Abstract
This work explores the effect of fractional derivative on the fourth-order nonlinear Schrödinger equation with Kerr law nonlinearity, a highly significant equation in the study of wave propagation in dispersive media. By employing the improved modified extended tanh-function method, a variety of optical soliton solutions are derived. These solutions including dark solitons, bright solitons, and singular solitons. Moreover, singular periodic solutions and exponential solutions are raised. These solutions offer valuable insights into the dynamic behavior of nonlinear wave phenomena. The impact of the fractional derivative is illustrated graphically using examples of some of the retrieved solutions with various values of fractional order. Bright and dark solitons, pivotal components of our findings, play a critical role in fiber optics by facilitating the transmission of high-power optical signals with exceptional attributes such as shape preservation. These properties eliminate the need for external pulse compression, simplifying the design and operation of optical systems. The outcomes of this study contribute in advancing our knowledge of wave propagation in dispersive media and have practical implications for the development of efficient and robust optical communication technologies.
Similar content being viewed by others
Data availibility
No datasets were generated or analysed during the current study.
References
Abbagari, S., Saliou, Y., Houwe, A., Akinyemi, L., Inc, M., Bouetou, T.B.: Modulated wave and modulation instability gain brought by the cross-phase modulation in birefringent fibers having anti-cubic nonlinearity. Phys. Lett. A 442, 128191 (2022)
Abdullah, F., Islam, T., Gómez-Aguilar, J.F., Akbar, M.: Impressive and innovative soliton shapes for nonlinear Konno-Oono system relating to electromagnetic field. Opt. Quantum Electron. 55, 69 (2022)
Ahmad, R., Javid, A.: Analytical soliton solutions for the (2+1)-perturbed and higher order cubic-quintic nonlinear Schrödinger equations. Opt. Quant. Electron. 55(10), 850 (2023)
Ahmed, N., Baber, M.Z., Iqbal, M.S., Annum, A., Ali, S.M., Ali, M., Akgül, A., El Din, S.M.: Analytical study of reaction diffusion Lengyel–Epstein system by generalized Riccati equation mapping method. Sci. Rep. 13(1), 20033 (2023)
Akbar, M.A., Wazwaz, A.-M., Mahmud, F., Baleanu, D., Roy, R., Barman, H.K., Mahmoud, W., Al Sharif, M.A., Osman, M.S.: Dynamical behavior of solitons of the perturbed nonlinear Schrödinger equation and microtubules through the generalized Kudryashov scheme. Results Phys. 43, 106079 (2022)
Akbar, M.A., Abdullah, F.A., Khatun, M.M.: An investigation of optical solitons of the fractional cubic-quintic nonlinear pulse propagation model: An analytic approach and the impact of fractional derivative. Opt. Quantum Electron. 56(1), 58 (2023)
Akbar, M.A., Abdullah, F.A., Islam, M.T., Al Sharif, M.A., Osman, M.S.: New solutions of the soliton type of shallow water waves and superconductivity models. Results Phys. 44, 106180 (2023)
Akram, G., Sadaf, M., Khan, M.A.U.: Abundant optical solitons for Lakshmanan-Porsezian-Daniel model by the modified auxiliary equation method. Optik 251, 168163 (2022)
Ali, M., Alquran, M., Salman, O.B.: A variety of new periodic solutions to the damped (2+1)-dimensional Schrödinger equation via the novel modified rational sine-cosine functions and the extended tanh-coth expansion methods. Results Phys. 37, 105462 (2022)
Aliyu, A.I., Alshomrani, A.S., Inc, M., Baleanu, D., Aliyu, A.I., Alshomrani, A.S., Inc, M., Baleanu, D.: Optical solitons for Triki–Biswas equation by two analytic approaches. AIMS Math. 5(2), 1001–1010 (2020)
Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model 1–8 (2016). arXiv preprint arXiv:1602.03408
Bakodah, H.O., Al Qarni, A.A., Banaja, M.A., Zhou, Q., Moshokoa, S.P., Biswas, A.: Bright and dark Thirring optical solitons with improved Adomian decomposition method. Optik 130, 1115–1123 (2017)
Bekir, A., Raza, N., Rezazadeh, H., Rafiq, M.H.: Optical solitons of the (2 + 1)-dimensional Kundu-Mukherjee-Naskar equation with Lokal M-derivative. Preprint, Preprints 1–17 (2022). https://doi.org/10.22541/au.164864681.13404176/v1
Biswas, A., Zhou, Q., Moshokoa, S., Houria, T., Belić, M., Alqahtani, R.: Existence of resonant 1-soliton in medium with anti-cubic nonlinearity and perturbations. Opt. Int. J. Light Electron Opt. 145, 14–17 (2017)
Ghanbari, B., Baleanu, D., Al Qurashi, M.: New Exact Solutions of the Generalized Benjamin-Bona-Mahony Equation. Symmetry 11(1), 1–12 (2019)
Ghanbari, B., Akgül, A.: Abundant new analytical and approximate solutions to the generalized Schamel equation. Phys. Scr. 95(7), 075201 (2020)
Ghanbari, B., Baleanu, D.: New solutions of Gardner’s equation using two analytical methods. Front. Phys. 7, 202 (2019)
Ghanbari, B., Kuo, C.-K.: New exact wave solutions of the variable-coefficient (1 + 1)-dimensional Benjamin–Bona–Mahony and (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov equations via the generalized exponential rational function method. Eur. Phys. J. Plus 134(7), 334 (2019)
Hosseini, K., Mirzazadeh, M., Ilie, M., Radmehr, S.: Dynamics of optical solitons in the perturbed Gerdjikov–Ivanov equation. Optik 206, 164350 (2020)
Hosseini, K., Mirzazadeh, M., Ilie, M., Gómez-Aguilar, J.F.: Biswas–Arshed equation with the beta time derivative: optical solitons and other solutions. Optik 217, 164801 (2020)
Houwe, A., Abbagari, S., Djorwe, P., Saliou, Y., Doka, S.Y., Inc, M.: W-shaped profile and breather-like soliton of the fractional nonlinear Schrödinger equation describing the polarization mode in optical fibers. Opt. Quant. Electron. 54(8), 483 (2022)
Islam, M.T., Akter, M.A., Ryehan, S., Gómez-Aguilar, J.F., Akbar, M.A.: A variety of solitons on the oceans exposed by the Kadomtsev Petviashvili-modified equal width equation adopting different techniques. J. Ocean Eng. Sci. 1–12 (2022). https://doi.org/10.1016/j.joes.2022.07.001
Islam, M.T., Akbar, M.A., Kalam Azad, M.A.: The exact traveling wave solutions to the nonlinear space-time fractional modified Benjamin–Bona–Mahony equation. J. Mech. Contin. Math. Sci. 13(2), 56–71 (2018)
Islam, M.T., Akbar, M.A., Azad, M.A.K.: Closed-form travelling wave solutions to the nonlinear space-time fractional coupled Burgers’ equation. Arab J. Basic Appl. Sci. 26(1), 1–11 (2019)
Islam, M.T., Akbar, M.A., Ahmad, H., Ilhan, O.A., Gepreel, K.A.: Diverse and novel soliton structures of coupled nonlinear Schrödinger type equations through two competent techniques. Modern Phys. Lett. B 36(11), 2250004 (2022)
Islam, M.T., Akbar, M.A., Gómez-Aguilar, J.F., Bonyah, E., Fernandez-Anaya, G.: Assorted soliton structures of solutions for fractional nonlinear Schrödinger types evolution equations. J. Ocean Eng. Sci. 7(6), 528–535 (2022)
Islam, M.T., Akter, M.A., Gómez-Aguilar, J.F., Akbar, M.A., Pérez-Careta, E.: Innovative and diverse soliton solutions of the dual core optical fiber nonlinear models via two competent techniques. J. Nonlinear Opt. Phys. Mater. 32(04), 2350037 (2023)
Jaradat, M.M.M., Batool, A., Butt, A.R., Raza, N.: New solitary wave and computational solitons for Kundu–Eckhaus equation. Results Phys. 43, 106084 (2022)
Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)
Khan, H., Kumam, P., Khan, Q., Khan, S., Arshad, M., Sitthithakerngkiet, K.: The solution comparison of time-fractional non-linear dynamical systems by using different techniques. Front. Phys. 10, 248 (2022)
Kılınç, S.ŞŞ, Çelik, E., Bulut, H.: Solitary wave solutions to some nonlinear conformable partial differential equations. Opt. Quant. Electron. 55(8), 693 (2023)
Kohl, R., Milovic, D., Zerrad, E., Biswas, A.: Optical solitons by He’s variational principle in a non-Kerr law media. J. Infrared Millim. Terahertz Waves 30, 526–537 (2009)
Kudryashov, N.A.: First integrals and solutions of the traveling wave reduction for the Triki–Biswas equation. Optik 185, 275–281 (2019)
Kudryashov, N.A.: Periodic and solitary waves of the Biswas–Arshed equation. Optik 200, 163442 (2020)
Kudryashov, N.A.: Traveling wave solutions of the generalized Gerdjikov–Ivanov equation. Optik 219, 165193 (2020)
Mirzazadeh, M., Eslami, M., Milovic, D., Biswas, A.: Topological solitons of resonant nonlinear Schrödinger’s equation with dual-power law nonlinearity by \(G^{\prime }/G\)-expansion technique. Optik 125(19), 5480–5489 (2014)
Muniyappan, A., Ravichandran, R., Manikandan, K.: A bookshelf layer model for anti-kink and kink pair solitons in the ferroelectric liquid crystal. J. Opt. 1–16 (2023). https://doi.org/10.1007/s12596-023-01488-0
Muniyappan, A., Priya, O.A., Amirthani, S., Brintha, K., Biswas, A., Ekici, M., Dakova, A., Alshehri, H.M., Belic, M.R.: Peakon and cuspon excitations in optical fibers for eighth order nonlinear Schrödinger’s model. Optik 243, 167509 (2021)
Muniyappan, A., Hemamalini, D., Akila, E., Elakkiya, V., Anitha, S., Devadharshini, S., Biswas, A., Yıldırım, Y., Alshehri, H.M.: Bright solitons with anti-cubic and generalized anti-cubic nonlinearities in an optical fiber. Optik 254, 168612 (2022)
Muniyappan, A., Amirthani, S., Chandrika, P., Biswas, A., Yıldırım, Y., Alshehri, H.M., Maturi, D.A.A., Al-Bogami, D.H.: Dark solitons with anti-cubic and generalized anti-cubic nonlinearities in an optical fiber. Optik 255, 168641 (2022)
Muniyappan, A., Manikandan, K., Seadawy, A.R., Parasuraman, E.: Dynamical characteristics and physical structure of cusp-like singular solitons in birefringent fibers. Results Phys. 56, 107241 (2024)
Podder, A., Arefin, M.A., Akbar, M.A., Uddin, M.H.: A study of the wave dynamics of the space-time fractional nonlinear evolution equations of beta derivative using the improved Bernoulli sub-equation function approach. Sci. Rep. 13, 20478 (2023)
Rafiq, M.H., Jhangeer, A., Raza, N.: The analysis of solitonic, super nonlinear, periodic, quasi-periodic, bifurcation and chaotic patterns of perturbed Gerdjikov-Ivanov model with full nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 116, 106818 (2023)
Rafiq, M.H., Raza, N., Jhangeer, A.: Nonlinear dynamics of the generalized unstable nonlinear Schrödinger equation: a graphical perspective. Opt. Quant. Electron. 55(7), 628 (2023)
Raza, N., Arshed, S., Javid, A.: Optical solitons and stability analysis for the generalized second-order nonlinear Schrödinger equation in an optical fiber. Int. J. Nonlinear Sci. Numer. Simul. 21(7–8), 855–863 (2020)
Raza, N., Arshed, S., Butt, A.R., Inc, M., Yao, S.-W.: Investigation of new solitons in nematic liquid crystals with Kerr and non-Kerr law nonlinearities. J. Nonlinear Opt. Phys. Mater. 32(02), 2350020 (2023)
Raza, N., Rani, B., Chahlaoui, Y., Shah, N.A.: A variety of new rogue wave patterns for three coupled nonlinear Maccari’s models in complex form. Nonlinear Dyn. 111(19), 18419–18437 (2023)
Saha, M., Sarma, A.K., Biswas, A.: Dark optical solitons in power law media with time-dependent coefficients. Phys. Lett. A 373(48), 4438–4441 (2009)
Samir, I., Badra, N., Seadawy, A.R., Ahmed, H.M., Arnous, A.H.: Computational extracting solutions for the perturbed Gerdjikov–Ivanov equation by using improved modified extended analytical approach. J. Geom. Phys. 176, 104514 (2022)
Samir, I., Badra, N., Ahmed, H.M., Arnous, A.H.: Optical soliton perturbation with Kudryashov’s generalized law of refractive index and generalized nonlocal laws by improved modified extended tanh method. Alex. Eng. J. 61(5), 3365–3374 (2022)
Tahir, M., Awan, A.U.: Optical singular and dark solitons with Biswas–Arshed model by modified simple equation method. Optik 202, 163523 (2020)
Triki, H., Biswas, A., Moshokoa, S.P., Belić, M.: Optical solitons and conservation laws with quadratic-cubic nonlinearity. Optik 128, 63–70 (2017)
Wazwaz, A.-M.: Bright and dark optical solitons of the (2+1)-dimensional perturbed nonlinear Schrödinger equation in nonlinear optical fibers. Optik 251, 168334 (2022)
Yang, Z., Hon, B.Y.C.: An improved modified extended tanh-function method. Zeitschrift für Naturforschung A 61(3–4), 103–115 (2006)
Zayed, E.M.E., Alngar, M.E.M., Al-Nowehy, A.-G.: On solving the nonlinear Schrödinger equation with an anti-cubic nonlinearity in presence of Hamiltonian perturbation terms. Optik 178, 488–508 (2019)
Zhang, Q., Xiong, M., Chen, L.: Exact travelling wave solutions of two nonlinear Schrödinger equations by using two methods. J. Appl. Math. Phys. 07(12), 3101 (2019)
Zhou, Q., Zhu, Q., Biswas, A.: Optical solitons in birefringent fibers with parabolic law nonlinearity. Opt. Appl. 44(3), 399–409 (2014)
Zhu, C., Al-Dossari, M., El-Gawaad, N.S.A., Alsallami, S.A.M., Shateyi, S.: Uncovering diverse soliton solutions in the modified Schrödinger’s equation via innovative approaches. Results Phys. 54, 107100 (2023)
Acknowledgements
Not applicable.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no competing interests.
Ethical approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Soliman, M., Ahmed, H.M., Badra, N. et al. Effects of fractional derivative on fiber optical solitons of (2 + 1) perturbed nonlinear Schrödinger equation using improved modified extended tanh-function method. Opt Quant Electron 56, 777 (2024). https://doi.org/10.1007/s11082-024-06593-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-024-06593-3