Abstract
In this paper, a comprehensive analysis of traveling wave solutions of two nonlinear Schrödinger type equations are carried out with help of three different integration techniques namely the tanh–coth, Kudryashov and sine–cosine methods. These equations include the (2 + 1)-dimensional perturbed nonlinear Schrödinger’s equation and cubic–quintic nonlinear Schrödinger’s equation. The obtained travelling wave solutions are in the form of rational function solutions, trigonometric function solutions, exponential function solutions and hyperbolic function solutions. Our proposed results showed that these techniques are reliable to study the nonlinear PDEs in fiber optics. The higher order cubic–quintic nonlinear Schrödinger equation (NLSE) explains the transmission of incredibly low signals and broadband communications that stretch into the spectral region, as well as the doping of optical fiber and the encryption of data in optical fibers.
Similar content being viewed by others
Availability of data and materials
There is no associated data.
References
Akram, G., Sadaf, M., Arshed, S., Sameen, F.: Bright, dark, kink, singular and periodic soliton solutions of Lakshmanan–Porsezian–Daniel model by generalized projective Riccati equations method. Optik 241, 167051 (2021)
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 Schrodinger equation via the novel modified rational sine–cosine functions and the extended tanh–coth expansion methods. Res. Phys. 37, 105462 (2022)
Alquran, M.: Optical bidirectional wave-solutions to new two-mode extension of the coupled KdV-Schrodinger equations. Opt. Quant. Electron. 53(10), 1–9 (2021)
Alquran, M., Alhami, R.: Analysis of lumps, single-stripe, breather-wave, and two-wave solutions to the generalized perturbed-KdV equation by means of Hirota’s bilinear method. Nonlinear Dyn. 109, 1985–1992 (2022a)
Alquran, M., Alhami, R.: Convex-periodic, kink-periodic, peakon-soliton and kink bidirectional wave-solutions to new established two-mode generalization of Cahn–Allen equation. Res. Phys. 34, 105257 (2022b)
Alquran, M., Alqawaqneh, A.: New bidirectional wave solutions with different physical structures to the complex coupled Higgs model via recent ansatze methods: applications in plasma physics and nonlinear optics. Opt. Quant. Electron. 54(5), 1–10 (2022)
Alquran, M., Jaradat, I.: Multiplicative of dual-waves generated upon increasing the phase velocity parameter embedded in dual-mode Schrodinger with nonlinearity Kerr laws. Nonlinear Dyn. 96(1), 115–121 (2019)
Alquran, M., Jaradat, I., Yusuf, A., Sulaiman, T.A.: Heart-cusp and bell-shaped-cusp optical solitons for an extended two-mode version of the complex Hirota model: application in optics. Opt. Quant. Electron. 53(1), 1–13 (2021)
Arshad, M., Lu, D., Wang, J.: (N + 1)-dimensional fractional reduced differential transform method for fractional order partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 48, 509–519 (2017)
Arshad, M., Seadawy, A.R., Lu, D.C., Ali, A.: Dispersive solitary wave solutions of strain wave dynamical model and its stability. Commun. Theor. Phys. 71(10), 1155 (2019)
Arshed, S., Biswas, A., Abdelaty, M., Zhou, Q., Moshokoa, S.P., Belic, M.: Optical soliton perturbation for Gerdjikov–Ivanov equation via two analytical techniques. Chin. J. Phys. 56(6), 2879–2886 (2018)
Bezgabadi, A.S., Bolorizadeh, M.A.: Analytic combined bright–dark, bright and dark solitons solutions of generalized nonlinear Schrodinger equation using extended sinh-Gordon equation expansion method. Res. Phys. 30, 104852 (2021)
Cinar, M., Onder, I., Secer, A., Yusuf, A., Sulaiman, T.A., Bayram, M., Aydin, H.: The analytical solutions of Zoomeron equation via extended rational sin–cos and sinh–cosh methods. Phys. Scr. 96(9), 094002 (2021)
Debin, K., Rezazadeh, H., Ullah, N., Vahidi, J., Tariq, K.U., Akinyemi, L.: New soliton wave solutions of a (2 + 1)-dimensional Sawada–Kotera equation. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.03.007
Dreiden, G.V., Khusnutdinova, K.R., Samsonov, A.M., Semenova, I.V.: Splitting induced generation of soliton trains in layered waveguides. J. Appl. Phys. 107(3), 034909 (2010)
Ekici, M., Mirzazadeh, M., Sonmezoglu, A., Ullah, M.Z., Asma, M., Zhou, Q., Belic, M.: Dispersive optical solitons with Schrodinger–Hirota equation by extended trial equation method. Optik 136, 451–461 (2017)
Esen, H., Ozdemir, N., Secer, A., Bayram, M., Sulaiman, T.A., Yusuf, A.: Solitary wave solutions of chiral nonlinear Schrodinger equations. Mod. Phys. Lett. B 35(30), 2150472 (2021)
Esen, H., Secer, A., Ozisik, M., Bayram, M.: Analytical soliton solutions of the higher order cubic–quintic nonlinear Schrodinger’s equation and the influence of the model’s parameters 132, 053103 (2022)
Eslami, M., Mirzazadeh, M.: First integral method to look for exact solutions of a variety of Boussinesq-like equations. Ocean Eng. 83, 133–137 (2014)
Ghanbari, B., Gómez-Aguilar, J.F.: Optical soliton solutions for the nonlinear Radhakrishnan–Kundu–Lakshmanan equation. Mod. Phys. Lett. B 33(32), 1950402 (2019)
Hosseini, K., Mirzazadeh, M., Ilie, M., Radmehr, S.: Dynamics of optical solitons in the perturbed Gerdjikov–Ivanov equation. Optik 206, 164350 (2020a)
Hosseini, K., Mirzazadeh, M., Vahidi, J., Asghari, R.: Optical wave structures to the Fokas–Lenells equation. Optik 207, 164450 (2020b)
Hu, X., Arshad, M., Xiao, L., Nasreen, N., Sarwar, A.: Bright-dark and multi wave novel solitons structures of Kaup–Newell Schrodinger equations and their applications. Alex. Eng. J. 60(4), 3621–3630 (2021)
Inc, M., Aliyu, A.I., Yusuf, A., Baleanu, D.: Optical solitons for complex Ginzburg–Landau model in nonlinear optics. Optik 158, 368–375 (2018)
Islam, M.T., Akbar, M.A., Gómez-Aguilar, J.F., Bonyah, E., Fernandez-Anaya, G.: Assorted soliton structures of solutions for fractional nonlinear Schrodinger types evolution equations. J. Ocean Eng. Sci. 7, 528–535 (2021)
Jaradat, H.M., Alquran, M., Jaradat, M.M.M., Mustafa, Z.: Mathematical analysis and physical interpretation on new multiple solitonic solutions of n-coupled modified KdV system. J. Math. Anal 7(6), 118–129 (2016)
Jaradat, I., Alquran, M., Momani, S., Biswas, A.: Dark and singular optical solutions with dual-mode nonlinear Schrodinger’s equation and Kerr-law nonlinearity. Optik 172, 822–825 (2018)
Jaradat, I., Alquran, M., Qureshi, S., Sulaiman, T.A., Yusuf, A.: Convex-rogue, half-kink, cusp-soliton and other bidirectional wave-solutions to the generalized Pochhammer–Chree equation. Phys. Scr. 97(5), 055203 (2022)
Karpman, V.I.: Radiation of solitons described by a higher-order nonlinear Schrodinger equation. Phys. Scr. 1999(T82), 44–47 (1999)
Kopçasız, B., Yaşar, E.: Novel exact solutions and bifurcation analysis to dual-mode nonlinear Schrodinger equation. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.06.007
Krishnan, E. V., Ghabshi, M. A., Alquran, M.: \((G^{\prime }/G)\)-Expansion Method and Weierstrass Elliptic Function Method Applied to Coupled Wave Equation (2019). arXiv preprint arXiv:1911.10586
Kudryashov, N.A.: Traveling wave solutions of the generalized Gerdjikov–Ivanov equation. Optik 219, 165193 (2020)
Li, Y., Geng, X., Xue, B., Li, R.: Darboux transformation and exact solutions for a four-component Fokas–Lenells equation. Res. Phys. 31, 105027 (2021)
Ling, L., Feng, B.F., Zhu, Z.: General soliton solutions to a coupled Fokas–Lenells equation. Nonlinear Anal. Real World Appl. 40, 185–214 (2018)
Liu, S., Fu, Z., Liu, S., Zhao, Q.: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 289(1–2), 69–74 (2001)
Liu, W.J., Tian, B., Zhang, H.Q., Li, L.L., Xue, Y.S.: Soliton interaction in the higher-order nonlinear Schrodinger equation investigated with Hirota’s bilinear method. Phys. Rev. E 77(6), 066605 (2008)
Martinez, J.C., Shutler, P.M.E., Toon, A.: Theory of dark optical solitons. J. Appl. Phys. 104(4), 043102 (2008)
Mirzazadeh, M., Ekici, M., Sonmezoglu, A., Eslami, M., Zhou, Q., Kara, A.H., Belić, M.: Optical solitons with complex Ginzburg–Landau equation. Nonlinear Dyn. 85(3), 1979–2016 (2016)
Molerón, M., Leonard, A., Daraio, C.: Solitary waves in a chain of repelling magnets. J. Appl. Phys. 115(18), 184901 (2014)
Naher, H., Abdullah, F.A.: New approach of \((G^{\prime }/G)\)-expansion method and new approach of generalized \((G^{\prime }/G)\)-expansion method for nonlinear evolution equation. AIP Adv. 3(3), 032116 (2013)
Nawaz, B., Ali, K., Rizvi, S.T.R., Younis, M.: Soliton solutions for quintic complex Ginzburg–Landau model. Superlattices Microstruct. 110, 49–56 (2017)
Oad, A., Arshad, M., Shoaib, M., Lu, D., Li, X.: Novel soliton solutions of two-mode Sawada–Kotera equation and its applications. IEEE Access 9, 127368–127381 (2021)
Osman, M.S., Korkmaz, A., Rezazadeh, H., Mirzazadeh, M., Eslami, M., Zhou, Q.: The unified method for conformable time fractional Schrodinger equation with perturbation terms. Chin. J. Phys. 56(5), 2500–2506 (2018)
Osman, M.S., Almusawa, H., Tariq, K.U., Anwar, S., Kumar, S., Younis, M., Ma, W.X.: On global behavior for complex soliton solutions of the perturbed nonlinear Schrödinger equation in nonlinear optical fibers. J. Ocean Eng. Sci. 7(5), 431–443 (2022)
Ozdemir, N., Esen, H., Secer, A., Bayram, M., Yusuf, A., Sulaiman, T.A.: Optical soliton solutions to Chen Lee Liu model by the modified extended tanh expansion scheme. Optik 245, 167643 (2021a)
Ozdemir, N., Esen, H., Secer, A., Bayram, M., Sulaiman, T.A., Yusuf, A., Aydin, H.: Optical solitons and other solutions to the Radhakrishnan–Kundu–Lakshmanan equation. Optik 242, 167363 (2021b)
Seadawy, A.R., Arnous, A.H., Biswas, A., Belic, M.: Optical solitons with Sasa–Satsuma equation by F-expansion scheme. Optoelectron. Adv. Mater. Rapid Commun. 13(1–2), 31–36 (2019)
Seadawy, A.R., Arshad, M., Lu, D.: The weakly nonlinear wave propagation theory for the Kelvin–Helmholtz instability in magnetohydrodynamics flows. Chaos Solitons Fractals 139, 110141 (2020)
Sebawe Abdalla, M., Eleuch, H.: Exact analytic solutions of the Schrodinger equations for some modified q-deformed potentials. J. Appl. Phys. 115(23), 234906 (2014)
Semrau, D., Sillekens, E., Killey, R.I., Bayvel, P.: Modelling the delayed nonlinear fiber response in coherent optical communications. J. Lightw. Technol. 39(7), 1937–1952 (2020)
Sulaiman, T.A., Yusuf, A., Alquran, M.: Dynamics of optical solitons and nonautonomous complex wave solutions to the nonlinear Schrodinger equation with variable coefficients. Nonlinear Dyn. 104(1), 639–648 (2021)
Taghizadeh, N., Mirzazadeh, M., Farahrooz, F.: Exact solutions of the nonlinear Schrodinger equation by the first integral method. J. Math. Anal. Appl. 374(2), 549–553 (2011)
Tang, L.: Dynamical behavior and traveling wave solutions in optical fibers with Schrodinger–Hirota equation. Optik 245, 167750 (2021)
Triki, H., Wazwaz, A.M.: Combined optical solitary waves of the Fokas–Lenells equation. Waves Random Complex Media 27(4), 587–593 (2017)
Wang, Z., Feng, Y., Zhu, B., Zhao, J., Jiang, T.: Dark Schrodinger solitons and harmonic generation in left-handed nonlinear transmission line. J. Appl. Phys. 107(9), 094907 (2010)
Wang, J., Shehzad, K., Seadawy, A.R., Arshad, M., Asmat, F.: Dynamic study of multi-peak solitons and other wave solutions of new coupled KdV and new coupled Zakharov–Kuznetsov systems with their stability. J. Taibah Univ. Sci. 17(1), 2163872 (2023)
Wazwaz, A.M.: The sine–cosine method for obtaining solutions with compact and noncompact structures. Appl. Math. Comput. 159(2), 559–576 (2004)
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)
Xian-Lin, Y., Jia-Shi, T.: Travelling wave solutions for Konopelchenko–Dubrovsky equation using an extended sinh-Gordon equation expansion method. Commun. Theor. Phys. 50(5), 1047 (2008)
Yıldırım, Y.: Optical solitons to Chen–Lee–Liu model in birefringent fibers with trial equation approach. Optik 183, 881–886 (2019)
Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9(2), 190–194 (1968)
Zayed, E.M., Gepreel, K.A., El-Horbaty, M., Yıldırım, Y.: Optical solitons in birefringent fibers with Kaup–Newell equation using two integration schemes. Optik 251, 167992 (2022)
Funding
There is no funding informed for this research.
Author information
Authors and Affiliations
Contributions
RA did conceptualization, methodology and writing of first draft. AJ did conceptualization, formal analysis, writing and editing.
Corresponding author
Ethics declarations
Conflict of interest
The authors mention that they have no conflict of interest.
Ethical approval
There was no need of ethical approval for this study.
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
Ahmad, R., Javid, A. Analytical soliton solutions for the (2 + 1)-perturbed and higher order cubic–quintic nonlinear Schrödinger equations. Opt Quant Electron 55, 850 (2023). https://doi.org/10.1007/s11082-023-05108-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-023-05108-w