Abstract
Nonlinear Schrödinger model is one of the important fundamental nonlinear physical model to describing the fluctuations development of optical solitons and play a important role in the demonstration of dynamical fiber optics. Therefore, in nonlinear dispersive media the propagation of waves are the area of considerable interest due to its large range of possibilities to the ultrafast light pulses and data processing in communication systems. In this research, we investigated the one of the important class of nonlinear Schrödinger equation named modified unstable nonlinear Schrödinger equation (mUNLSE), which describe time periods disturbances in slightly stable and unstable medium, and also manage the instabilities of train form modulated waves. The mUNLSE is a valuable model for understanding wave behavior in fiber optics, aiding engineers in optimizing optical fiber design and predicting various conditions. We explored the collection of optical solitons and solitary wave structures to examine the dynamical properties of the governing model on the base of powerful unified approach. The explored optical solitons demonstrated in 2D, 3D, and contour plots under the aid of computing software Mathematica and may helpful for studying the waves phenomena in nonlinear optics, solitons wave theory, optical fiber, fluid dynamics, communication system, signal transmission, computer networking, sound and heat processing system. Also, the present work provide understanding into the fundamental properties of optical solitons solutions in mUNLSE, and also its experimental consequences in the physically system. On the base of this study, the proposed approach can be utilized to study the other real and complex nonlinear evolution equations.
Similar content being viewed by others
References
Akbar, M.A., Ali, N.H.M.: The alternative \((G^{^{\prime }}/G)\)-expansion method and its applications to nonlinear partial differential equations. Int. J. Phys. Sci. 6(35), 7910–7920 (2011)
Akbar, M.A., Ali, N.H.M., Tanjim, T.: Outset of multiple soliton solutions to the nonlinear Schrödinger equation and the coupled Burgers equation. J. Phys. Commun. 3(9), 095013 (2019)
Alam, M.N., Ilhan, O.A., Manafian, J., Asjad, M.I., Rezazadeh, H., Baskonus, H.M., Macias-Diaz, J.E.: New results of some of the conformable models arising in dynamical systems. Adv. Math. Phys. 2022, 1–13 (2022)
Alam, M.N., Akash, H.S., Saha, U., Hasan, M.S., Parvin, M.W., Tunç, C.: Bifurcation analysis and solitary wave analysis of the nonlinear fractional soliton neuron model. Iran. J. Sci. 47(5), 1797 (2023)
Alam, M.N., Talib, I., Tunç, C.: The new soliton configurations of the 3D fractional model in arising shallow water waves. Int. J. Appl. Computat. Math. 9(5), 75 (2023)
Al-Amin, M., Islam, M.N., Akbar, M.A.: Adequate wide-ranging closed-form wave solutions to a nonlinear biological model. Partial Diff. Equ. Appl. Math. 4, 100042 (2021)
Al-Amr, M.O.: Exact solutions of the generalized (2+ 1)-dimensional nonlinear evolution equations via the modified simple equation method. Comput. Math. Appl. 69(5), 390–397 (2015)
Ali, A., Seadawy, A.R., Baleanu, D.: Computational solutions of conformable space-time derivatives dynamical wave equations: analytical mathematical techniques. Results Phys. 19, 103419 (2020)
Arshad, M., Seadawy, A.R., Lu, D., Jun, W.: Modulation instability analysis of modify unstable nonlinear Schrödinger dynamical equation and its optical soliton solutions. Results Phys. 7, 4153–4161 (2017)
Aslan, I.: Exact solutions for fractional DDEs via auxiliary equation method coupled with the fractional complex transform. Math. Methods Appl. Sci. 39(18), 5619–5625 (2016)
Gao, W., Rezazadeh, H., Pinar, Z., Baskonus, H.M., Sarwar, S., Yel, G.: Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique. Opt. Quant. Electron. 52, 1–13 (2020)
Ghanbari, B.: Abundant soliton solutions for the Hirota-Maccari equation via the generalized exponential rational function method. Modern Phys. Lett. B 33(09), 1950106 (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., Baleanu, D.: New optical solutions of the fractional Gerdjikov-Ivanov equation with conformable derivative. Front. Phys. 8, 167 (2020)
Ghanbari, B., Gómez-Aguilar, J.F.: Optical soliton solutions for the nonlinear Radhakrishnan-Kundu-Lakshmanan equation. Modern Phys. Lett. B 33(32), 1950402 (2019)
Ghanbari, B., Gómez-Aguilar, J.F.: New exact optical soliton solutions for nonlinear Schrödinger equation with second-order spatio-temporal dispersion involving M-derivative. Modern Phys. Lett. B 33(20), 1950235 (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. Europ. Phys. J. Plus 134(7), 334 (2019)
Ghanbari, B., Baleanu, D., Al Qurashi, M.: New exact solutions of the generalized Benjamin-Bona-Mahony equation. Symmetry 11(1), 20 (2018)
Hosseini, K., Ansari, R.: New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method. Waves Random Compl. Media 27(4), 628–636 (2017)
Hosseini, K., Seadawy, A.R., Mirzazadeh, M., Eslami, M., Radmehr, S., Baleanu, D.: Multiwave, multicomplexiton, and positive multicomplexiton solutions to a (3+ 1)-dimensional generalized breaking soliton equation. Alex. Eng. J. 59(5), 3473–3479 (2020)
Ibrahim, S.: Solitary wave solutions for the (2+1) CBS equation. Adv. Diff. Equ. Contr. Process. 29, 117–126 (2022)
Ibrahim, S.: Optical soliton solutions for the nonlinear third-order partial differential equation. Advances in Differential Equations and Control Processes 29, 127–138 (2022)
Ibrahim, S., Baleanu, D.: Classes of solitary solution for nonlinear Schrödinger equation arising in optical fibers and their stability analysis. Opt. Quant. Electron. 55(13), 1158 (2023)
Ilhan, O.A., Islam, M.N., Akbar, M.A.: Construction of functional closed form wave solutions to the ZKBBM equation and the Schrödinger equation. Iran. J. Sci. Technol. Trans. Mechan. Eng. 45, 827–840 (2021)
Inc, M., Aliyu, A.I., Yusuf, A., Baleanu, D.: Optical solitons for Biswas-Milovic model in nonlinear optics by sine-gordon equation method. Optik 157, 267–274 (2018)
Iqbal, M., Seadawy, A.R., Lu, D., Xia, X.: Construction of bright-dark solitons and ion-acoustic solitary wave solutions of dynamical system of nonlinear wave propagation. Modern Phys. Lett. A 34(37), 1950309 (2019)
Iqbal, M., Seadawy, A.R., Khalil, O.H., Lu, D.: Propagation of long internal waves in density stratified ocean for the (2+ 1)-dimensional nonlinear Nizhnik-Novikov-Vesselov dynamical equation. Results Phys. 16, 102838 (2020)
Iqbal, M., Seadawy, A.R., Althobaiti, S.: Mixed soliton solutions for the (2+ 1)-dimensional generalized breaking soliton system via new analytical mathematical method. Results Phys. 32, 105030 (2022)
Iqbal, M., Seadawy, A.R., Lu, D., Zhang, Z.: Computational approach and dynamical analysis of multiple solitary wave solutions for nonlinear coupled Drinfeld-Sokolov-Wilson equation. Results Phys. 54, 107099 (2023)
Iqbal, M., Seadawy, A.R., Lu, D., Zhang, Z.: Physical structure and multiple solitary wave solutions for the nonlinear Jaulent-Miodek hierarchy equation. Modern Phys. Lett. B 47(5), 1797 (2023)
Iqbal, M., Seadawy, A.R., Lu, D., Zhang, Z.: Multiple optical soliton solutions for wave propagation in nonlinear low-pass electrical transmission lines under analytical approach. Opt. Quant. Electron. 56(1), 35 (2024)
Iqbal, M., Seadawy, A.R., Lu, D., Zhang, Z.: Multiple optical soliton solutions for wave propagation in nonlinear low-pass electrical transmission lines under analytical approach. Opt. Quant. Electron. 56(1), 35 (2024)
Iqbal, M., Lu, D., Seadawy, A.R., et al.: Investigation of solitons structures for nonlinear ionic currents microtubule and Mikhaillov-Novikov-Wang dynamical equations. Opt. Quant. Electron. 56, 361 (2024). https://doi.org/10.1007/s11082-023-05984-2
Ismael, H.F., Baskonus, H.M., Bulut, H.: Abundant novel solutions of the conformable Lakshmanan-Porsezian-Daniel model. Discr. Contin. Dyn. Syst. S 14(7), 2311–2333 (2021)
Jiang, Y., Wang, F., Salama, S.A., Botmart, T., Khater, M.M.: Computational investigation on a nonlinear dispersion model with the weak non-local nonlinearity in quantum mechanics. Results Phys. 38, 105583 (2022)
Khater, M., Ghanbari, B.: On the solitary wave solutions and physical characterization of gas diffusion in a homogeneous medium via some efficient techniques. Europ. Phys. J. Plus 136(4), 1–28 (2021)
Liao, B., Dong, G., Ma, Y., Ma, X., Perlin, M.: Modified nonlinear Schrödinger equation for gravity waves with the influence of wind, currents, and dissipation. Phys. Fluids 35(3), 102 (2023)
Liu, J.G., Eslami, M., Rezazadeh, H., Mirzazadeh, M.: The dynamical behavior of mixed type lump solutions on the (3+ 1)-dimensional generalized Kadomtsev-Petviashvili-Boussinesq equation. Int. J. Nonlinear Sci. Numer. Simul. 21(7–8), 661–665 (2020)
Lu, D., Seadawy, A., Arshad, M.: Applications of extended simple equation method on unstable nonlinear Schrödinger equations. Optik 140, 136–144 (2017)
Lu, D., Seadawy, A.R., Iqbal, M.: Mathematical methods via construction of traveling and solitary wave solutions of three coupled system of nonlinear partial differential equations and their applications. Results Phys. 11, 1161–1171 (2018)
Lu, D., Seadawy, A.R., Iqbal, M.: Construction of new solitary wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Holm equations. Open Phys. 16(1), 896–909 (2018)
Mohyud-Din, S.T., Noor, M.A., Noor, K.I.: Exp-function method for traveling wave solutions of modified Zakharov-Kuznetsov equation. J. King Saud Univ. Sci. 22(4), 213–216 (2010)
Rezazadeh, H., Korkmaz, A., Eslami, M., Mirhosseini-Alizamini, S.M.: A large family of optical solutions to Kundu-Eckhaus model by a new auxiliary equation method. Opt. Quant. Electron. 51, 1–12 (2019)
Sarwar, A., Arshad, M., Farman, M., Akgül, A., Ahmed, I., Bayram, M., De la Sen, M.: Construction of novel bright-dark solitons and breather waves of unstable nonlinear Schrödinger equations with applications. Symmetry 15(1), 99 (2022)
Sarwar, A., Gang, T., Arshad, M., Ahmed, I., Ahmad, M.O.: Abundant solitary wave solutions for space-time fractional unstable nonlinear Schrödinger equations and their applications. Ain Shams Eng. J. 14(2), 101839 (2023)
Seadawy, A.R., Iqbal, M.: Propagation of the nonlinear damped Korteweg-de Vries equation in an unmagnetized collisional dusty plasma via analytical mathematical methods. Math. Methods Appl. Sci. 44(1), 737–748 (2021)
Seadawy, A.R., Iqbal, M.: Dispersive propagation of optical solitions and solitary wave solutions of Kundu-Eckhaus dynamical equation via modified mathematical method. Appl. Math. A J. Chin. Univ. 38(1), 16–26 (2023)
Seadawy, A.R., Iqbal, M., Lu, D.: Applications of propagation of long-wave with dissipation and dispersion in nonlinear media via solitary wave solutions of generalized Kadomtsev-Petviashvili modified equal width dynamical equation. Comput. Math. Appl. 78(11), 3620–3632 (2019)
Seadawy, A.R., Iqbal, M., Lu, D.: Applications of propagation of long-wave with dissipation and dispersion in nonlinear media via solitary wave solutions of generalized Kadomtsev-Petviashvili modified equal width dynamical equation. Comput. Math. Appl. 78(11), 3620–3632 (2019)
Seadawy, A.R., Iqbal, M., Lu, D.: The nonlinear diffusion reaction dynamical system with quadratic and cubic nonlinearities with analytical investigations. Int. J. Modern Phys. B 34(09), 2050085 (2020)
Seadawy, A.R., Iqbal, M., Lu, D.: Propagation of long-wave with dissipation and dispersion in nonlinear media via generalized Kadomtsive-Petviashvili modified equal width-Burgers equation. Indian J. Phys. 94, 675–687 (2020)
Seadawy, A.R., Iqbal, M., Lu, D.: Propagation of kink and anti-kink wave solitons for the nonlinear damped modified Korteweg-de Vries equation arising in ion-acoustic wave in an unmagnetized collisional dusty plasma. Phys. A Statist. Mechan. Appl. 544, 123560 (2020)
Seadawy, A.R., Iqbal, M., Lu, D.: Construction of soliton solutions of the modify unstable nonlinear Schrödinger dynamical equation in fiber optics. Indian J. Phys. 94, 823–832 (2020)
Seadawy, A.R., Iqbal, M., Althobaiti, S., Sayed, S.: Wave propagation for the nonlinear modified Kortewege-de Vries Zakharov-Kuznetsov and extended Zakharov-Kuznetsov dynamical equations arising in nonlinear wave media. Opt. Quant. Electron. 53, 1–20 (2021)
Seadawy, A.R., Iqbal, M., Lu, D.: Ion-acoustic solitary wave solutions of nonlinear damped Korteweg-de Vries and damped modified Korteweg-de Vries dynamical equations. Indian J. Phys. 95, 1479–1489 (2021)
Tala-Tebue, E., Seadawy, A.R., Djoufack, Z.I.: The modify unstable nonlinear Schrödinger dynamical equation and its optical soliton solutions. Opt. Quant. Electron. 50, 1–11 (2018)
Tascan, F., Bekir, A., Koparan, M.: Travelling wave solutions of nonlinear evolution equations by using the first integral method. Commun. Nonlinear Sci. Numer. Simulat. 14(5), 1810–1815 (2009)
Wazwaz, A.M.: The extended tanh method for the Zakharov-Kuznetsov (ZK) equation, the modified ZK equation, and its generalized forms. Commun. Nonlinear Sci. Numer. Simulat. 13(6), 1039–1047 (2008)
Wazwaz, A.M., Kaur, L.: Optical solitons and Peregrine solitons for nonlinear Schrödinger equation by variational iteration method. Optik 179, 804–809 (2019)
Younis, M., Sulaiman, T.A., Bilal, M., Rehman, S.U., Younas, U.: Modulation instability analysis, optical and other solutions to the modified nonlinear Schrödinger equation. Commun. Theoret. Phys. 72(6), 065001 (2020)
Zahran, E.H., Khater, M.M.: Modified extended tanh-function method and its applications to the Bogoyavlenskii equation. Appl. Math. Modell. 40(3), 1769–1775 (2016)
Acknowledgements
The authors would like to acknowledge the Deanship of Scientific Research of Taif University, for funding this work.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
MI: Writing-Original draft preparation, formal analysis, methodology. MNA: Writing-reviewing & editing, visualization, investigation. DL: Software, conceptualization, resources. ARS: Data curation, supervision. NEA: Acquisition, resources, validation. SI: Revised, analysis.
Corresponding author
Ethics declarations
Conflict of interest
This manuscript has no conflicts of interest and was created through the collaborative efforts.
Availability of data and materials
All the data can be found within the paper.
Ethical approval
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
Iqbal, M., Nur Alam, M., Lu, D. et al. On the exploration of dynamical optical solitons to the modify unstable nonlinear Schrödinger equation arising in optical fibers. Opt Quant Electron 56, 765 (2024). https://doi.org/10.1007/s11082-024-06468-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-024-06468-7