Abstract
The first good prediction of the multipole soliton solution for the non-integrable equation, i.e., the saturable nonlinear Schrödinger equation under the PT-symmetric potential, is achieved using the physical information neural network. In addition, we construct multipole (tripole to sextupole) soliton families in saturable nonlinear media with fractional diffraction under the PT-symmetric potential, and quadrupole, pentapole and sextupole solitons can coexist for the same parameters. The existence of multipole solitons is modulated by the modulation intensity of the PT-symmetric potential and Lévy index altogether, while the stable domain of multipole solitons is modulated by both the power and Lévy index together. With the increase in the modulation intensity of the PT-symmetric potential and Lévy index, the existence domain of multipole solitons gradually enlarges. When the soliton power is conserved, with the add of the Lévy index, the peak amplitudes at the outermost part of the profiles of real and imaginary parts for the multipole soliton increase, while the peak amplitudes at other positions decrease, and yet the soliton width increases. In addition, the strong saturable nonlinearity not only reduces the stability of tripole solitons but also inhibits the instability of quadrupole and pentapole solitons. However, the saturable nonlinear intensity exists a threshold for the stability modulation of sextupole solitons, beyond which the stability of sextupole solitons is no longer modulated by the saturable nonlinearity.
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References
Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268, 298–305 (2000)
Laskin, N.: Fractional quantum mechanics. Phys. Rev. E 62, 3135 (2000)
Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E 66, 056108 (2002)
Longhi, S.: Fractional Schrödinger equation in optics. Opt. Lett. 40, 1117–1120 (2015)
Dong, L., Huang, C.: Vortex solitons in fractional systems with partially parity-time-symmetric azimuthal potentials. Nonlinear Dyn. 98, 1019–1028 (2019)
Zeng, L., Zeng, J.: One-dimensional gap solitons in quintic and cubic-quintic fractional nonlinear Schrödinger equations with a periodically modulated linear potential. Nonlinear Dyn. 98, 985–995 (2019)
Cao, Q.H., Dai, C.Q.: Symmetric and anti-symmetric solitons of the fractional second- and third-order nonlinear schrodinger equation. Chin. Phys. Lett. 38, 090501 (2021)
Li, P., Li, R., Dai, C.: Existence, symmetry breaking bifurcation and stability of two-dimensional optical solitons supported by fractional diffraction. Opt. Express 29, 3193–3210 (2021)
Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)
Bender, C.M., Brody, D.C., Jones, H.F.: Complex extension of quantum mechanics. Phys. Rev. Lett. 89, 270401 (2002)
Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007)
Mock, A.: Parity-time–symmetry breaking in two-dimensional photonic crystals: square lattice. Phys. Rev. A. 93, 063812 (2016)
Zhang, J., Liu, J., Zhang, H., Gong, Z., Zhang, S., Yan, L.: Topological optomechanical amplifier in synthetic PT-symmetry. Nanophotonics 11, 1149–1158 (2022)
Joglekar, Y.N., Marathe, R., Durganandini, P., Pathak, R.K.: PT spectroscopy of the Rabi problem. Phys. Rev. A 90, 040101 (2014)
Zhu, X., Yang, F., Cao, S., Xie, J., He, Y.: Multipole gap solitons in fractional Schrödinger equation with parity-time-symmetric optical lattices. Opt. Express 28, 1631–1639 (2020)
Huang, C., Lin, Z., Dong, L., Li, C., Gao, P., Su, W.: Fundamental and multipole solitons in amplitude-modulated Fibonacci lattices. Opt. Express 29, 35327–35335 (2021)
Bo, W., Liu, W., Wang, Y.: Symmetric and antisymmetric solitons in the fractional nonlinear Schrödinger equation with saturable nonlinearity and PT-symmetric potential: Stability and dynamics. Optik 255, 168697 (2022)
Li, P., Dai, C., Li, R., Gao, Y.: Symmetric and asymmetric solitons supported by a PT-symmetric potential with saturable nonlinearity: bifurcation, stability and dynamics. Opt. Express 26, 6949–6961 (2018)
Yang, J.: Symmetry breaking of solitons in two-dimensional complex potentials. Phys. Rev. E 91, 023201 (2015)
Dmitriev, S.V., Sukhorukov, A.A., Kivshar, Y.S.: Binary parity-time-symmetric nonlinear lattices with balanced gain and loss. Opt. Lett. 2010(35), 2976–2978 (2010)
Driben, R., Malomed, B.A.: Stability of solitons in parity-time-symmetric couplers. Opt. Lett. 36, 4323–4325 (2011)
Pannian, J.C., Alberucci, A., Brazhnyi, V.A., Assanto, G.: Nonlocal gap solitons in PT-symmetric periodic potentials with defocusing nonlinearity. Phys. Rev. A 89, 013812 (2014)
Achilleos, V., Kevrekidis, P.G., Frantzeskakis, D.J., Carretero-González, 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)
Zhen, S., Zhang, Y., Chen, Y., Sun, F., Zou, X., Guo, G.: Reconfigurable optomechanical circulator and directional amplifier. Nat. Commun. 9, 1797 (2018)
Su, S., Gou, S., Chew, L., Chang, Y., Yu, I., Kalachev, A.: Setting a disordered password on a photonic memory. Phys. Rev. A 95, 061805 (2017)
Dong, L., Huang, C., Qi, W.: Nonlocal solitons in fractional dimensions. Opt. Lett. 44, 4917–4920 (2019)
Zeng, L., Mihalache, D., Malomed, B.A., Lu, X., Li, J.: Families of fundamental and multipole solitons in a cubic-quintic nonlinear lattice in fractional dimension. Chaos Solitons & Fract. 144, 110589 (2021)
Desyatnikov, A.S., Neshev, D., Ostrovskaya, E.A., Kivshar, Y.S., Krolikowski, W.: Multipole spatial vector solitons. Opt. Lett. 26, 435–437 (2001)
Desyatnikov, A.S., Neshev, D., Ostrovskaya, E.A., Kivshar, Y.S., Mccarthy, G., Krolikowski, W.: Multipole composite spatial solitons: theory and experiment. Opt. Soc. Am. J. B 19, 586–595 (2002)
Huang, C., Li, C., Dong, L.: Stabilization of multipole-mode solitons in mixed linear-nonlinear lattices with a PT symmetry. Opt. Express 21, 3917–3125 (2013)
Porras, M.A., Ruiz-Jimenez, C., Carvalho, M.: Stationary and stable light-beam propagation in Kerr media with nonlinear absorption with controllable dissipation patterns. Phys. Rev. A 95, 043816 (2017)
Wen, X., Wu, G., Liu, W., Dai, C.: Dynamics of diverse data-driven solitons for the three-component coupled nonlinear Schrödinger model by the MPS-PINN method. Nonlinear Dyn. 109, 3041–3050 (2022)
Fang, Y., Wu, G., Wen, X., Wang, Y., Dai, C.: Predicting certain vector optical solitons via the conservation-law deep-learning method. Opt. Laser Tech. 155, 108428 (2022)
Fang, Y., Wu, G., Kudryashov, N.A., Wang, Y., Dai, C.: Data-driven soliton solutions and model parameters of nonlinear wave models via the conservation-law constrained neural network method. Chaos Solitons Fract. 158, 112118 (2022)
Li, P., Mihalache, D., Malomed, B.A.: Optical solitons in media with focusing and defocusing saturable nonlinearity and a parity-time-symmetric external potential. Philos. Trans. R. Soc. Math. Phys. Eng. Sci. 376, 2124 (2018)
Zhong, M., Chen, Y., Yan, Z., Tian, S.: 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)
Yaroslav, V.K., Boris, A.M., Lluis, T.: Solitons in nonlinear lattices. Rev. of Modern Phys. 83, 247–306 (2011)
Zezyulin, D.A., Konotop, V.V.: Nonlinear modes in the harmonic PT-symmetric potential. Phys. Rev. A 85, 043840 (2012)
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 PT-symmetric systems. Phys. Rev. A 90, 043855 (2014)
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Zhejiang Provincial Natural Science Foundation of China (Grant No. LR20A050001); National Natural Science Foundation of China (Grant Nos. 12075210 and 11874324); the Scientific Research and Developed Fund of Zhejiang A&F University (Grant No. 2021FR0009).
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Bo, WB., Wang, RR., Fang, Y. et al. Prediction and dynamical evolution of multipole soliton families in fractional Schrödinger equation with the PT-symmetric potential and saturable nonlinearity. Nonlinear Dyn 111, 1577–1588 (2023). https://doi.org/10.1007/s11071-022-07884-8
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DOI: https://doi.org/10.1007/s11071-022-07884-8