Abstract
The nonlinear space–time fractional Sasa–Satsuma and Schrödinger–Hirota equations with beta derivative describe optical soliton, photonics, plasmas, neutral scalar masons, and long-surface gravitational waves in the real world. Through the fractional wave transform, the models are converted into a single wave variable equation. In this article, we examine a range of compatible, useful, and typical wave solutions expressed in the forms of hyperbolic, trigonometric, and rational functions uniformly through the (\(Q^{\prime}/Q,1/Q\))-expansion approach. When specific parameter values are set, the generalized wave solutions exhibit a wide range of shapes, including asymptotic, anti-asymptotic, dark-optical, breather, lump-periodic, kink, kink-bell-shaped, homoclinic-breather, bright, dark, and periodic solitons that resemble periodic breathing patterns. We also investigate the effect of the fractional parameter \(\delta\) into the wave profile, revealing a clear correlation between changes in the fractional order derivative \(\delta\) and variation in the soliton’s shape. The results underscore the use of this approach for the exploration of diverse nonlinear fractional systems within the context of beta derivatives. Varying the fractional-order \(\delta\) and maintaining specific fixed parameter values, we depict 3D-surface, 2D-surface, density, and contour plots to visualize some of the derived solutions.
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References
Akram, G., Arshed, S., Sadaf, M.: Soliton solutions of generalized time-fractional Boussinesq-like equation via three techniques. Chaos Solitons Fractals 173, 1–10 (2023a)
Akram, G., Sadaf, M., Zainab, I.: Effect of a new local derivative on space-time fractional nonlinear Schrödinger equation and its stability analysis. Opt. Quant. Electron. 55, 1–18 (2023b)
Akram, G., Arshed, S., Sadaf, M., Maqbool, M.: Comparison of fractional effects for Phi-4 equation using beta and M-truncated derivatives. Opt. Quant. Electron. 55, 1–14 (2023c)
Alain, M., Fernand, M.L., Enyegue, A.N., Bioule, M.: Exact solitary waves for the 2D Sasa–Satsuma equation. Chaos Solitons Fractals 133, 1–9 (2020)
Aly, R.S., Naila, N., Dian-Chen, L.: Optical soliton and elliptic functions solutions of Sasa–Satsuma dynamical equation and its applications. Appl. Math-A J. Chin. Univ. 36, 229–242 (2021)
Arnous, A.H., Mirzazadeh, M., Zhou, Q., Moshokoa, S.P., Biswas, A., Belic, M.: Optical solitons with higher order dispersions in parabolic law medium by trial solution approach. Optik 127, 11306–11310 (2016)
Asghari, Y., Eslami, M., Rezazadeh, H.: Soliton solutions for the time-fractional nonlinear differential-difference equation with conformable derivatives in the ferroelectric materials. Opt. Quant. Electron. 55(4), 1–11 (2023a)
Asghari, Y., Eslami, M., Rezazadeh, H.: Exact solutions to the conformable time-fractional discretized mKdv lattice system using the fractional transformation method. Opt. Quant. Electron. 55(4), 1–10 (2023b)
Asghari, Y., Eslami, M., Rezazadeh, H.: Novel optical solitons for the Ablowitz-Ladik lattice equation with conformable derivatives in the optical fibers. Opt. Quant. Electron. 55(10), 1–12 (2023c)
Asma, M., Othman, W.A.M., Wong, B.R., Biswas, A.: Optical soliton perturbation with quadratic-cubic nonlinearity by Adomian decomposition method. Optik 164, 632–641 (2018)
Baleanu, D., Osman, M.S., Zubair, A., Raza, N., Arqub, O.A., Wen-Xiu, M.A.: Soliton solutions of a nonlinear fractional Sasa–Satsuma equation in monomode optical fibers. Appl. Math. Inform. Sci. 14, 365–374 (2020)
Bao-Feng, F., Changyan, S., Guangxiong, Z., Chengfaw, W.: Higher-order rogue wave solutions of the Sasa–Satsuma equation. J Phys A Math. Theor. 55, 1–25 (2022)
Bhrawy, A., Aisha, A., Hilal, E., Khan, K., Mahmood, M., Biswas, A.: Optical soliton in nonlinear directional couplers with spatio-temporal dispersion. J. Mod. Opt. 61, 442–459 (2014)
Bobo, H., Wendi, X.: Existence of ground state solutions to some nonlinear Schrödinger equations on the lattice graphs. Calc. Var. Partial. Differ. Equ. 62, 1–17 (2023)
Ekici, M., Zhou, Q., Sonmezoglu, A., Moshokoa, S.P., Ullah, M.Z., Biswas, A.: Solitons in magneto-optic waveguides by extended trial function scheme. Superlattices Microstruct. 107, 197–218 (2017)
Eslami, M., Rezazadeh, H.: The first integral method for Wu-Zhang system with conformable time-fractional derivative. Calcolo 53, 475–485 (2016)
Eslami, M., Neyrame, A., Ebrahimi, M.: Explicit solutions of nonlinear (2+1)-dimensional dispersive long wave equation. J. King Saudi Univ. Sci. 24(1), 69–71 (2012)
Eslami, M., Fathi Vajargah, B., Mirzazadeh, M., Biswas, A.: Application of first integral method to fractional partial differential equations. Indian J. Phys. 88, 177–184 (2014)
Fan, S.: Optical solutions of Sasa–Satsuma equation in optical fibers. Optik 228, 1–9 (2021)
Ferdous, F., Hafez, M.G.: Oblique closed form solutions of some important fractional evolution equations via the modified Kudryashov method arising in physical problems. J. Ocean Eng. Sci. 3, 244–252 (2018)
Ghazala, A., Maasoomah, S., Atta, M.U.K.: Soliton solutions of the resonant nonlinear Schrödinger equation using modified auxiliary equation method with three different nonlinearities. Math. Comput. Simul 206, 1–20 (2023)
Hai, N.F., Bin, L.Z.: Fractional Schrödinger equations with logarithmic and critical nonlinearities. Acta Math. Sin. Eng. Ser. 39, 285–325 (2023)
Hosseini, K., Sadri, K., Salahshour, S., Baleanu, D., Mirzazadeh, M., Mustafa Inc.: The generalized Sasa-Satsuma equation and its optical solitons. Opt. Quant. Electron. 54(11), 1–15 (2022)
Khatun, M.M., Akbar, M.A.: Analytical soliton solutions of the beta time-fractional simplified modified Camassa-Holm equation in shallow water wave propagation. J. Umm Al-Qura Univ. Appl. Sci. (2023). https://doi.org/10.1007/s43994-023-00085-y
Khodadad, F.S., Nazari, F., Eslami, M., Rezazadeh, H.: Soliton solutions of the conformable fractional Zakharov-Kuznetsov equation with dual-power law nonlinearity. Opt. Quant. Electron. 49, 1–12 (2017)
Kottakkaran, S.N., Khalid, A.K., Mustafa, I., Mehanna, M.S., Hadi, R., Lanre, A.: New solutions for the generalized resonant nonlinear Schrödinger equation. Results Phys. 33, 1–8 (2022)
Mirzazadeh, M., Ekici, M., Sonomezoglu, A., Zhou, Q., Triki, H., Moshokoa, S.P., Biswas, A., Belic, M.: Optical solitons in birefringent fibers by extended trial equation method. Optik 127, 11311–11325 (2016a)
Mirzazadeh, M., Ekici, M., Sonmegoglu, A., Ortakaya, S., Eslami, M., Biswas, A.: Soliton solutions to a few fractional nonlinear evolution equations in shallow water wave dynamics. Eur. Phys. J. plus 131, 1–1 (2016b)
Mostafa, A.M.K., Seadawy, A.R., Dianchen, L.: Dispersive optical soliton solutions for higher order nonlinear Sasa–Satsuma equation in mono mode fibers via auxiliary equation method. Superlattices Microstruct. 113, 346–358 (2017)
Odabasi, M.: Traveling wave solutions of conformable time-fractional Zakharov Kuznetsov and Zoomeron equations. Chin. J. Phys. 64, 194–202 (2020)
Pengde, W., Zhiguo, X., Jin, Y.: Simple high-order boundary conditions for computing rouge waves in the nonlinear Schrödinger equation. Comput. Phys. Commun. 251, 1–15 (2020)
Sadaf, M., Akram, G., Inc, M., Dawood, M., Rezazadeh, H., Akgül, A.: Exact special solutions of space-time fractional Cahn–Allen equation by beta and M-truncated derivatives. Int. J. Mod. Phys. B (2023). https://doi.org/10.1142/S0217979224501212
Sajjad, M.H., Mohammad, M.: Optical solitons of the perturbed nonlinear Schrödinger equation using lie symmetry method. Optik 281, 1–7 (2023)
Sakar, M.G., Ergoren, H.: Alternative variational iteration method for solving the time-fractional Fornberg–Whitham equation. Appl. Math. Mod. 39, 3972–3979 (2015)
Samir, I., Ahmed, H.M., Mirzazadeh, M., Triki, H.: Derivation new solitons and other solutions for higher order Sasa–Satsuma equation by using the improved modified extended tanh scheme. Optik 274, 1–13 (2023)
Samsamik, F., Mirhosseini-AI, S.M., Gunay, B., Lanre, A.I., Hadi, R., Mustafa, I.: Abundant optical solitons to the Sasa–Satsuma higher-order nonlinear Schrödinger equation. Opt. Quantum Electron. 53, 1–17 (2021)
Seadawy, A.R.: The generalized nonlinear higher order of Kdv equations from the higher order nonlinear Schrödinger equation and its solutions. Optik 139, 31–43 (2017)
Shao-Wen, Y., Lanre, A., Mohammad, M., Mustafa, I., Kamyar, H., Mehmet, S.: Dynamics of optical solutions in higher-order Sasa–Satsuma equation. Results Phys. 30, 1–10 (2021)
Shihua, C.: Twisted rouge-wave pairs in the Sasa–Satsuma equation. Phys. Rev. E 88, 1–5 (2013)
Tantawy, M., Abdel-Gawad, H.I.: On multi-geometric structures optical waves propagation in self-phase modulation medium: Sasa–Satsuma equation. Eur. Phys. J. plus 135, 1–10 (2020)
Tariq, H., Ghazala Akram, G., Sadaf, M., Iftikhar, M., Guran, L.: Computational study for fiber Bragg gratings with dispersive reflectivity using fractional derivative. Fractal Fract. 7(8), 1–24 (2023)
Tian, S.-F.: Initial-boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method. J. Differ. Equ. 262, 506–588 (2017)
Triki, H., Mirzazadeh, M., Ahmed, H.M., Samir, I., Hashemi, M.S.: Higher-order Sasa–Satsuma equation: Nucci’s reduction and soliton solutions. Eur. Phys. J. plus 138(50), 1–10 (2023)
Wen, B.B., Row, R., Yin, F., Yue-Yue, W., Chao-oign, D.: Prediction and dynamical evolution of multiple soliton families in fractional Schrödinger equation with the PT-Symmetric potential and saturable nonlinearity. Nonlinear Dyn. 111, 1577–1588 (2023)
Xianguo, G., Jianping, W.: Riemann-Hilbert approach and N-soliton solutions for a generalized Sasa–Satsuma equation. Wave Motion 60, 62–72 (2016)
Yildirim, Y.: Optical solitons to Sasa–Satsuma model with trial equation approach. Optik 184, 70–74 (2019)
Yildridirm, Y., Biswas, A., Jawad, A., Ekici, M., Zhou, Q., Khan, S.: Cubic-quartic optical solitons in birefringent fibers with four forms of nonlinear refractive index by exp-function expansion. Results Phys. 16, 1–8 (2020)
Zafar, A., Seadawy, A.R.: The conformable space-time fractional mKDV equations and their exact solutions. J. King Saud Univ. Sci. 31, 1478–1484 (2019)
Zainab, I., Akram, G.: Effect of β-derivative on time fractional Jaulent-Miodek system under modified auxiliary equation method and exp(−G(Ω))-expansion method. Chaos Solitons Fractals 168, 1–11 (2023)
Zehar, P.I.: Rouge waves and solitons of the generalized modified nonlinear Schrödinger equations. Math. Comput. Simul 208, 535–549 (2023)
Zhon, A., Liu, L., Zhang, H., Mirzazadeh, M., Bhrawy, A., Biswas, A.: Dark and singular optical solutions with competing nonlocal nonlinearities. Opt. Appl. 46, 79–86 (2016)
Zhou, Q., Zhong, Y., Mirzazadeh, M., Bhrawy, A.H., Zerrad, E., Biswas, A.: Thirring combo-solitons with cubic nonlinearity and spatio-temporal dispersion. Waves Rand. Complex Media 26(2), 294–210 (2016)
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Asaduzzaman, Akbar, M.A. Dynamics of optical solitons of nonlinear fractional models: a comprehensive analysis of space–time fractional equations. Opt Quant Electron 56, 783 (2024). https://doi.org/10.1007/s11082-024-06490-9
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DOI: https://doi.org/10.1007/s11082-024-06490-9