Abstract
This paper is devoted to seek new optical soliton solutions of nonlinear Schrödinger equation (NLSE) with time-dependent coefficients which describes the dispersion decreasing fiber. To achieve optical soliton solutions of NLSE, the basic idea of homogenous balance approach has been used to propose Bernoulli \((G'/G)\)-expansion method, where \(G =G(\zeta )\) satisfies Bernoulli equation, which is easier to solve than previous studies. By applying some transformations and using this method, some periodic wave, bright and dark soliton solutions are successfully obtained. Moreover, 3D surfaces, standard deviation line plots and contour maps graphs of the obtained results under effect of different values of coefficients are illustrated to have acceptable image of dynamic structures and to find the relation between the parameters and wave behaviors.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abdullaev, F.K., Caputo, J.G., Kraenkel, R.A., Malomed, B.A.: Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length. Phys. Rev. A 67(1), 013605, 1–10 (2003). https://doi.org/10.1103/physreva.67.013605
Bao, W., Cai, Y.: Mathematical theory and numerical methods for Bose-Einstein condensation. Kinetic Related Models 6, 1–135 (2012). https://doi.org/10.3934/krm.2013.6.1
Bergé, L., Mezentsev, V.K., Rasmussen, J.J., Christiansen, P.L., Gaididei, Y.B.: Self-guiding light in layered nonlinear media. Opt. Lett. 25(14), 1037, 1–3 (2000). https://doi.org/10.1364/ol.25.001037
Bulut, H., Sulaiman, T.A., Baskonus, H.M., Aktürk, T.: On the bright and singular optical solitons to the (2+1)-dimensional NLS and the Hirota equations. Opt. Quant. Electron. 50(3), 134, 1–12 (2018). https://doi.org/10.1007/s11082-018-1411-6
El-Shiekh, R.M.: Classes of new exact solutions for nonlinear schrödinger equations with variable coefficients arising in optical fiber. Results Phys. 13, 102214, 1–5 (2019). https://doi.org/10.1016/j.rinp.2019.102214
Feng, B.: Sharp threshold of global existence and instability of standing wave for the Schrödinger-Hartree equation with a harmonic potential. Nonlinear Anal. Real World Appl. 31, 132–145 (2016). https://doi.org/10.1016/j.nonrwa.2016.01.012
Feng, B., Zhang, H.: Stability of standing waves for the fractional Schrödinger-Choquard equation. Comput. Math. Appl. 75(7), 2499–2507 (2018). https://doi.org/10.1016/j.camwa.2017.12.025
Gu, Y., Aminakbari, N.: Two different systematic methods for constructing meromorphic exact solutions to the KdV-Sawada-Kotera equation. AIMS Math. 5(4), 3990–4010 (2020). https://doi.org/10.3934/math.2020257
Gu, Y., Yuan, W., Aminakbari, N., Lin, J.: Meromorphic solutions of some algebraic differential equations related Painlevé equation IV and its applications. Math. Methods Appl. Sci. 41(10), 3832–3840 (2018). https://doi.org/10.1002/mma.4869
Gu, Y., Wu, C., Yao, X., Yuan, W.: Characterizations of all real solutions for the KdV equation and WR. Appl. Math. Lett. 107, 106446, 1–8 (2020). https://doi.org/10.1016/j.aml.2020.106446
Guo, Q., Liu, J.: New exact solutions to the nonlinear Schrödinger equation with variable coefficients. Results Phys. 16, 102857, 1–5 (2020). https://doi.org/10.1016/j.rinp.2019.102857
He, X.-G., Zhao, D., Li, L., Luo, H.-G.: Engineering integrable nonautonomous nonlinear Schrödinger equations. Phys. Rev. E 79(5), 056610, 1–9 (2009). https://doi.org/10.1103/physreve.79.056610
He, J., Charalampidis, E., Kevrekidis, P., Frantzeskakis, D.: Rogue waves in nonlinear Schrödinger models with variable coefficients: Application to bose–einstein condensates. Phys. Lett. A 378(5–6), 577–583 (2014). https://doi.org/10.1016/j.physleta.2013.12.002
Hosseini, K., Mirzazadeh, M., Rabiei, F., Baskonus, H.M., Yel, G.: Dark optical solitons to the Biswas-Arshed equation with high order dispersions and absence of the self-phase modulation. Optik 209, 164576, 1–6 (2020). https://doi.org/10.1016/j.ijleo.2020.164576
Hossen, M.B., Roshid, H.-O., Ali, M.Z.: Characteristics of the solitary waves and rogue waves with interaction phenomena in a (2+1)-dimensional Breaking Soliton equation. Phys. Lett. A 382(19), 1268–1274 (2018). https://doi.org/10.1016/j.physleta.2018.03.016
Kaur, L., Wazwaz, A.-M.: Bright – dark optical solitons for Schrödinger-Hirota equation with variable coefficients. Optik 179, 479–484 (2019). https://doi.org/10.1016/j.ijleo.2018.09.035
Kedziora, D.J., Ankiewicz, A., Chowdury, A., Akhmediev, N.: Integrable equations of the infinite nonlinear Schrödinger equation hierarchy with time variable coefficients. Chaos: Interdiscip. J. Nonlinear Sci. (2015). https://doi.org/10.1063/1.4931710
Khan, K., Akbar, M.A.: Solving unsteady Korteweg-de Vries equation and its two alternatives. Math. Methods Appl. Sci. 39(10), 2752–2760 (2016). https://doi.org/10.1002/mma.3727
Khatun, M.S., Hoque, M.F., Rahman, M.A.: Multisoliton solutions, completely elastic collisions and non-elastic fusion phenomena of two PDEs. Pramana (2017). https://doi.org/10.1007/s12043-017-1390-3
Kruglov, V.I., Peacock, A.C., Harvey, J.D.: Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients. Phys. Rev. E 71(5), 056619, 1–11 (2005). https://doi.org/10.1103/physreve.71.056619
Liu, J.-G., Osman, M., Wazwaz, A.-M.: A variety of nonautonomous complex wave solutions for the (2\(+\)1)-dimensional nonlinear Schrödinger equation with variable coefficients in nonlinear optical fibers. Optik 180, 917–923 (2019). https://doi.org/10.1016/j.ijleo.2018.12.002
Montesinos, G.: Stabilization of solitons of the multidimensional nonlinear Schrödinger equation: matter-wave breathers. Physica D 191(3–4), 193–210 (2004). https://doi.org/10.1016/j.physd.2003.12.001
Özkan, Y.S., Yaşar, E., Seadawy, A.R.: On the multi-waves, interaction and Peregrine-like rational solutions of perturbed Radhakrishnan-Kundu-Lakshmanan equation. Phys. Scr. 95(8), 085205, 1–13 (2020). https://doi.org/10.1088/1402-4896/ab9af4
Pérez-García, V.M., Torres, P.J., Konotop, V.V.: Similarity transformations for nonlinear Schrödinger equations with time-dependent coefficients. Physica D 221(1), 31–36 (2006). https://doi.org/10.1016/j.physd.2006.07.002
Ponomarenko, S.A., Agrawal, G.P.: Optical similaritons in nonlinear waveguides. Opt. Lett. 32(12), 1659, 1–3 (2007). https://doi.org/10.1364/ol.32.001659
Rahman, Z., Ali, M.Z., Roshid, H.-O.: Closed form soliton solutions of three nonlinear fractional models through proposed improved Kudryashov method. Chin. Phys. B 30(5), 050202, 1–14 (2021). https://doi.org/10.1088/1674-1056/abd165
Rizvi, S., Seadawy, A.R., Ashraf, F., Younis, M., Iqbal, H., Baleanu, D.: Lump and Interaction solutions of a geophysical Korteweg-de Vries equation. Results Phys. 19, 103661, 1–8 (2020). https://doi.org/10.1016/j.rinp.2020.103661
Roshid, H., Rahman, M.A.: The exp\((-\phi (\xi ))\)-expansion method with application in the (1+1)-dimensional classical Boussinesq equations. Results. Phys. 4, 150–155 (2014). https://doi.org/10.1016/j.rinp.2014.07.006
Roshid, H.-O., Kabir, M.R., Bhowmik, R.C., Datta, B.K.: Investigation of Solitary wave solutions for Vakhnenko-Parkes equation via exp-function and Exp\((-\phi (\xi ))\)-expansion method. SpringerPlus. 3(1), 692, 1–10 (2014). https://doi.org/10.1186/2193-1801-3-692
Roshid, M.M., Roshid, H.-O., Ali, M.Z., Rezazadeh, H.: Kinky periodic pulse and interaction of bell wave with kink pulse wave propagation in nerve fibers and wall motion in liquid crystals. Partial Differ. Eq. Appl. Math. 2, 100012, 1–10 (2020). https://doi.org/10.1016/j.padiff.2020.100012
Saito, H., Ueda, M.: Dynamically stabilized bright solitons in a two-dimensional Bose-Einstein condensate. Phys. Rev. Lett. 90(4), 040403, 1–4 (2003). https://doi.org/10.1103/physrevlett.90.040403
Seadawy, A.R., Lu, D., Iqbal, M.: Application of mathematical methods on the system of dynamical equations for the ion sound and Langmuir waves. Pramana (2019b). https://doi.org/10.1007/s12043-019-1771-x
Seadawy, A.R., Cheemaa, N.: Propagation of nonlinear complex waves for the coupled nonlinear Schrödinger Equations in two core optical fibers. Physica A 529, 121330, 1–16 (2019). https://doi.org/10.1016/j.physa.2019.121330
Seadawy, A.R., Cheemaa, N.: Applications of extended modified auxiliary equation mapping method for high-order dispersive extended nonlinear Schrödinger equation in nonlinear optics. Mod. Phys. Lett. B 33(18), 1950203, 1–10 (2019). https://doi.org/10.1142/s0217984919502038
Seadawy, A.R., Cheemaa, N.: Some new families of spiky solitary waves of one-dimensional higher-order K-dV equation with power law nonlinearity in plasma physics. Indian J. Phys. 94(1), 117–126 (2019). https://doi.org/10.1007/s12648-019-01442-6
Seadawy, A.R., Ali, A., Albarakati, W.A.: Analytical wave solutions of the (2+1)-dimensional first integro-differential Kadomtsev-Petviashivili hierarchy equation by using modified mathematical methods. Results Phys. 15, 102775, 1–7 (2019). https://doi.org/10.1016/j.rinp.2019.102775
Serkin, V.N., Hasegawa, A.: Novel soliton solutions of the nonlinear Schrödinger equation model. Phys. Rev. Lett. 85(21), 4502–4505 (2000). https://doi.org/10.1103/physrevlett.85.4502
Ullah, M.S., Ali, M.Z., Roshid, H.-O., Seadawy, A.R., Baleanu, D.: Collision phenomena among lump, periodic and soliton solutions to a (2+1)-dimensional Bogoyavlenskii’s breaking soliton model. Phys. Lett. A 397, 127263, 1–6 (2021). https://doi.org/10.1016/j.physleta.2021.127263
Wang, M., Li, X., Zhang, J.: The (G\(^{\prime }\)/\({G}\))-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372(4), 417–423 (2008). https://doi.org/10.1016/j.physleta.2007.07.051
Wazwaz, A.-M.: Partial Differential Equations and Solitary Waves Theory. Springer, Berlin Heidelberg (2009)
Yan, Z., Wen, Z., Konotop, V.V.: Solitons in a nonlinear Schrödinger equation with PT-symmetric potentials and inhomogeneous nonlinearity: Stability and excitation of nonlinear modes. Phys. Rev. A 92(2), 023821, 1–9 (2015). https://doi.org/10.1103/physreva.92.023821
Yao, Y.-Q., Li, J., Han, W., Wang, D.-S., Liu, W.-M.: Localized spatially nonlinear matter waves in atomic-molecular Bose-Einstein condensates with space-modulated nonlinearity. Sci. Rep. 6(1), 29566, 1–12 (2016). https://doi.org/10.1038/srep29566
Younas, U., Seadawy, A.R., Younis, M., Rizvi, S.: Dispersive of propagation wave structures to the dullin-Gottwald-Holm dynamical equation in a shallow water waves. Chin. J. Phys. 68, 348–364 (2020). https://doi.org/10.1016/j.cjph.2020.09.021
Younas, U., Younis, M., Seadawy, A.R., Rizvi, S., Althobaiti, S., Sayed, S.: Diverse exact solutions for modified nonlinear Schrödinger equation with conformable fractional derivative. Results Phys. 20, 103776, 1–10 (2021). https://doi.org/10.1016/j.rinp.2020.103766
Yu, F., Yan, Z.: New rogue waves and dark-bright soliton solutions for a coupled nonlinear Schrödinger equation with variable coefficients. Appl. Math. Comput. 233, 351–358 (2014). https://doi.org/10.1016/j.amc.2014.02.023
Zeng, Z.-F., Liu, J.-G., Jiang, Y., Nie, B.: Transformations and soliton solutions for a variable-coefficient nonlinear Schrödinger equation in the dispersion decreasing fiber with symbolic computation. Fundamenta Informaticae, 145(2), 207–219 (2016) ISSN 0169-2968. https://doi.org/10.3233/FI-2016-1355
Zhang, J.: Stability of attractive Bose-Einstein condensates. J. Stat. Phys. 101(3/4), 731–746 (2000). https://doi.org/10.1023/a:1026437923987
Zhang, S., Tong, J.-L., Wang, W.: A generalized (G\(^{\prime }\)/\({G}\))-expansion method for the mKdV equation with variable coefficients. Phys. Lett. A 372(13), 2254–2257 (2008). https://doi.org/10.1016/j.physleta.2007.11.026
Acknowledgements
This work is supported by the National Natural Science Foundation of China (11901111, 12171232); National Statistical Science Research Project of China (2021LY014); Jiangsu Postdoctoral Research Funding Program (2021K359C).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gu, Y., Aminakbari, N. New optical soliton solutions for the variable coefficients nonlinear Schrödinger equation. Opt Quant Electron 54, 255 (2022). https://doi.org/10.1007/s11082-022-03645-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-022-03645-4