail]

References

• Akbulut, A., Kumar, D., Conservation laws and optical solutions of the complex modified Korteweg-de Vries equation, Journal of Ocean Engineering and Science.
• Akinyemi, L., Anol, M., Akpan, U., Oluwasegun, K., The optical soliton solutions of generalized coupled nonlinear Schrodinger-Korteweg-de Vries equations, Optical and Quantum Electronics, 53(7)(2021).
• Akinyemi, L., Rezazadeh, H., Yao, S.W., Akbar, M.A., Khater, M.M. et al., Nonlinear dispersion in parabolic law medium and its optical solitons, Results in Physics, 26(2021), Article ID: 104411.
• Akinyemi, L., Houwe, A., Abbagari, S.,Wazwaz, A.M., Alshehri, H.M., Osman, M. S., Effects of the higher-order dispersion on solitary waves and modulation instability in a monomode fiber, Optik 288(2023).
• Arafat, S.M.Y., Islam, S.M.R., Bashar, M.H., Influence of the free parameters and obtained Wave solutions from CBS equation, International Journal of Applied and Computational Mathematics, 8(2022).
• Bekhouche, F., Alquran, M., Komashynska, I., Explicit rational solutions for time-space fractional nonlinear equation describing the propagation of bidirectional Waves in low-pass electrical lines, Romanian Journal of Physics, 66(7-8)(2021), 1–14.
• Bekhouche, F., Komashynska, I., Traveling wave solutions for the space-time fractional (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff equation via two different methods, International Journal of Mathematics and Computer Science, 16(4)(2021), 1729—1744.
• Bilal, M., Ren, J., Younas, U., Stability analysis and optical soliton solutions to the nonlinear Schr¨odinger model with efficient computational techniques, Optical and Quantum Electronics, 53(7)(2021).
• Bilal, M., Ren, J., Inc, M., Almohsen, B., Akinyemi, L., Dynamics of diverse wave propagation to integrable Kraenkel-Manna-Merle system under zero damping effect in ferrites materials, Optical and Quantum Electronics, 55(7)(2023).
• Biswas, A., Zhou, Q., Ullah, M.Z., Triki, H., Moshokoa, S.P. et al., Optical soliton perturbation with anti-cubic nonlinearity by semi-inverse variational principle, Optik-International Journal for Light and Electron Optics, 143(2017), 131–134.
• Biswas, A., Ekici, M.A., Sonmezoglu, A., Zhou, Q., Moshokoa, S.P. et al., Optical soliton perturbation with full nonlinearity for Kundu- Eckhaus equation by extended trial function scheme, Optik-Int. J. Light and Electron Optics, 160(2018), 17–23.
• Bulut, H., Sulaiman, T.A., Demirdag, B., Dynamics of soliton solutions in the chiral nonlinear Schr¨odinger equations, Nonlinear Dynamics, 91(3)(2018), 1985–1991.
• Bulut, H., Yel, G., Baskonus, H.M., An application of improved Bernoulli sub-equation function method to the nonlinear time-fractional Burgers equation, Turkish Journal of Mathematics and Computer Science, 5(2016), 1–7.
• Darvishi, M.T., Najafi, M., Seadawy, A.R., Dispersive bright, dark and singular optical soliton solutions in conformable fractional optical fiber Schrodinger models and its applications, Optical and Quantum Electronics, 50(4)(2018).
• Demiray, S.T., Kastal, S., New exact solutions of (3+1)-dimensional modified Quantum Zakharov-Kuznetsov equation, Turkish Journal of Mathematics and Computer Science, 11(2019), 56–59.
• Eslami, M., Neirameh, A., New exact solutions for higher order nonlinear Schr¨odinger equation in optical fibers, Optical and Quantum Electronics, 50(1)(2018).
• Gundogdu, H., Gozukizil, O.F., Cubic nonlinear fractional Schr¨odinger equation with conformable derivative and its new travelling Wave solutions, Journal of Applied Mathematics and Computational Mechanics, 20(2)(2021), 29–41.
• Islam, S.M.R., Bashar, M.H., Arafat, S.M.Y.,Wang, H., Roshid, M.M., Effect of the free parameters on the Biswas-Arshed model with a unified technique, Chinese Journal of Physics, 77(2022), 2501–2519.
• Islam, S.M.R., Kumar, D., Donfack, E.F., Inc, M., Impact of nonlinearity and wave dispersion parameters on the soliton pulses of the (2+1)- dimensional Kundu-Mukherjee-Naskar equation, Revista Mexicana de Fisica, 68(6)(2022).
• Khalil, R., Al-Horani, M., Yousef, A., Sababheh, M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264(2014), 65–70.
• Kudryashov, N.A., Optical solitons of the resonant nonlinear Schr¨odinger equation with arbitrary index, Optik, 235(2021).
• Lan, Z.Z., Guo, B.L., Nonlinear waves behaviors for a coupled generalized nonlinear Schr¨odinger–Boussinesq system in a homogeneous magnetized plasma, Nonlinear Dynamics, 100(4)(2020), 3771-–3784.
• Mirzazadeh, M., Akbulut, A., Taşcan, F., Akinyemi, L., A novel integration approach to study the perturbed Biswas-Milovic equation with Kudryashov’s law of refractive index, Optik, 252(2022).
• Pandir, Y., Ulusoy, H., Solutions of nonlinear partial differential equations using generalized hyperbolic functions, Turkish Journal of Mathematics and Computer Science, 1(2016), 38–46.
• Pandir, Y., Agir, T., New exact solutions of cubic nonlinear Schr¨odinger equation by using extended trial equation method, Afyon Kocatepe University Journal of Science and Engineering, 20(2020), 582–588.
• Salim, B.J., Jasim, O.A., Ali, Z.Y., Numerical solution of Drinfeld-Sokolov-Wilso system by using modified adomian decomposition method, Indonesian Journal of Electrical Engineering and Computer Science, 23(1)(2021), 590-599.
• Seadawy, A.R., Tariq, K.U., On some novel solitons to the generalized (1+ 1)-dimensional unstable space–time fractional nonlinear Schrodinger model emerging in the optical fibers, Optical and Quantum Electronics, 53(2021), 1–16.
• Senol, M., Akinyemi, L., Nkansah, H., Adel, W., New solutions for four novel generalized nonlinear fractional fifth-order equations, Journal of Ocean Engineering and Science, 9(1)(2022), 59–65.
• Senol, M., Gencyigit, M., Ntiamoah, D., Akinyemi, L., New (3+ )-dimensional conformable KdV equation and its analytical and numerical solutions, International Journal of Modern Physics B, (2023).
• Sulaiman, T.A., Akturk, T., Bulut H., Baskonus, H.M., Investigation of various soliton solutions to the Heisenberg ferromagnetic spin chain equation, Journal of Electromagnetic Waves and Applications, 32(9)(2017), 1–13.
• Wazwaz, A.M., A study on linear and nonlinear Schr¨odinger equations by the variational iteration method, Chaos, Solitons and Fractals, 37(4)(2008), 1136–1142.
• Zhang, W.M., Solitary solutions and singular periodic solutions of the Drinfeld-Sokolov-Wilson equation by variational approach, Applied Mathematical Sciences, 5(38)(2011), 1887–1894.