Abstract
The M-shaped solitons—a special type of solitons—have been reported in many models, while most of them are not focused on the commonly used nonlinear Schrödinger equation (NLSE), which is widely employed to describe the propagation of laser beams in nonlinear media or the evolution of Bose–Einstein condensates. We demonstrate that families of M-shaped solitons can be generated in the NLSE with cubic (Kerr) nonlinear media and the help of a composite linear potential. It should be mentioned that these M-shaped solitons are completely stable for all values of the parameters of the model, a result obtained by the linear stability analysis and verified by the direct numerical simulations. The M-shaped solitons quickly disperse if the linear potential is taken away. We also report robust propagations of breather-like solitons in our model, when we use a specific soliton modulation procedure in the propagation of input M-shaped waveforms. The self-adaptive processes for M-shaped solitons are also investigated, and we show that these solitons will turn into irregular oscillations if the parameters of the external potential suffer a sudden variation. Interestingly, the solitons will keep their M-shaped waveforms and experience regular propagations if the parameters of the potential change gradually.
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The datasets generated during and analyzed during the current study are available from the corresponding author on reasonable request.
References
Kivshar, Y.S., Malomed, B.A.: Dynamics of solitons in nearly integrable systems. Rev. Mod. Phys. 61, 763–915 (1989)
Malomed, B.A., Mihalache, D., Wise, F., Torner, L.: Spatiotemporal optical solitons. J. Opt. B 7, R53–R72 (2005)
Kartashov, Y.V., Malomed, B.A., Torner, L.: Solitons in nonlinear lattices. Rev. Mod. Phys. 83, 247–305 (2011)
Leblond, H., Mihalache, D.: Models of few optical cycle solitons beyond the slowly varying envelope approximation. Phys. Rep. 523, 61–126 (2013)
Konotop, V.V., Yang, J., Zezyulin, D.A.: Nonlinear waves in \(\cal{PT} \)-symmetric system. Rev. Mod. Phys. 88, 035002 (2016)
Malomed, B.A., Mihalache, D.: Nonlinear waves in optical and matter-wave media: a topical survey of recent theoretical and experimental results. Rom. J. Phys. 64, 106 (2019)
Kartashov, Y.V., Astrakharchik, G.E., Malomed, B.A., Torner, L.: Frontiers in multidimensional self-trapping of nonlinear fields and matter. Nat. Rev. Phys. 1, 185–197 (2019)
Mihalache, D.: Localized structures in optical and matter-wave media: a selection of recent studies. Rom. Rep. Phys. 73, 403 (2021)
Akhmediev, N.N., Ankiewicz, A., Soto-Crespo, J.M.: Multisoliton solutions of the complex Ginzburg–Landau equation. Phys. Rev. Lett. 79, 4047–4051 (1997)
He, Y., Mihalache, D.: Lattice solitons in optical media described by the complex Ginzburg–Landau model with \(\cal{PT} \)-symmetric periodic potentials. Phys. Rev. A 87, 013812 (2013)
Leblond, H., Niang, A., Amrani, F., Salhi, M., Sanchez, F.: Motion of solitons of the complex Ginzburg–Landau equation: the effect of an external frequency-shifted source. Phys. Rev. A 88, 033809 (2013)
Zanga, D., Fewo, S.I., Tabi, C.B., Kofané, T.C.: Generation of dissipative solitons in a doped optical fiber modeled by the higher-order dispersive cubic-quintic-septic complex Ginzburg-Landau equation. Phys. Rev. A 105, 023502 (2022)
Chen, S., Ye, Y., Soto-Crespo, J.M., Grelu, P., Baronio, F.: Peregrine solitons beyond the threefold limit and their two-soliton interactions. Phys. Rev. Lett. 121, 104101 (2018)
Lashkin, V.M.: Perturbation theory for solitons of the Fokas–Lenells equation: inverse scattering transform approach. Phys. Rev. E 103, 042203 (2021)
Dieu-donne, G., Hubert, M.B., Seadawy, A., Etienne, T., Betchewe, G., Doka, S.Y.: Chirped soliton solutions of Fokas–Lenells equation with perturbation terms and the effect of spatio-temporal dispersion in the modulational instability analysis. Eur. Phys. J. Plus 135, 212 (2020)
Dieu-donne, G., Tiofack, C.G.L., Seadawy, A., Hubert, M.B., Betchewe, G., Serge, D.Y.: Propagation of W-shaped, M-shaped and other exotic optical solitons in the perturbed Fokas–Lenells equation. Eur. Phys. J. Plus 135, 371 (2020)
Rao, J., He, J., Kanna, T., Mihalache, D.: Nonlocal M-component nonlinear Schrödinger equations: Bright solitons, energy-sharing collisions, and positons. Phys. Rev. E 102, 032201 (2020)
Rao, J., Kanna, T., Sakkaravarthi, K., He, J.: Multiple double-pole bright-bright and bright-dark solitons and energy-exchanging collision in the M-component nonlinear Schrödinger equations. Phys. Rev. E 103, 062214 (2021)
Zeng, L., Shi, J., Belić, M.R., Mihalache, D., Chen, J., Lu, X., Cai, Y., Li, J.: Multipole solitons in saturable nonlinear lattices. Nonlinear Dyn. 111, 3665–3678 (2023)
Zeng, L., Belić, M.R., Mihalache, D., Xiang, D., Wang, Q., Yang, J., Zhu, X.: Triangular bright solitons in nonlinear optics and Bose–Einstein condensates. Opt. Express 31, 9563–9578 (2023)
Zhang, Y., Qiu, D.Q., Cheng, Y., He, J.: Rational solution of the nonlocal nonlinear Schrödinger equation and its application in optics. Rom. J. Phys. 62, 108 (2017)
Kartashov, Y.V., Skryabin, D.V.: Two-dimensional topological polariton laser. Phys. Rev. Lett. 122, 083902 (2019)
Zeng, L., Konotop, V.V., Lu, X., Cai, Y., Zhu, Q., Li, J.: Localized modes and dark solitons sustained by nonlinear defects. Opt. Lett. 46, 2216–2219 (2021)
Kartashov, Y.V., Ye, F., Konotop, V.V., Torner, L.: Multifrequency solitons in commensurate-incommensurate photonic Moiré lattices. Phys. Rev. Lett. 127, 163902 (2021)
Kartashov, Y.V., Zezyulin, D.A.: Stable multiring and rotating solitons in two-dimensional spin-orbit-coupled Bose–Einstein condensates with a radially periodic potential. Phys. Rev. Lett. 122, 123201 (2019)
Kartashov, Y.V., Konotop, V.V.: Stable nonlinear modes sustained by gauge fields. Phys. Rev. Lett. 125, 054101 (2020)
Zeng, L., Zhu, Y., Malomed, B.A., Mihalache, D., Wang, Q., Long, H., Cai, Y., Lu, X., Li, J.: Quadratic fractional solitons. Chaos Solitons Fract. 154, 111586 (2022)
Zhu, X., Xiang, D., Zeng, L.: Fundamental and multipole gap solitons in spin-orbit-coupled Bose–Einstein condensates with parity-time-symmetric Zeeman lattices. Chaos Solitons Fract. 169, 113317 (2023)
Zeng, L., Zhu, X., Belić, M.R., Mihalache, D., Shi, J., Chen, J.: Multiple-peak and multiple-ring solitons in the nonlinear Schrödinger equation with inhomogeneous self-defocusing nonlinearity. Nonlinear Dyn. 111, 5671–5680 (2023)
Wang, P., Zheng, Y., Chen, X., Huang, C., Kartashov, Y.V., Torner, L., Konotop, V.V., Ye, F.: Localization and delocalization of light in photonic Moiré lattices. Nature 577, 42–46 (2020)
Fu, Q., Wang, P., Huang, C., Kartashov, Y.V., Torner, L., Konotop, V.V., Ye, F.: Optical soliton formation controlled by angle twisting in photonic Moiré lattices. Nat. Photon. 14, 663–668 (2020)
Farolfi, A., Trypogeorgos, D., Mordini, C., Lamporesi, G., Ferrari, G.: Observation of magnetic solitons in two-component Bose–Einstein condensates. Phys. Rev. Lett. 125, 030401 (2020)
Lannig, S., Schmied, C., Prüfer, M., Kunkel, P., Strohmaier, R., Strobel, H., Gasenzer, T., Kevrekidis, P.G., Oberthaler, M.K.: Collisions of three-component vector solitons in Bose–Einstein condensates. Phys. Rev. Lett. 125, 170401 (2020)
Jiang, Y., Rao, J., Mihalache, D., He, J., Cheng, Y.: Rogue breathers and rogue lumps on a background of dark line solitons for the Maccari system. Commun. Nonlinear Sci. Numer. Simul. 102, 105943 (2021)
Luo, D., Jin, Y., Nguyen, J.H.V., Malomed, B.A., Marchukov, O.V., Yurovsky, V.A., Dunjko, V., Olshanii, M., Hulet, R.G.: Creation and characterization of matter-wave breathers. Phys. Rev. Lett. 125, 183902 (2020)
Yuan, F., Cheng, Y., He, J.: Degeneration of breathers in the Kadomtsev–Petviashvili I equation. Commun. Nonlinear Sci. Numer. Simul. 83, 105027 (2020)
Kibler, B., Fatome, J., Finot, C., Millot, G., Dias, F., Genty, G., Akhmediev, N., Dudley, J.M.: The Peregrine soliton in nonlinear fibre optics. Nat. Photon. 6, 790–795 (2010)
Borovkova, O.V., Kartashov, Y.V., Torner, L., Malomed, B.A.: Bright solitons from defocusing nonlinearities. Phys. Rev. E 84, 035602 (2011)
Lobanov, V.E., Borovkova, O.V., Kartashov, Y.V., Malomed, B.A., Torner, L.: Stable bright and vortex solitons in photonic crystal fibers with inhomogeneous defocusing nonlinearity. Opt. Lett. 37, 1799–1801 (2012)
Zeng, L., Belić, M.R., Mihalache, D., Li, J., Xiang, D., Zeng, X., Zhu, X.: Solitons in a coupled system of fractional nonlinear Schrödinger equations. Physica D 456, 133924 (2023)
Driben, R., Dror, N., Malomed, B.A., Meier, T.: Multipoles and vortex multiplets in multidimensional media with inhomogeneous defocusing nonlinearity. New J. Phys. 17, 083043 (2015)
Zeng, L., Zeng, J.: One-dimensional solitons in fractional Schrödinger equation with a spatially periodical modulated nonlinearity: nonlinear lattice. Opt. Lett. 44, 2661–2664 (2019)
Driben, R., Kartashov, Y.V., Malomed, B.A., Meier, T., Torner, L.: Three-dimensional hybrid vortex solitons. New J. Phys. 16, 063035 (2014)
Lobanov, V.E., Kartashov, Y.V., Konotop, V.V.: Fundamental, multipole, and half-vortex gap solitons in spin-orbit coupled Bose–Einstein condensates. Phys. Rev. Lett. 112, 180403 (2014)
Shi, J., Zeng, L., Chen, J.: Two-dimensional localized modes in saturable quintic nonlinear lattices. Nonlinear Dyn. 111, 13415–13424 (2023)
Zeng, L., Belić, M.R., Mihalache, D., Shi, J., Li, J., Li, S., Lu, X., Cai, Y., Li, J.: Families of gap solitons and their complexes in media with saturable nonlinearity and fractional diffraction. Nonlinear Dyn. 108, 1671–1680 (2022)
Zeng, L., Shi, J., Li, J., Li, J., Wang, Q.: Dark soliton families in quintic nonlinear lattices. Opt. Express 30, 42504–42511 (2022)
Kartashov, Y.V., Malomed, B.A., Vysloukh, V.A., Belić, M.R., Torner, L.: Rotating vortex clusters in media with inhomogeneous defocusing nonlinearity. Opt. Lett. 42, 446–449 (2017)
Zeng, L., Zeng, J.: Modulated solitons, soliton and vortex clusters in purely nonlinear defocusing media. Ann. Phys. 421, 168284 (2020)
Kartashov, Y.V., Malomed, B.A., Shnir, Y., Torner, L.: Twisted toroidal vortex solitons in inhomogeneous media with repulsive nonlinearity. Phys. Rev. Lett. 113, 264101 (2014)
Driben, R., Kartashov, Y.V., Malomed, B.A., Meier, T., Torner, L.: Soliton gyroscopes in media with spatially growing repulsive nonlinearity. Phys. Rev. Lett. 112, 020404 (2014)
Zeng, L., Zeng, J., Kartashov, Y.V., Malomed, B.A.: Purely Kerr nonlinear model admitting flat-top solitons. Opt. Lett. 44, 1206–1209 (2019)
Zeng, L., Zeng, J.: Gaussian-like and flat-top solitons of atoms with spatially modulated repulsive interactions. J. Opt. Soc. Am. B 36, 2278–2284 (2019)
Zeng, L., Malomed, B.A., Mihalache, D., Cai, Y., Lu, X., Zhu, Q., Li, J.: Flat-floor bubbles, dark solitons, and vortices stabilized by inhomogeneous nonlinear media. Nonlinear Dyn. 106, 815–830 (2021)
Chen, S., Mihalache, D., Jin, K., Li, J., Rao, J.: Bright solitons in the space-shifted \(\cal{PT} \)-symmetric nonlocal nonlinear Schrödinger equation. Rom. Rep. Phys. 75, 108 (2023)
Ahmed, I., Seadawy, A.R., Lu, D.: M-shaped rational solitons and their interaction with kink waves in the Fokas-Lenells equation. Phys. Scripta 94, 055205 (2019)
Rizvi, S.T.R., Seadawy, A.R., Ashraf, M.A., Younis, M., Khaliq, A., Baleanu, D.: Rogue, multi-wave, homoclinic breather, M-shaped rational and periodic-kink solutions for a nonlinear model describing vibrations. Results Phys. 29, 104654 (2021)
Rizvi, S.T.R., Seadawy, A.R., Ashraf, M.A., Bashir, A., Younis, M., Baleanu, D.: Multi-wave, homoclinic breather, M-shaped rational and other solitary wave solutions for coupled-Higgs equation. Eur. Phys. J. Spec. Top. 230, 3519–3532 (2021)
Essama, B.G.O., Bisse, J.T.N., Essiane, S.N., Atangana, J.: M-shaped and other exotic solitons generated by cubic-quintic saturable nonlinearities in a nonlinear electrical transmission network with higher-order dispersion effects. Chaos Solitons Fract. 161, 112320 (2022)
Zhao, L.-C., Li, S.-C., Ling, L.: Rational W-shaped solitons on a continuous-wave background in the Sasa–Satsuma equation. Phys. Rev. E 89, 023210 (2014)
Zhao, L.-C., Li, S.-C., Ling, L.: W-shaped solitons generated from a weak modulation in the Sasa–Satsuma equation. Phys. Rev. E 93, 032215 (2016)
Zeng, L., Malomed, B.A., Mihalache, D., Cai, Y., Lu, X., Zhu, Q., Li, J.: Bubbles and W-shaped solitons in Kerr media with fractional diffraction. Nonlinear Dyn. 104, 4253–4264 (2021)
Mahalingam, A., Rajan, M.S.M.: Influence of generalized external potentials on nonlinear tunneling of nonautonomous solitons: soliton management. Opt. Fiber Technol. 25, 44–50 (2015)
Vijayalekshmi, S., Mahalingam, A., Uthayakumar, A., Rajan, M.S.M.: Multi-soliton propagation in generalized inhomogeneous NLS equation with symmetric potentials. Optik 181, 948–955 (2019)
Mani Rajan, M.S.: Dynamics of optical soliton in a tapered erbium-doped fiber under periodic distributed amplification system. Nonlinear Dyn. 85, 599–606 (2016)
Rajan, M.S.M.: Transition from bird to butterfly shaped nonautonomous soliton and soliton switching in erbium doped resonant fiber. Phys. Scr. 95, 105203 (2020)
Veni, S.S., Rajan, M.S.M.: Attosecond soliton switching through the interactions of two and three solitons in an inhomogeneous fiber. Chaos Solitons Fract. 152, 111390 (2021)
Houwe, A., Inc, M., Doka, S.Y.: Breather-like soliton, M-shaped profile, W-shaped profile, and modulation instability conducted by self-frequency shift of the nonlinear Schrödinger equation. J. Comput. Electron. 21, 733–743 (2022)
Matthews, M.R., Anderson, B.P., Haljan, P.C., Hall, D.S., Wieman, C.E., Cornell, E.A.: Vortices in a Bose–Einstein condensate. Phys. Rev. Lett. 83, 2498–2501 (1999)
Kanamoto, R., Wright, E.M., Meystre, P.: Quantum dynamics of Raman-coupled Bose–Einstein condensates using Laguerre–Gaussian beams. Phys. Rev. A 75, 063623 (2007)
Lim, Y., Goo, J., Kwak, H., Shin, Y.: Large-area \(^{87}\)Rb Bose–Einstein condensate in a clipped-Gaussian optical dipole trap. Phys. Rev. A 103, 063319 (2021)
Szameit, A., Kartashov, Y.V., Dreisow, F., Pertsch, T., Nolte, S., Tünnermann, A., Torner, L.: Observation of two-dimensional surface solitons in asymmetric waveguide arrays. Phys. Rev. Lett. 98, 173903 (2007)
Kirsch, M.S., Zhang, Y., Kremer, M., Maczewsky, L.J., Ivanov, S.K., Kartashov, Y.V., Torner, L., Bauer, D., Szameit, A., Heinrich, M.: Nonlinear second-order photonic topological insulators. Nat. Phys. 17, 995–1000 (2021)
Vakhitov, M., Kolokolov, A.: Stationary solutions of the wave equation in a medium with nonlinearity saturation. Radiophys. Quantum Electron. 16, 783–789 (1973)
Sakaguchi, H., Malomed, B.A.: Solitons in combined linear and nonlinear lattice potentials. Phys. Rev. A 81, 013624 (2010)
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Funding for this study was obtained from the National Natural Science Foundation of China (62205224, 12071304); the Guangdong Basic and Applied Basic Research Foundation (2023A1515010865); the Shenzhen Natural Science Fund (the Stable Support Plan Program 20220809163103001); and the Romanian Ministry of Research, Innovation, and Digitization (PN 23210101/2023).
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Zeng, L., Mihalache, D., Zhu, X. et al. M-shaped solitons in cubic nonlinear media with a composite linear potential. Nonlinear Dyn 112, 3811–3822 (2024). https://doi.org/10.1007/s11071-023-09263-3
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DOI: https://doi.org/10.1007/s11071-023-09263-3