Abstract
The generalized nonlinear Schrödinger equation with anti-cubic and cubic-quintic-septic nonlinearities is considered. A series of direct transformations is performed for the traveling wave reduction of the original equation. The implicit function method is used for nonlinear ordinary differential equation with some constraints on the parameters. Some analytical solutions are found in the form of periodic and solitary waves. Three conservation laws, corresponding to the generalized nonlinear Schrödinger equation are obtained and conserved quantities corresponding to the solution in implicit form are found
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Not applicable.
References
Abdel Kader, A.H., Abdel Latif, M.S., Zhou, Q.: Exact optical solitons in metamaterials with anti-cubic law of nonlinearity by Lie group method. Opt. Quant. Electron. 51(1), 30 (2019). https://doi.org/10.1007/s11082-019-1748-5
Arnous, A.H., Ekici, M., Biswas, A., Alshomrani, A.S., Belic, M.R.: Optical solitons having anti-cubic nonlinearity with two integration architectures. Chin. J. Phys. 60, 659–664 (2019). https://doi.org/10.1016/j.cjph.2019.06.006
Asjad, M.I., Ullah, N., Rehman, H.U., Inc, M.: Construction of optical solitons of magneto-optic waveguides with anti-cubic law nonlinearity. Opt. Quant. Electron. 53(11), 646 (2021). https://doi.org/10.1007/s11082-021-03288-x
Bölükbasi, H., Ekici, M., Biswas, A.: Optical solitons in birefringent fibers having anti-cubic nonlinearity with Jacobi’s elliptic function expansions. Opt. Quant. Electron. 53(10), 590 (2021). https://doi.org/10.1007/s11082-021-03237-8
Biswas, A., Ekici, M., Sonmezoglu, A., Zhou, Q., Alshomrani, A.S., Moshokoa, S.P., Belic, M.: Solitons in optical metamaterials with anti-cubic nonlinearity. Eur. Phys. J. Plus 133(5), 204 (2018a). https://doi.org/10.1140/epjp/i2018-12046-6
Biswas, A., Ekici, M., Sonmezoglu, A., Belic, M.R.: Optical solitons in birefringent fibers having anti-cubic nonlinearity with exp-function. Optik 186, 363–368 (2019). https://doi.org/10.1016/j.ijleo.2019.04.121
Biswas, A.: Conservation laws for optical solitons with anti-cubic and generalized anti-cubic nonlinearities. Optik 176, 198–201 (2019). https://doi.org/10.1016/j.ijleo.2018.09.074
Biswas, A., Mohamad Jawad, A.J., Zhou, Q.: Resonant optical solitons with anti-cubic nonlinearity. Optik 157, 525–531 (2018b). https://doi.org/10.1016/j.ijleo.2017.11.125
Biswas, A., Zhou, Q., Moshokoa, S.P., Triki, H., Belic, M., Alqahtani, R.T.: Resonant 1-soliton solution in anti-cubic nonlinear medium with perturbations. Optik 145, 14–17 (2017a). https://doi.org/10.1016/j.ijleo.2017.07.036
Biswas, A., Zhou, Q., Ullah, M.Z., Asma, M., Moshokoa, S.P., Belic, M.: Perturbation theory and optical soliton cooling with anti-cubic nonlinearity. Optik 142, 73–76 (2017b). https://doi.org/10.1016/j.ijleo.2017.05.060
Biswas, A., Zhou, Q., Ullah, M.Z., Triki, H., Moshokoa, S.P., Belic, M.: Optical soliton perturbation with anti-cubic nonlinearity by semi-inverse variational principle. Optik 143, 131–134 (2017c). https://doi.org/10.1016/j.ijleo.2017.06.087
Ekici, M., Mirzazadeh, M., Sonmezoglu, A., Ullah, M.Z., Zhou, Q., Triki, H., Moshokoa, S.P., Biswas, A.: Optical solitons with anti-cubic nonlinearity by extended trial equation method. Optik 136, 368–373 (2017). https://doi.org/10.1016/j.ijleo.2017.02.004
Ekici, M., Sonmezoglu, A., Zhou, Q., Moshokoa, S.P., Ullah, M.Z., Arnous, A.H., Biswas, A., Belic, M.: Analysis of optical solitons in nonlinear negative-indexed materials with anti-cubic nonlinearity. Opt. Quant. Electron. 50(2), 75 (2018). https://doi.org/10.1007/s11082-018-1341-3
Fedele, R., Schamel, H., Karpman, V., Shukla, P.K.: Envelope solitons of nonlinear Schrödinger equation with an anti-cubic nonlinearity. J. Phys. A Math. Gen. 36(4), 1169 (2003). https://doi.org/10.1088/0305-4470/36/4/322
Jawad, A.J.M., Mirzazadeh, M., Zhou, Q., Biswas, A.: Optical solitons with anti-cubic nonlinearity using three integration schemes. Superlattices Microstruct. 105, 1–10 (2017). https://doi.org/10.1016/j.spmi.2017.03.015
Kivshar, Y.S., Agrawal, G.P.: Optical solitons: from fibers to photonic crystals, pp. 1–540. Academic press, New York (2003). https://doi.org/10.1016/B978-0-12-410590-4.X5000-1
Kaplan, A.: Bistable solitons. Phys. Rev. Lett. 55(12), 1291 (1985). https://doi.org/10.1103/PhysRevLett.55.1291
Kumar, S., Biswas, A., Ekici, M., Zhou, Q., Moshokoa, S.P., Belic, M.R.: Optical solitons and other solutions with anti-cubic nonlinearity by Lie symmetry analysis and additional integration architectures. Optik 185, 30–38 (2019). https://doi.org/10.1016/j.ijleo.2019.03.080
Krishnan, E.V., Biswas, A., Zhou, Q., Babatin, M.M.: Optical solitons with anti-cubic nonlinearity by map** methods. Optik 170, 520–526 (2018). https://doi.org/10.1016/j.ijleo.2018.06.010
Kudryashov, N.A., Nifontov, D.R.: Conservation laws and hamiltonians of the mathematical model with unrestricted dispersion and polynomial nonlinearity. Chaos Solitons Fractals 175, 114076 (2023). https://doi.org/10.1016/j.chaos.2023.114076
Kudryashov, N.A.: First integrals and general solution of the traveling wave reduction for Schrödinger equation with anti-cubic nonlinearity. Optik 185, 665–671 (2019). https://doi.org/10.1016/j.ijleo.2019.03.167
Kudryashov, N.A.: highly dispersive optical solitons of an equation with arbitrary refractive index. Regul. Chaotic Dyn. 25(6), 537–543 (2020a). https://doi.org/10.1134/S1560354720060039
Kudryashov, N.A.: Highly dispersive optical solitons of equation with various polynomial nonlinearity law. Chaos Solitons Fractals 140, 110202 (2020b). https://doi.org/10.1016/j.chaos.2020.110202
Kudryashov, N.A.: Method for finding highly dispersive optical solitons of nonlinear differential equations. Optik 206, 163550 (2020c). https://doi.org/10.1016/j.ijleo.2019.163550
Kudryashov, N.A.: Optical solitons of the model with arbitrary refractive index. Optik 224, 165767 (2020d). https://doi.org/10.1016/j.ijleo.2020.165767
Kudryashov, N.A.: Traveling wave solutions of the generalized Gerdjikov–Ivanov equation. Optik 219, 165193 (2020e). https://doi.org/10.1016/j.ijleo.2020.165193
Kudryashov, N.A.: Method for finding optical solitons of generalized nonlinear Schrödinger equations. Optik 261, 169163 (2022a). https://doi.org/10.1016/j.ijleo.2022.169163
Kudryashov, N.A.: Optical solitons of the model with generalized anti-cubic nonlinearity. Optik 257, 168746 (2022b). https://doi.org/10.1016/j.ijleo.2022.168746
Malomed, B.A.: Soliton management in periodic systems. Springer, Boston (2006). https://doi.org/10.1007/0-387-29334-5
Messouber, A., Triki, H., Liu, Y., Biswas, A., Yildirim, Y., Alghamdi, A.A., Zhou, Q.: Chirped spatial solitons on a continuous-wave background in weak nonlocal media with polynomial law of nonlinearity. Phys. Lett. A 467, 128731 (2023). https://doi.org/10.1016/j.physleta.2023.128731
Ozisik, M., Secer, A., Bayram, M., Biswas, A., González-Gaxiola, O., Moraru, L., Moldovanu, S., Iticescu, C., Bibicu, D., Alghamdi, A.A.: Retrieval of optical solitons with anti-cubic nonlinearity. Mathematics 11(5), 1215 (2023). https://doi.org/10.3390/math11051215
Qiu, Y., Malomed, B.A., Mihalache, D., Zhu, X., Peng, J., He, Y.: Generation of stable multi-vortex clusters in a dissipative medium with anti-cubic nonlinearity. Phys. Lett. A 383(22), 2579–2583 (2019). https://doi.org/10.1016/j.physleta.2019.05.022
Sun, Y., Hu, Z., Triki, H., Mirzazadeh, M., Liu, W., Biswas, A., Zhou, Q.: Analytical study of three-soliton interactions with different phases in nonlinear optics. Nonlinear Dyn. 111(19), 18391–18400 (2023). https://doi.org/10.1007/s11071-023-08786-z
Triki, H., Bouguerra, A., Gao, X., Biswas, A., Yildirim, Y., Alshomrani, A.S.: Propagation dynamics of nonautonomous solitons in a temporally modulated cubic–quintic–septimal nonlinear medium. Eur. Phys. J. Plus 139(3), 287 (2024). https://doi.org/10.1140/epjp/s13360-024-05062-w
Triki, H., Kara, A.H., Biswas, A., Moshokoa, S.P., Belic, M.: Optical solitons and conservation laws with anti-cubic nonlinearity. Optik 127(24), 12056–12062 (2016). https://doi.org/10.1016/j.ijleo.2016.09.122
Triki, H., Porsezian, K., Dinda, P.T., Grelu, P.: Dark spatial solitary waves in a cubic-quintic-septimal nonlinear medium. Phys. Rev. A 95(2), 023837 (2017). https://doi.org/10.1103/PhysRevA.95.023837
Tang, J.-X., Su, X.: Traveling wave solutions, dynamic properties and chaotic behaviors of Schrödinger equation in magneto-optic waveguide with anti-cubic nonlinearity. Res. Phys. 54, 107106 (2023). https://doi.org/10.1016/j.rinp.2023.107106
Zayed, E.M., Alngar, M.E., 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 (2019a). https://doi.org/10.1016/j.ijleo.2018.09.064
Zayed, E.M., Alngar, M.E., El-Horbaty, M., Biswas, A., Ekici, M., Triki, H., Zhou, Q., Moshokoa, S.P., Belic, M.R.: Optical solitons having anti-cubic nonlinearity with strategically sound integration architectures. Optik 185, 57–70 (2019b). https://doi.org/10.1016/j.ijleo.2019.03.078
Zayed, E.M.E., Alngar, M.E.M., El-Horbaty, M.M., Biswas, A., Asma, M., Ekici, M., Alzahrani, A.K., Belic, M.R.: Solitons in magneto-optic waveguides with generalized anti-cubic nonlinearity. Optik 223, 165456 (2020). https://doi.org/10.1016/j.ijleo.2020.165456
Zhou, Q., Huang, Z., Sun, Y., Triki, H., Liu, W., Biswas, A.: Collision dynamics of three-solitons in an optical communication system with third-order dispersion and nonlinearity. Nonlinear Dyn. 111(6), 5757–5765 (2023). https://doi.org/10.1007/s11071-022-08138-3
Zhong, Y., Yu, K., Sun, Y., Triki, H., Zhou, Q.: Stability of solitons in Bose-Einstein condensates with cubic–quintic–septic nonlinearity and non-PT-symmetric complex potentials. Eur. Phys. J. Plus 139(2), 119 (2024). https://doi.org/10.1140/epjp/s13360-024-04930-9
Zhang, Q., Zhou, Y.: Bifurcations and obtained exact solutions of the optical soliton model in metamaterials dominated by anti-cubic nonlinearity. J. Appl Anal. Comput. 13(4), 1931–1971 (2023). https://doi.org/10.11948/20220289
Funding
This research was supported by Russian Science Foundation Grant No. 23-41-00070, https://rscf.ru/en/project/23-41-00070/.
Author information
Authors and Affiliations
Contributions
N.A. Kudryashov: Conceptualization, Supervision, Writing - review & editing. A.A. Kutukov: Formal analysis, Investigation, Writing - Sections 1–3, 6. D.R. Nifontov: Formal analysis, Investigation, Writing - Sections 4,5.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no Conflict of interest.
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
Kudryashov, N.A., Kutukov, A.A. & Nifontov, D.R. Analytical solutions and conservation laws of the generalized nonlinear Schrödinger equation with anti-cubic and cubic-quintic-septic nonlinearities. Opt Quant Electron 56, 1157 (2024). https://doi.org/10.1007/s11082-024-07092-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-024-07092-1