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FRACTAL LAKSHMANAN–PORSEZIAN–DANIEL MODEL WITH DIFFERENT FORMS OF NONLINEARITY AND ITS NOVEL SOLITON SOLUTIONS

Department of Mathematics, University of Hafr Al-Batin, Hafr Al-Batin 31991, Saudi Arabia

https://doi.org/10.1142/S0218348X21500328Cited by:7 (Source: Crossref)

Abstract

The nonlinear Schrödinger equation (NLSE) can well identify the development of waves in deep water and optical fibers towards the least-order approximation. This study addresses the Lakshmanan–Porsezian–Daniel (LPD) fractal model which emerges from the application of the Heisenberg spin chain and fiber optics. This paper analyzes three types of nonlinear rules, namely Kerr law, quadratic law, and parabolic law. The variational approach to the combination of the Ritz idea is used to discover the new optical soliton solutions for the LPD-equation. It poses the requisite novel conditions for ensuring the existence of valid solitons. Three- and two-dimensional configurations are demonstrated by choosing the correct values for the parameters. This study focused on the pioneering research boundaries of the LPD-equation and other associated nonlinear evolution models in the field of communications network technology and optical fiber.

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References

1. E. P. Gross, Structure of a quantized vortex in boson systems, Il Nuovo Cimento 20 (1961) 454–477. Crossref, Web of Science, Google Scholar
2. L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, J. Exptl. Theor. Phys. 13 (1961) 451–454. Google Scholar
3. H. Bailung, S. K. Sharma and Y. Nakamura, Observation of peregrine solitons in a multicomponent plasma with negative ions, Phys. Rev. Lett. 107 (2011) 255005. Crossref, Web of Science, Google Scholar
4. M. J. Potasek and M. Tabor, Exact solutions for an extended nonlinear Schrödinger equation, Phys. Lett. A 154 (1991) 449–452. Crossref, Web of Science, Google Scholar
5. S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz and A. S. Gouveia-Neto, Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation, Phys. Rev. A 43 (1991) 6162–6165. Crossref, Web of Science, Google Scholar
6. Y. Kodama, A. V. Mikhailov and S. Wabnitz, Input pulse optimization in wavelength-division-multiplexed soliton transmissions, Opt. Commun. 143 (1997) 53–56. Crossref, Web of Science, Google Scholar
7. M. Lakshmanan, K. Porsezian and M. Daniel, Effect of discreteness on the continuum limit of the Heisenberg spin chain, Phys. Lett. A 133 (1988) 483–488. Crossref, Web of Science, Google Scholar
8. K. Porsezian, Completely integrable nonlinear Schrödinger type equations on moving space curves, Phys. Rev. E 55 (1997) 3785–3788. Crossref, Web of Science, Google Scholar
9. H. Q. Zhang, B. Tian, X. H. Meng, X. Lü and W. J. Liu, Conservation laws, soliton solutions and modulational instability for the higher-order dispersive nonlinear Schrödinger equation, Eur. Phys. J. B 72 (2009) 233–239. Crossref, Web of Science, Google Scholar
10. Y. Shen and J. H. He, Variational principle for a generalized KdV equation in a fractal space, Fractals, 28 (2020) 2050069. Link, Web of Science, Google Scholar
11. Y. Khan, Fractal modification of complex Ginzburg–Landau model arising in the oscillating phenomena, Results Phys. 18 (2020) 103324. Crossref, Web of Science, Google Scholar
12. K Nouri, M Nazari and L Torkzadeh, Numerical approximation of the system of fractional differential equations with delay and its applications, Eur. Phys. J. Plus 135 (2020) 341. Crossref, Web of Science, Google Scholar
13. Y Khan, A variational approach for novel solitary solutions of FitzHugh–Nagumo equation arising in the nonlinear reaction–diffusion equation, Int. J. Numer. Meth. Heat Fluid Flow (2020), https://doi.org/10.1108/HFF-05-2020-0299. Crossref, Web of Science, Google Scholar
14. Ş. Yüzbaşı and M. Karaçayır, A Galerkin-type fractional approach for solutions of Bagley-torvik equations, CMES 123 (2020) 941–956. Crossref, Web of Science, Google Scholar
15. F. Y. Ji, C. H. He, J. J. Zhang and J. H. He, A fractal Boussinesq equation for nonlinear transverse vibration of a nanofiber-reinforced concrete pillar, Appl. Math. Model. 82 (2020) 437–448. Crossref, Web of Science, Google Scholar
16. J. Calatayud, J. C. Cortés and M. Jornet, A modified perturbation method for mathematical models with randomness: An analysis through the steady-state solution to Burgers’ partial differential equation, Math. Meth. Appl. Sci. (2020), https://doi.org/10.1002/mma.6420. Crossref, Web of Science, Google Scholar
17. A. Izadkhah, K. Nouri and A. Nikoobin, Proportional Integral derivative control of fractional-order for a quarter-vechicle active suspension system, Roman. J. Phys. 65 (2020) 103. Web of Science, Google Scholar
18. Y. Khan, Maclaurin series method for fractal differential-difference models arising in coupled nonlinear optical waveguides, Fractals 28 (2021) 2150004. Link, Web of Science, Google Scholar
19. C. Burgos, J. C. Cortés, L. Villafuerte and R. J. Villanueva, Mean square convergent numerical solutions of random fractional differential equations: Approximations of moments and density, J. Comput. Appl. Math. 378 (2020) 112925. Crossref, Web of Science, Google Scholar
20. Ş. Yüzbaşı and M. Karaçayır, A Galerkin-like method for solving linear functional differential equations under initial conditions, Turk. J. Math. 44 (2020) 85–97. Crossref, Web of Science, Google Scholar
21. D. Słota, Exact solution of the heat equation with boundary condition of the fourth kind by He’s variational iteration method, Comput. Math. Appl. 58 (2009) 2495–2503. Crossref, Web of Science, Google Scholar
22. K. Nouri, D. Baleanu and L. Torkzadeh, Study on application of hybrid functions to fractional differential equations, Iran. J. Sci. Tech. Trans. A. Sci. 42 (2018) 1343–1350. Crossref, Web of Science, Google Scholar
23. D. Slota, E. Hetmaniok, R. Witula, K. Gromysz and T. Trawinski, Homotopy approach for integrodifferential equations, Mathematics 7 (2019) 904. Crossref, Web of Science, Google Scholar
24. J. H. He, Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos Solitons Fractals 19 (2004) 847–851. Crossref, Web of Science, Google Scholar
25. Y. Khan, A new necessary condition of soliton solutions for Kawahara equation arising in physics, Optik 155 (2018) 273–275. Crossref, Web of Science, Google Scholar
26. J. H. He, A fractal variational theory for one-dimensional compressible flow in a microgravity space, Fractals 28 (2020) 2050024. Link, Web of Science, Google Scholar
27. Y. Khan, N. Faraz and A. Yildirim, New soliton solutions of the generalized Zakharov equations using He’s variational approach, Appl. Math. Lett. 24 (2011) 965–968. Crossref, Web of Science, Google Scholar
28. W. Liu, D. Q. Qiu, Z. W. Wu and J. S. He, Dynamical behavior of solution in integrable nonlocal Lakshmanan–Porsezian–Daniel equation, Commun. Theor. Phys. 65 (2016) 671–676. Crossref, Web of Science, Google Scholar
29. J. Manafian, M. Foroutan and A. Guzali, Applications of the ETEM for obtaining optical soliton solutions for the Lakshmanan–Porsezian–Daniel model, Eur. Phys. J. Plus 132 (2017) 494. Crossref, Web of Science, Google Scholar
30. J. V. Guzman, R. T. Alqahtani, Q. Zhou, M. F. Mahmood, S. P. Moshokoa, M. Z. Ullah, A. Biswas and M. Belic, Optical solitons for Lakshmanan–Porsezian–Daniel model with spatio-temporal dispersion using method of undetermined coefficients, Optik 144 (2017) 115–123. Crossref, Web of Science, Google Scholar
31. R. T. Alqahtani, M. M. Babatin and A. Biswas, Bright optical solitons for Lakshmanan-Porsezian-Daniel model by semi-inverse variational principle, Optik 154 (2018) 109–114. Crossref, Web of Science, Google Scholar
32. A. Biswas, A. H. Kara, R. T. Alqahtani, M. Z. Ullah, H. Triki and M. Belic, Conservation laws for optical solitons of Lakshmanan–Porsezian–Daniel model, Proc. Roman. Acad. Ser. A 19 (2018) 39–44. Web of Science, Google Scholar
33. A. Biswas, M. Ekici, A. Sonmezoglu, H. Triki, F. B. Majid, Q. Zhou, S. P. Moshokoa, M. Mirzazadeh and M. Belic, Optical solitons with Lakshmanan–Porsezian–Daniel model using a couple of integration schemes, Optik 158 (2018) 705–711. Crossref, Web of Science, Google Scholar
34. A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, S. P. Moshokoa and M. Belic, Optical solitons for Lakshmanan–Porsezian–Daniel model by modified simple equation method, Optik 160 (2018) 24–32. Crossref, Web of Science, Google Scholar
35. M. B. Hubert, G. Betchewea, M. Justin, S. Y. Doka, K. T. Crepin, A. Biswas, Q. Zhouh, A. S. Alshomrani, M. Ekici, S. P. Moshokoa and M. Belic, Optical solitons with Lakshmanan–Porsezian–Daniel model by modified extended direct algebraic method, Optik 162 (2018) 228–236. Crossref, Web of Science, Google Scholar
36. A. J. M. Jawad, M. J. Abu-AlShaeer, A. Biswas, Q. Zhou, S. P. Moshokoa and M. Belic, Optical solitons to Lakshmanan–Porsezian–Daniel model for three nonlinear forms, Optik 160 (2018) 197–202. Crossref, Web of Science, Google Scholar
37. J. H. He, Fractal calculus and its geometrical explanation, Results Phys. 10 (2018) 272–276. Crossref, Web of Science, Google Scholar
38. J. H. He and F. Y. Ji, Two-scale mathematics and fractal calculus for thermo dynamics, Therm. Sci. 23 (2019) 2131–2133. Crossref, Web of Science, Google Scholar