Spontaneous symmetry breaking and ghost states supported by the fractional \(\mathcal{PT}\)-symmetric saturable nonlinear Schrödinger equation. (English) Zbl 07880548
References:
| [1] | Bender, C. M.; Boettcher, S., Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett., 80, 5243, 1998 · Zbl 0947.81018 · doi:10.1103/PhysRevLett.80.5243 |
| [2] | Ahmed, Z., Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential, Phys. Lett. A, 282, 343, 2001 · Zbl 0984.81043 · doi:10.1016/S0375-9601(01)00218-3 |
| [3] | Dorey, P.; Dunning, C.; Tateo, R., Spectral equivalences, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics, J. Phys. A, 34, 5679, 2001 · Zbl 0982.81021 · doi:10.1088/0305-4470/34/28/305 |
| [4] | Bender, C. M., Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys., 70, 947, 2007 · doi:10.1088/0034-4885/70/6/R03 |
| [5] | Bender, C. M., Rigorous backbone of PT-symmetric quantum mechanics, J. Phys. A, 49, 401002, 2016 · Zbl 1349.81109 · doi:10.1088/1751-8113/49/40/401002 |
| [6] | Longhi, S., Bloch oscillations in complex crystals with PT symmetry, Phys. Rev. Lett., 103, 123601, 2009 · doi:10.1103/PhysRevLett.103.123601 |
| [7] | Midya, B.; Roy, B.; Roychoudhury, R., A note on the PT invariant periodic potential \(V(x)=4\cos^2x+4iV_0\sin2x\), Phys. Lett. A, 374, 2605, 2010 · Zbl 1237.81071 · doi:10.1016/j.physleta.2010.04.046 |
| [8] | Musslimani, Z.; Makris, K. G.; El-Ganainy, R.; Christodoulides, D. N., Optical solitons in PT periodic potentials, Phys. Rev. Lett., 100, 030402, 2008 · doi:10.1103/PhysRevLett.100.030402 |
| [9] | Dissipative Solitons, edited by N. Akhmediev and A. Ankiewicz (Springer, Berlin, 2005). · Zbl 1061.35003 |
| [10] | Abdullaev, F. K.; Konotop, V. V.; Salerno, M.; Yulin, A. V., Dissipative periodic waves, solitons, and breathers of the nonlinear Schrödinger equation with complex potentials, Phys. Rev. E, 82, 056606, 2010 · doi:10.1103/PhysRevE.82.056606 |
| [11] | Abdullaev, F. K.; Kartashov, Y. V.; Konotop, V. V.; Zezyulin, D. A., Solitons in PT-symmetric nonlinear lattices, Phys. Rev. A, 83, 041805(R), 2011 · doi:10.1103/PhysRevA.83.041805 |
| [12] | Zezyulin, D. A.; Konotop, V. V., Nonlinear modes in finite-dimensional PT-symmetric systems, Phys. Rev. Lett., 108, 213906, 2012 · doi:10.1103/PhysRevLett.108.213906 |
| [13] | Nixon, S.; Zhu, Y.; Yang, J., Nonlinear dynamics of wave packets in parity-time-symmetric optical lattices near the phase transition point, Opt. Lett., 37, 4874, 2012 · doi:10.1364/OL.37.004874 |
| [14] | Achilleos, V.; Kevrekidis, P.; Frantzeskakis, D.; Carretero-Gonzalez, R., Dark solitons and vortices in PT-symmetric nonlinear media: From spontaneous symmetry breaking to nonlinear PT phase transitions, Phys. Rev. A, 86, 013808, 2012 · doi:10.1103/PhysRevA.86.013808 |
| [15] | Zezyulin, D. A.; Konotop, V. V., Nonlinear modes in the harmonic PT-symmetric potential, Phys. Rev. A, 85, 043840, 2012 · doi:10.1103/PhysRevA.85.043840 |
| [16] | Bludov, Y. V.; Konotop, V. V.; Malomed, B. A., Stable dark solitons in PT-symmetric dual-core waveguides, Phys. Rev. A, 87, 013816, 2013 · doi:10.1103/PhysRevA.87.013816 |
| [17] | Yan, Z.; Wen, Z.; Hang, C., Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized parity-time-symmetric Scarf-II potentials, Phys. Rev. E, 92, 022913, 2015 · doi:10.1103/PhysRevE.92.022913 |
| [18] | 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, 023821, 2015 · doi:10.1103/PhysRevA.92.023821 |
| [19] | Konotop, V. V.; Yang, J.; Zezyulin, D. A., Nonlinear waves in PT-symmetric systems, Rev. Mod. Phys., 88, 035002, 2016 · doi:10.1103/RevModPhys.88.035002 |
| [20] | Zhan, K.; Tian, H.; Li, X.; Xu, X.; Jiao, Z.; Jia, Y., Solitons in PT-symmetric periodic systems with the logarithmically saturable nonlinearity, Sci. Rep., 6, 32990, 2016 · doi:10.1038/srep32990 |
| [21] | Yan, Z.; Chen, Y., The nonlinear Schrödinger equation with generalized nonlinearities and PT-symmetric potentials: Stable solitons, interactions, and excitations, Chaos, 27, 073114, 2017 · Zbl 1390.35344 · doi:10.1063/1.4995363 |
| [22] | Chen, Y.; Yan, Z.; Mihalache, D., Stable flat-top solitons and peakons in the PT-symmetric \(\delta \)-signum potentials and nonlinear media, Chaos, 29, 083108, 2019 · Zbl 1428.35496 · doi:10.1063/1.5100294 |
| [23] | Wang, L.; Malomed, B. A.; Yan, Z., Attraction centers and parity-time-symmetric delta-functional dipoles in critical and supercritical self-focusing media, Phys. Rev. E, 99, 052206, 2019 · doi:10.1103/PhysRevE.99.052206 |
| [24] | Chen, Y.; Song, J.; Li, X.; Yan, Z., Stability and modulation of optical peakons in self-focusing/defocusing Kerr nonlinear media with PT-\( \delta \)-hyperbolic-function potentials, Chaos, 32, 023122, 2022 · Zbl 1542.35370 · doi:10.1063/5.0080485 |
| [25] | Zhong, M.; Chen, Y.; Yan, Z.; Tian, S. F., Formation, stability, and adiabatic excitation of peakons and double-hump solitons in parity-time-symmetric Dirac-\( \delta(x)\)-Scarf-II optical potentials, Phys. Rev. E, 105, 014204, 2022 · doi:10.1103/PhysRevE.105.014204 |
| [26] | Song, J.; Zhou, Z.; Weng, W.; Yan, Z., PT-symmetric peakon solutions in self-focusing/defocusing power-law nonlinear media: Stability, interactions and adiabatic excitations, Phys. D, 435, 133266, 2022 · Zbl 1490.35453 · doi:10.1016/j.physd.2022.133266 |
| [27] | Burlak, G.; Chen, Z.; Malomed, B. A., Solitons in PT-symmetric systems with spin-orbit coupling and critical nonlinearity, Commun. Nonlinear Sci. Numer. Simulat., 109, 106282, 2022 · Zbl 07840925 · doi:10.1016/j.cnsns.2022.106282 |
| [28] | Guo, A.; Salamo, G.; Duchesne, D.; Morandotti, R.; Volatier-Ravat, M.; Aimez, V.; Siviloglou, G.; Christodoulides, D., Observation of PT-symmetry breaking in complex optical potentials, Phys. Rev. Lett., 103, 093902, 2009 · doi:10.1103/PhysRevLett.103.093902 |
| [29] | 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, 2010 · doi:10.1038/nphys1515 |
| [30] | Regensburger, A.; Bersch, C.; Miri, M.-A.; Onishchukov, G.; Christodoulides, D. N.; Peschel, U., Parity-time synthetic photonic lattices, Nature, 488, 167, 2012 · doi:10.1038/nature11298 |
| [31] | Peng, B.; Ödemir, S. K.; Lei, F.; Monifi, F.; Gianfreda, M.; Long, G. L.; Fan, S.; Nori, F.; Bender, C. M.; Yang, L., Parity-time-symmetric whispering-gallery microcavities, Nat. Phys., 10, 394-398, 2014 · doi:10.1038/nphys2927 |
| [32] | Hodaei, H.; Hassan, A. U.; Wittek, S.; Garcia-Gracia, H.; ElGanainy, R.; Christodoulides, D. N.; Khajavikhan, M., Enhanced sensitivity at higher-order exceptional points, Nature, 548, 187, 2017 · doi:10.1038/nature23280 |
| [33] | Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations, edited by B. A. Malomed (Springer, Berlin, 2013). · Zbl 1267.00027 |
| [34] | 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 |
| [35] | Sacchetti, A., Universal critical power for nonlinear Schrödinger equations with a symmetric double well potential, Phys. Rev. Lett., 103, 194101, 2009 · doi:10.1103/PhysRevLett.103.194101 |
| [36] | Yang, J., No stability switching at saddle-node bifurcations of solitary waves in generalized nonlinear Schrödinger equations, Phys. Rev. E, 85, 037602, 2012 · doi:10.1103/PhysRevE.85.037602 |
| [37] | Yang, J., Can parity-time-symmetric potentials support families of non-parity-time-symmetric solitons?, Stud. Appl. Math., 132, 332, 2014 · Zbl 1298.35208 · doi:10.1111/sapm.12032 |
| [38] | Yang, J., Symmetry breaking of solitons in one-dimensional parity-time-symmetric optical potentials, Opt. Lett., 39, 5547, 2014 · doi:10.1364/OL.39.005547 |
| [39] | Yang, J., Symmetry breaking of solitons in two-dimensional complex potentials, Phys. Rev. E, 91, 023201, 2015 · doi:10.1103/PhysRevE.91.023201 |
| [40] | Li, P.; Mihalache, D., Symmetry breaking of solitons in PT-symmetric potentials with competing cubic-quintic nonlinearity, Proc. Rom. Acad. A, 19, 61, 2018 |
| [41] | Li, P.; Dai, C.; Li, R.; Gao, Y., Symmetric and asymmetric solitons supported by a \(\mathcal{P}\mathcal{T} \)-symmetric potential with saturable nonlinearity: Bifurcation, stability and dynamics, Opt. Exp., 26, 6949, 2018 · doi:10.1364/OE.26.006949 |
| [42] | Dong, L.; Huang, C.; Qi, W., Symmetry breaking and restoration of symmetric solitons in partially parity-time-symmetric potentials, Nonlinear Dyn., 98, 1701, 2019 · Zbl 1430.37089 · doi:10.1007/s11071-019-05280-3 |
| [43] | Bo, W. B.; Wang, R. R.; Liu, W.; Wang, Y. Y., Symmetry breaking of solitons in the PT-symmetric nonlinear Schrödinger equation with the cubic-quintic competing saturable nonlinearity, Chaos, 32, 093104, 2022 · Zbl 1541.35123 · doi:10.1063/5.0091738 |
| [44] | Laskin, N., Fractional quantum mechanics, Phys. Rev. E, 62, 3135, 2000 · doi:10.1103/PhysRevE.62.3135 |
| [45] | Laskin, N., Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268, 298, 2020 · Zbl 0948.81595 · doi:10.1016/S0375-9601(00)00201-2 |
| [46] | Laskin, N., Fractional Schrödinger equation, Phys. Rev. E, 66, 056108, 2002 · doi:10.1103/PhysRevE.66.056108 |
| [47] | Wei, Y. C., Comment on fractional quantum mechanics and fractional Schrödinger equation, Phys. Rev. E, 93, 066103, 2016 · doi:10.1103/PhysRevE.93.066103 |
| [48] | Laskin, N., Reply to ‘comment on fractional quantum mechanics and fractional Schrödinger equation’, Phys. Rev. E, 93, 066104, 2016 · doi:10.1103/PhysRevE.93.066104 |
| [49] | Stickler, B., Potential condensed-matter realization of space-fractional quantum mechanics: The one-dimensional Lévy crystal, Phys. Rev. E, 88, 12120, 2013 · doi:10.1103/PhysRevE.88.012120 |
| [50] | Longhi, S., Fractional Schrödinger equation in optics, Opt. Lett., 40, 1117, 2015 · doi:10.1364/OL.40.001117 |
| [51] | Huang, C.; Dong, L., Gap solitons in the nonlinear fractional Schrödinger equation with an optical lattice, Opt. Lett., 41, 5636, 2016 · doi:10.1364/OL.41.005636 |
| [52] | Yao, X.; Liu, X., Off-site and on-site vortex solitons in space-fractional photonic lattices, Opt. Lett., 43, 5749, 2018 · doi:10.1364/OL.43.005749 |
| [53] | Yao, X.; Liu, X., Solitons in the fractional Schrödinger equation with parity-time-symmetric lattice potential, Photonics Res., 6, 875, 2018 · doi:10.1364/PRJ.6.000875 |
| [54] | Chen, M.; Zeng, S.; Lu, D.; Hu, W.; Guo, Q., Optical solitons, self-focusing, and wave collapse in a space-fractional Schrödinger equation with a Kerr-type nonlinearity, Phys. Rev. E, 98, 022211, 2018 · doi:10.1103/PhysRevE.98.022211 |
| [55] | Dong, L.; Huang, C.; Qi, W., Nonlocal solitons in fractional dimensions, Opt. Lett., 44, 4917, 2019 · doi:10.1364/OL.44.004917 |
| [56] | Xie, J.; Zhu, X.; He, Y., Vector solitons in nonlinear fractional Schrödinger equations with parity-time-symmetric optical lattices, Nonlinear Dyn., 97, 1287, 2019 · Zbl 1430.35214 · doi:10.1007/s11071-019-05048-9 |
| [57] | Malomed, B. A., Optical solitons and vortices in fractional media: A mini-review of recent results, Photonics, 8, 353, 2021 · doi:10.3390/photonics8090353 |
| [58] | Li, P.; Dai, C., Double loops and pitchfork symmetry breaking bifurcations of optical solitons in nonlinear fractional Schrödinger equation with competing cubic-quintic nonlinearities, Ann. Phys., 532, 2000048, 2020 · Zbl 07764334 · doi:10.1002/andp.202000048 |
| [59] | Li, P.; Malomed, B. A.; Mihalache, D., Symmetry breaking of spatial Kerr solitons in fractional dimension, Chaos, Solitons Fractals, 132, 109602, 2020 · Zbl 1434.35008 · doi:10.1016/j.chaos.2020.109602 |
| [60] | Li, P.; Li., R.; Dai, C., Existence, symmetry breaking bifurcation and stability of two-dimensional optical solitons supported by fractional diffraction, Opt. Exp., 29, 3193, 2021 · doi:10.1364/OE.415028 |
| [61] | Li, P.; Malomed, B. A.; Mihalache, D., Symmetry-breaking bifurcations and ghost states in the fractional nonlinear Schrödinger equation with a PT-symmetric potential, Opt. Lett., 46, 3267, 2021 · doi:10.1364/OL.428254 |
| [62] | Coutaz, J. L.; Kull, M., Saturation of the nonlinear index of refraction in semiconductor-doped glass, J. Opt. Soc. Am. B, 8, 95-98, 1991 · doi:10.1364/JOSAB.8.000095 |
| [63] | Tikhonenko, V.; Christou, J.; Luther-Davies, B., Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium, Phys. Rev. Lett., 76, 2698-2701, 1996 · doi:10.1103/PhysRevLett.76.2698 |
| [64] | Li, K.; Kevrekidis, P.; Frantzeskakis, D. J.; Rüter, C. E.; Kip, D., Revisiting the-symmetric trimer: Bifurcations, ghost states and associated dynamics, J. Phys. A: Math. Theor., 46, 375304, 2013 · Zbl 1275.81096 · doi:10.1088/1751-8113/46/37/375304 |
| [65] | Cartarius, H.; Haag, D.; Dast, D.; Wunner, G., Nonlinear Schrödinger equation for a-symmetric delta-function double well, J. Phys. A Math. Theor., 45, 444008, 2012 · Zbl 1282.35325 · doi:10.1088/1751-8113/45/44/444008 |
| [66] | Susanto, H.; Kusdiantara, R.; Li, N.; Kirikchi, O. B.; Adzkiya, D.; Putri, E. R. M.; Asfihani, T., Snakes and ghosts in a parity-time-symmetric chain of dimers, Phys. Rev. E, 97, 062204, 2018 · doi:10.1103/PhysRevE.97.062204 |
| [67] | Yang, J., Nonlinear Waves in Integrable and Nonintegrable Systems, 2010, SIAM · Zbl 1234.35006 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
