Abstract
This work focuses on the propagation behaviors and stabilities of solitons of a nonlinear Schrödinger equation with temporally inhomogeneous dispersion, cubic–quintic–septic–nonic nonlinearity, and group delay under Scarff II, Rosen–Morse, Harmonic, and periodic (x, t) plane rotated parity-time (\( \mathcal{P}\mathcal{T}\))-symmetric potentials, which governs the soliton wave transmission in a temporally inhomogeneous quasi-1D Bose-Einstein condensate system with the above effects. The novel analytical solutions of the bright solitons on zero or cw backgrounds and kink solitons under rotated \(\mathcal{P}\mathcal{T}\)-symmetric potentials in this system are presented for the first time. The simulation results of these three kinds of solitons show perfect consistency with the analytical ones. The bright solitons on zero or continuous-wave backgrounds show high noise immunity. While the anti-interference capability of kink soliton is connected with the form of the dispersion. These results would be advantageous for experimentally realizing stable propagation of localized nonlinear waves in highly nonlinear mediums.
Similar content being viewed by others
Data Availability
All data generated or analyzed during this study are included in this published article.
References
Chen, S., Shi, D., Yi, L.: Timing jitter of femtosecond solitons in single-mode optical fibers: a perturbation model. Phys. Rev. E 69, 046602 (2004)
Mollenauer, L.F., Stolen, R.H., Gordon, J.P.: Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys. Rev. Lett. 45, 1095 (1980)
Aitchison, J.S., Weiner, A.M., Silberberg, Y., Oliver, M.K., Jackel, J.L., Leaird, D.E., Vogel, E.M., Smith, P.W.E.: Observation of spatial optical solitons in a nonlinear glass waveguide. Opt. Lett. 15, 471–473 (1990)
Salin, F., Grangier, P., Roger, G., Brun, A.: Observation of high-order solitons directly produced by a femtosecond ring laser. Phys. Rev. Lett. 56, 1132 (1986)
Stegeman, G.I., Segev, M.: Optical spatial solitons and their interactions: universality and diversity. Science 286, 1518–1523 (1999)
Geng, K.-L., Zhu, B.-W., Cao, Q.-H., Dai, C.-Q., Wang, Y.-Y.: Nondegenerate soliton dynamics of nonlocal nonlinear Schrödinger equation. Nonlinear Dyn. 111, 16483–16496 (2023)
Kivshar, Y.S., Agrawal, G.P.: Optical Solitons: From Fibers to Photonic Crystals. Academic Press, London (2003)
Akhmediev, N.N., Ankiewicz, A., eds.: Dissipative Solitons: From Optics to Biology and Medicine. Vol. 751 of Lecture Notes in Physics, Springer (2008)
Burger, S., Bongs, K., Dettmer, S., Ertmer, W., Sengstock, K., Sanpera, A., Shlyapnikov, G.V., Lewenstein, M.: Dark solitons in Bose-Einstein condensates. Phys. Rev. Lett. 83, 5198–5201 (1999)
Denschlag, J., Simsarian, J.E., Feder, D.L., Clark, C.W., Collins, L.A., Cubizolles, J., Deng, L., Hagley, E.W., Helmerson, K., Reinhardt, W.P., Rolston, S.L., Schneider, B.I., Phillips, W.D.: Generating solitons by phase engineering of a Bose-Einstein condensate. Science 287, 97–100 (2000)
Khaykovich, L., Schreck, F., Ferrari, G., Bourdel, T., Cubizolles, J., Carr, L.D., Castin, Y., Salomon, C.: Formation of a matter-wave bright soliton. Science 296, 1290–1293 (2002)
Kartashov, Y.V., Vysloukh, V.A., Torner, L.: Surface gap solitons. Phys. Rev. Lett. 96, 073901 (2006)
Wang, D.S., Song, S.W., Xiong, B., Liu, W.M.: Quantized vortices in a rotating Bose-Einstein condensate with spatiotemporally modulated interaction. Phys. Rev. A 84, 053607 (2011)
Pitaevskii, L., Stringari, S.: Bose-Einstein Condensation. Oxford University Press, Oxford, England (2003)
Dalfovo, F., Giorgini, S., Pitaevskii, L.P., Stringari, S.: Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys. 71, 463–512 (1999)
Ruprecht, P.A., Holland, M.J., Burnett, K., Edwards, M.: Time-dependent solution of the nonlinear Schrodinger equation for Bose-condensed trapped neutral atoms. Phys. Rev. A 51, 4704 (1995)
Burger, S., Bongs, K., Dettmer, S., Ertmer, W., Sengstock, K., Sanpera, A., Shlyapnikov, G.V., Lewenstein, M.: Dark solitons in Bose-Einstein condensates. Phys. Rev. Lett. 83, 5198 (1999)
Inouye, S., Andrews, M.R., Stenger, J., Miesner, H.J., Stamper-Kurn, D.M., Ketterle, W.: Observation of Feshbach resonances in a Bose-Einstein condensate. Nature 392, 151–154 (1998)
Theis, M., Thalhammer, G., Winkler, K., Hellwig, M., Ruff, G., Grimm, R., Denschlag, J.H.: Tuning the scattering length with an optically induced Feshbach resonance. Phys. Rev. Lett. 93, 123001 (2004)
Malomed, B.A.: Soliton Management in Periodic Systems. Springer, New York (2006)
Abdullaev, F.K., Kamchatnov, A.M., Konotop, V.V., Brazhnyi, V.A.: Adiabatic dynamics of periodic waves in Bose-Einstein condensates with time dependent atomic scattering length. Phys. Rev. Lett. 90, 230402 (2003)
Rodas-Verde, M.I., Michinel, H., Pérez-García, V.M.: Controllable soliton emission from a Bose-Einstein condensate. Phys. Rev Lett. 95, 153903 (2005)
Rajendran, S., Muruganandam, P., Lakshmanan, M.: Bright and dark solitons in a quasi-1D Bose-Einstein condensates modelled by 1D GRoss-Pitaevskii equation with time-dependent parameters. Phys. D 239, 366–386 (2010)
Wang, D.S., Hu, X.H., Hu, J., Liu, W.M.: Quantized quasi-two-dimensional Bose-Einstein condensates with spatially modulated nonlinearity. Phys. Rev. A 81, 025604 (2010)
Zheng, Y., Meng, Y.J., Liu, Y.W.: Solitons in Gaussian potential with spatially modulated nonlinearity. Opt. Commun. 315, 63–68 (2014)
Loomba, S., Pal, R., Kumar, C.N.: Bright solitons of the nonautonomous cubic-quintic nonlinear Schrödinger equation with sign-reversal nonlinearity. Phys. Rev. A 92, 033811 (2015)
Triki, H., Choudhuri, A., Zhou, Q., Biswas, A., Alshomrani, S.A.: Nonautonomous matter wave bright solitons in a quasi-1D Bose-Einstein condensate system with contact repulsion and dipole-dipole attraction. Appl. Math. Comput. 371, 124951 (2020)
Liu, X., Zeng, J.: One-dimensional purely Lee-Huang-Yang fluids dominated by quantum fluctuations in two-component Bose-Einstein condensates. Chaos Solitons Fractals 160, 112240 (2022)
Zhou, Z., Shi, Y., Ye, F., Chen, H., Tang, S., Deng, H., Zhong, H.: Self-bound states induced by the Lee-Huang-Yang effect in non-\( \cal{PT} \)-symmetric complex potentials. Nonlinear Dyn. 110, 3769–3778 (2022)
Ding, C.-C., Zhou, Q., Xu, S.-L., Sun, Y.Z., Liu, W.-J., Mihalache, D., Malomed, B.A.: Controlled nonautonomous matter-wave solitons in spinor Bose-Einstein condensates with spatiotemporal modulation. Chaos Solitons Fractals 169, 113247 (2023)
Belmonte-Beitia, J., Pérez-García, V.M., Vekslerchik, V., Konotop, V.V.: Localized nonlinear waves in systems with time- and space-modulated nonlinearities. Phys. Rev. Lett. 100, 164102 (2008)
Pal, R., Loomba, S., Kumar, C.N., Milovic, D., Maluckov, A.: Matter wave soliton solutions for driven Gross-Pitaevskii equation with distributed coefficients. Ann. Phys. 401, 116–129 (2019)
Chen, Y.X., Xiao, X.: Vector soliton pairs for a coupled nonautonomous NLS model with partially nonlocal coupled nonlinearities under the external potentials. Nonlinear Dyn. 109, 2003–2012 (2022)
Lin, Z., Ramezani, H., Eichelkraut, T., Kottos, T., Cao, H., Christodoulides, D.N.: Unidirectional invisibility induced by \(\cal{PT} \) -symmetric periodic structures. Phys. Rev. Lett. 106, 213901 (2011)
Wu, M., Liu, F., Zhao, D., Wang, Y.: Unidirectional invisibility in PT-symmetric cantor photonic crystals. Crystals 12, 199 (2022)
Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Musslimani, Z.H.: Beam Dynamics in \(\cal{PT} \) Symmetric Optical Lattices. Phys. Rev. Lett. 100, 103904 (2008)
Rüter, C.E., Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Segev, M., Kip, D.: Observation of parity-time symmetry in optics. Nat. Phys. 6, 192–195 (2010)
Zheng, M.C., Christodoulides, D.N., Fleischmann, R., Kottos, T.: \(\cal{PT} \) optical lattices and universality in beam dynamics. Phys. Rev. A 82, 010103(R) (2010)
Guo, A., Salamo, G.J., Duchesne, D., Morandotti, R., Volatier-Ravat, M., Aimez, V., Siviloglou, G.A., Christodoulides, D.N.: Observation of \(\cal{PT} \)-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009)
Sheng, J., Miri, M.A., Christodoulides, D.N., Xiao, M.: \( \cal{PT} \)-symmetric optical potentials in a coherent atomic medium. Phys. Rev. A 88, 041803(R) (2013)
Regensburger, A., Bersch, C., Miri, M.A., Onishchukov, G., Christodoulides, D.N., Peschel, U.: Parity-time synthetic photonic lattices. Nature 488, 167–171 (2012)
Regensburger, A., Miri, M.A., Bersch, C., Näger, G., Onishchukov, J., Christodoulides, D.N., Peschel, U.: Observation of defect states in \(\cal{PT} \)-symmetric optical lattices. Phys. Rev. Lett. 110, 223902 (2013)
Benisty, H., Degiron, A., Lupu, A., de Lustrac, A., Ché nais, S., Forget, S., Besbes, M., Barbillon, G., Bruyant, A., Blaize, S., L érondel, G.: Implementation of \(mathcal PT \) symmetric devices using plasmonics: principle and applications. Opt. Express. 19, 18004–18019 (2011)
Bittner, S., Dietz, B., Günther, U., Harney, H.L., Miski-Oglu, M., Richter, A., Schäfer, F.: \(\cal{PT} \) symmetry and spontaneous symmetry breaking in a Microwave Billiard. Phys. Rev. Lett. 108, 024101 (2012)
Bender, C.M., Berntson, B.K., Parker, D., Samuel, E.: Observation of \(\cal{PT} \) phase transition in a simple mechanical system. Am. J. Phys. 81, 173–179 (2013)
Feng, L., Xu, Y.L., Fegadolli, W.S., Lu, M.H., Oliveira, J.E.B., Almeida, V.R., Chen, Y.F., Scherer, A.: Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies. Nat. Mater. 12, 108–113 (2013)
Musslimani, Z.H., Makris, K.G., El-Ganainy, R., Christodoulide, D.N.: Optical Solitons in \(\cal{PT} \) Periodic Potentials. Phys. Rev. Lett. 100, 030402 (2008)
Zhou, Q., Triki, H., Xu, J., Zeng, Z., Liu, W., Biswas, A.: Perturbation of chirped localized waves in a dual-power law nonlinear medium. Chaos Solitons Fractals 160, 112198 (2022)
Chen, Y.X., Xu, F.Q.: Higher dimensional Gaussian-type solitons of nonlinear Schrödinger equation with cubic and power-law nonlinearities in \(\cal{PT} \)-symmetric potentials. Plos One 9, e115935 (2014)
Wang, Y.Y., Dai, C.Q., Wang, X.G.: Stable localized spatial solitons in \(\cal{PT} \)-symmetric potentials with power-law nonlinearity. Nonlinear Dyn. 77, 1323–1330 (2014)
Chen, Y.X.: One-dimensional optical solitons in cubic-quintic-septimal media with \(\cal{PT} \)-symmetric potentials. Nonlinear Dyn. 87, 1629–1635 (2017)
Chen, Y.X., Xu, F.Q., Hu, Y.L.: Bright spatial solitons in quintic-septimal nonlinear media with two families of \(\cal{PT} \) -symmetric potentials. Eur. Phys. J. Plus 132, 533 (2017)
Chow, K.W., Rogers, C.: Localized and periodic wave patterns for a nonic nonlinear Schrödinger equation. Phys. Lett. A 377, 2546 (2013)
Triki, H., Porsezian, K., Choudhuri, A., Dinda, P.T.: Chirped solitary pulses for a nonic nonlinear Schrödinger equation on a continuous-wave background. Phys. Rev. A 93, 063810 (2016)
Loriot, V., Hertz, E., Faucher, O., Lavorel, B.: Measurement of high order Kerr refractive index of major air components. Opt. Express 17, 13429–13434 (2009)
Novoa, D., Michinel, H., Tommasini, D.: Fermionic light in common optical media. Phys. Rev. Lett. 105, 203904 (2010)
Manikandan, K., Aravinthan, D., Sudharsan, J.B., Reddy, S.R.R.: Soliton and rogue wave solutions of the space-time fractional nonlinear Schrödinger equation with \(\cal{PT} \)-symmetric and time-dependent potentials. Optik 266, 169594 (2022)
Midya, B., Roychoudhury, R.: Nonlinear localized modes in \(\cal{PT} \)-symmetric Rosen-Morse potential wells. Phys. Rev. A 87, 045803 (2013)
Jisha, C.P., Devassy, L., Alberucci, A., Kuriakose, V.C.: Influence of the imaginary component of the photonic potential on the properties of solitons in \(\cal{PT} \)-symmetric systems. Phy. Rev. A 90, 043855 (2014)
Zezyulin, D.A., Konotop, V.V.: Nonlinear modes in the harmonic \(\cal{PT} \)-symmetric potential. Phys. Rev. A 85, 043840 (2012)
Suneera, T.P., Subha, P.A.: Single-hump and double-hump solitons in a symmetric complex potential. Waves Random Complex Media 20, 241–254 (2016)
Triki, H., Biswas, A., Milović, D., Belić, M.: Chirped femtosecond pulses in the higher-order nonlinear Schrödinger equation with non-Kerr nonlinear terms and cubic-quintic-septic nonlinearities. Opt. Commun. 366, 362–369 (2016)
Mansouri, F., Aouadi, S., Triki, H., Sun, Y., Yıldırım, Y., Biswas, A., Alshehri, H.M., Zhou, Q.: Chirped localized pulses in a highly nonlinear optical fiber with quintic non-Kerr nonlinearities. Results Phys. 43, 106040 (2022)
Wang, R.-R., Wang, Y.-Y., Dai, C.-Q.: Influence of higher-order nonlinear effects on optical solitons of the complex Swift-Hohenberg model in the mode-locked fiber laser. Opt. Laser Technol. 152, 108103 (2022)
Funding
This work was supported by the National Natural Science Foundation of China (Grant No. 11975172).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no conflicts of interest.
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
Zhong, Y., Triki, H. & Zhou, Q. Bright and kink solitons of time-modulated cubic–quintic–septic–nonic nonlinear Schrödinger equation under space-time rotated \(\mathcal{P}\mathcal{T}\)-symmetric potentials. Nonlinear Dyn 112, 1349–1364 (2024). https://doi.org/10.1007/s11071-023-09116-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-023-09116-z