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扰动振幅和扰动频率对Fermi-Pasta-Ulam-Tsingou回归现象的影响

郑州 李金花 马佑桥 任海东

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扰动振幅和扰动频率对Fermi-Pasta-Ulam-Tsingou回归现象的影响

郑州, 李金花, 马佑桥, 任海东

Influence of perturbation amplitude and perturbation frequency on Fermi-Pasta-Ulam-Tsingou recurrence phenomenon

Zheng Zhou, Li Jin-Hua, Ma You-Qiao, Ren Hai-Dong
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  • Fermi-Pasta-Ulam-Tsingou (FPUT)回归现象指一个多模非线性系统能周期性回到初始激发态的一个复杂的非线性过程, 与该非线性系统的调制不稳定性密切相关. 针对实验中能如何较为方便地观察到FPUT回归现象以及能如何观察到更多FPUT循环的问题, 本文基于调制不稳定性重点分析研究了施加在平面波上的扰动振幅和扰动频率对所观察到的FPUT循环的影响. 我们发现, 扰动振幅可以极大程度地影响所观察到的FPUT现象: 1) FPUT循环数对扰动振幅的值非常敏感, 扰动振幅越大, FPUT循环数越多; 2) 扰动振幅较小(较大)时, 相应的FPUT循环频谱就比较规则(很不规则). 相比之下, 扰动频率对FPUT循环数的影响不是很大(在最佳调制频率附近的一个小范围内, 可观察到FPUT循环最多), 但是它对脉冲周期性振幅最大位置处所激发的高阶频率成分的影响比较大, 扰动频率越大(越小), 其可以激发的高阶频率成分越少(越多). 本文的研究结果将对FPUT实验的观测和理论发展提供一定的帮助.
    Fermi-Pasta-Ulam-Tsingou recurrence (FPUT) phenomenon refers to the property of a multimode nonlinear system returning to the initial states after complex stages of evolution. The FPUT recurrence phenomenon closely links with modulation instability (MI) by employing the perturbed continuous waves as the initial condition. When the perturbation frequency is located inside the MI spectra, then the perturbed CWs are unstable and the perturbations will grow up with evolution. This nonlinear MI evolution results in the FPUT phenomenon. In this work, we explore in detail the effects of perturbation amplitude and perturbation frequency on the FPUT recurrence phenomena numerically, which has never been studied systematically, to the best of our knowledge. Using the results of our studies, we find that the perturbation amplitude can significantly affect the FPUT phenomenon. Firstly, the number of FPUT cycles is very sensitive to the perturbation amplitude. Large (small) perturbation amplitude can result in much more (much less) FPUT cycles. Secondly, very irregular (regular) FPUT wave evolution together with the corresponding spectra evolution can be observed at relatively large (small) values of perturbation amplitude, where the unequal (equal) distances are observed between adjacent maximum wave amplitudes spatially in the background of optical fibers. In contrast, the effects of perturbation frequency on the FPUT cycles are relatively minor, and the maximum FPUT cycles are observed at perturbation frequencies around the optimal modulation frequency generating the peak MI gain. However, the perturbation frequency can drastically affect the number of high-order sidebands excited at the distances of periodic maximum wave amplitude formation. We find that larger perturbation frequency leads to much fewer high-order sidebands. According to our studies, for observing FPUT conveniently and observing more FPUT cycles, the perturbation amplitude of the input signal should be as large as possible and the perturbation frequency should be around the optimum modulation frequency. We should also emphasize that the large perturbation amplitude results in irregular FPUT patterns with unequal distances between adjacent maximum wave amplitude formations spatially in the background of optical fibers, and large perturbation frequency results in much less high-order sidebands. Our results will provide very helpful information for the FPUT observation in experiment, and should arouse the interest of the readers in nonlinear physics.
      Corresponding author: Zheng Zhou, 20201217020@nuist.edu.cn ; Li Jin-Hua, lijinhua@nuist.edu.cn
    [1]

    Fermi E, Pasta P, Ulam S, Tsingou M 1955 Studies of the Nonlinear Problems Los Alamos, May 1, 1955 pLA-1940

    [2]

    Zabusky N J, Kruskal M D 1965 Phys. Rev. Lett. 15 240Google Scholar

    [3]

    Bagchi D 2021 Phys. Rev. E 104 054108Google Scholar

    [4]

    Friesecke G, Mikikits-Leitner A 2015 J. Dyn. Differ. Equ. 27 627Google Scholar

    [5]

    Pan Q, Yin H M, Chow K W 2021 J. Mar. Sci. Eng. 9 577Google Scholar

    [6]

    Van Simaeys G, Emplit P, Haelterman M 2002 J. Opt. Soc. Am. B 19 477Google Scholar

    [7]

    Conforti M, Mussot A, Kudlinski A, Trillo S, Akhmediev N 2020 Phys. Rev. A 101 023843Google Scholar

    [8]

    Vanderhaegen G, Naveau C, Szriftgiser P, Kudlinski A, Conforti M, Mussot A, Onorato M, Trillo S, Chabchoub A, Akhmediev N 2021 Proc. Natl. Acad. Sci. USA 118 2019348118Google Scholar

    [9]

    Chen S C, Liu C, Yao X, Zhao L C, Akhmediev N 2021 Phys. Rev. E 104 024215Google Scholar

    [10]

    Akhmediev N, Korneev V 1986 Theor. Math. Phys. 69 1089Google Scholar

    [11]

    Ablowitz M J, Herbst B M 1990 Siam J. Appl. Math. 50 339Google Scholar

    [12]

    Wabnitz S, Wetzel B 2014 Phys. Lett. A 378 2750Google Scholar

    [13]

    Wang X, Dong Z, Deng Z 2021 Results Phys. 29 104715Google Scholar

    [14]

    Chabchoub A, Hoffmann N, Tobisch E, Waseda T, Akhmediev N 2019 Wave Motion 90 168Google Scholar

    [15]

    Yao X, Yang Z Y, Yang W L 2021 Nonlinear Dyn. 103 1035Google Scholar

    [16]

    Kuznetsov E A 2017 JETP Lett. 105 125Google Scholar

    [17]

    Erkintalo M, Genty G, Wetzel B, Dudley J M 2011 Phys. Lett. A 375 2029Google Scholar

    [18]

    Liu C, Chen S C, Yao X, Akhmediev N 2022 Physica D 433 133192Google Scholar

    [19]

    Che W J, Chen S C, Liu C, Zhao L C, Akhmediev N 2022 Phys. Rev. A 105 043526Google Scholar

    [20]

    Liu C, Wu Y H, Chen S C, Yao X, Akhmediev N 2021 Phys. Rev. Lett. 127 094102Google Scholar

    [21]

    Yao X, Liu C, Yang Z Y, Yang W L 2022 Phys. Rev. Res. 4 013246Google Scholar

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    Akhmediev N N 2001 Nature 413 267Google Scholar

    [23]

    Chin S A, Ashour O A, Belić M R 2015 Phys. Rev. E 92 063202Google Scholar

    [24]

    Grinevich P, Santini P 2018 Phys. Lett. A 382 973Google Scholar

    [25]

    Wabnitz S, Akhmediev N 2010 Opt. Commun. 283 1152Google Scholar

    [26]

    Fatome J, El-Mansouri I, Blanchet J L, Pitois S, Millot G, Trillo S, Wabnitz S 2013 J. Opt. Soc. Am. B 30 99Google Scholar

    [27]

    Erkintalo M, Hammani K, Kibler B, Finot C, Akhmediev N, Dudley J M, Genty G 2011 Phys. Rev. Lett. 107 253901Google Scholar

    [28]

    Soto-Crespo J M, Ankiewicz A, Devine N, Akhmediev N 2012 J. Opt. Soc. Am. B 29 1930Google Scholar

    [29]

    Mussot A, Kudlinski A, Droques M, Szriftgiser P, Akhmediev N 2014 Phys. Rev. X 4 011054Google Scholar

    [30]

    Mussot A, Naveau C, Conforti M, Kudlinski A, Copie F, Szriftgiser P, Trillo S 2018 Nat. Photonics 12 303Google Scholar

    [31]

    Kimmoun O, Hsu H, Branger H, Li M, Chen Y, Kharif C, Onorato M, Kelleher E J, Kibler B, Akhmediev N 2016 SCI. REP-UK 6 1Google Scholar

    [32]

    Naveau C, Szriftgiser P, Kudlinski A, Conforti M, Trillo S, Mussot A 2019 Opt. Lett. 44 5426Google Scholar

    [33]

    Naveau C, Szriftgiser P, Kudlinski A, Conforti M, Trillo S, Mussot A 2019 Opt. Lett. 44 763Google Scholar

    [34]

    Pierangeli D, Flammini M, Zhang L, Marcucci G, Agranat A J, Grinevich P G, Santini P M, Conti C, DelRe E 2018 Phys. Rev. X 8 041017Google Scholar

    [35]

    Goossens J W, Hafermann H, Jaouen Y 2019 Sci. Rep. 9 1Google Scholar

    [36]

    Vanderhaegen G, Szriftgiser P, Kudlinski A, Conforti M, Trillo S, Droques M, Mussot A 2020 Opt. Express 28 17773Google Scholar

    [37]

    Bao C, Jaramillo-Villegas J A, Xuan Y, Leaird D E, Qi M, Weiner A M 2016 Phys. Rev. Lett. 117 163901Google Scholar

    [38]

    Agrawal G P 2000 Nonlinear Science at the Dawn of the 21st Century (Berlin Heidelberg: Springer-Verlag) pp195–211

    [39]

    Yin H M, Chow K W 2021 Physica D 428 133033Google Scholar

  • 图 1  非线性光纤中, 扰动平面波(a)和相应频谱(b)随传输距离的演化; 平面波演化至z1FPUT = 3.2 m处的波形(c)及频谱(d). 图中P0 = 1 kW, δ = 0.001, $ \varOmega = {\varOmega _{\max }} = 5\sqrt {10}\; {\text{rad/ps}} $

    Fig. 1.  Evolution of perturbed plane wave (a) and corresponding spectra (b) with transmission distance; wave form (c) and corresponding spectra (d) at z1FPUT = 3.2 m in typical single-core fibers for P0 = 1 kW, δ = 0.001, $ \varOmega = {\varOmega _{\max }} = 5\sqrt {10} \;{\text{rad/ps}} $.

    图 2  非线性光纤中, 扰动平面波(a)和相应频谱(b)随传输距离的演化; 平面波演化至z1FPUT = 1.3 m处的波形(c)及频谱(d). 图中P0 = 1 kW, δ = 0.1, $ \varOmega = {\varOmega _{\max }} = 5\sqrt {10} \;{\text{rad/ps}} $

    Fig. 2.  Evolution of perturbed plane wave (a) and corresponding spectra (b) with transmission distance; wave form (c) and corresponding spectra (d) at z1FPUT =1.3 m in typical single-core fibers for P0 = 1 kW, δ = 0. 1, $ \varOmega = {\varOmega _{\max }} = 5\sqrt {10}\; {\text{rad/ps}} $.

    图 3  非线性光纤中FPUT循环数随扰动振幅δ的演化

    Fig. 3.  Variation of the number of FPUT cycle with the perturbation amplitude δ in nonlinear fibers.

    图 4  非线性光纤中, 扰动平面波(a)和相应频谱(b)随传输距离的演化; 平面波演化至z1FPUT = 0.4 m处的波形(c)及频谱(d). 图中P0 = 1 kW, δ = 0.7, $\varOmega = {\varOmega _{\max }} = 5\sqrt {10} \;{\text{rad/ps}}$

    Fig. 4.  Evolution of perturbed plane wave (a) and corresponding spectra (b) with transmission distance; wave form (c) and corresponding spectra (d) at z1FPUT = 0.4 m in typical single-core fibers for P0 = 1 kW, δ = 0.7, $\varOmega = {\varOmega _{\max }} = 5\sqrt {10} \;{\text{rad/ps}}$.

    图 5  非线性光纤中, 扰动平面波(a)和相应频谱(b)随传输距离的演化; 平面波演化至z1FPUT = 3.4 m处的波形(c)及频谱(d). 图中P0 = 1 kW, δ = 0.001, $ \varOmega = {\varOmega _{\text{c}}}/2 = 5\sqrt 5 \;{\text{rad/ps}} $

    Fig. 5.  Evolution of perturbed plane wave (a) and corresponding spectra (b) with transmission distance; wave form (c) and corresponding spectra (d) at z1FPUT = 3.4 m in typical single-core fibers for P0 = 1 kW, δ = 0.001, $ \varOmega = {\varOmega _{\text{c}}}/2 = 5\sqrt 5 \;{\text{rad/ps}} $.

    图 6  高阶谐波数随扰动频率Ω (Ωc/2 < Ω < Ωc)的变化关系图

    Fig. 6.  Variation of mode numbers of high-order sidebands with perturbation frequency Ω (Ωc/2 < Ω < Ωc).

    图 7  非线性光纤中, 扰动平面波(a)和相应频谱(b)随传输距离的演化; 平面波演化至z1FPUT = 8.22 m处的波形(c)及频谱(d). 图中P0 = 1 kW, δ = 0.001, Ω = 22 rad/ps

    Fig. 7.  Evolution of perturbed plane wave (a) and corresponding spectra (b) with transmission distance; wave form (c) and corresponding spectra (d) at z1FPUT = 8.22 m in typical single-core fibers for P0 = 1 kW, δ = 0.001, Ω = 22 rad/ps.

    图 8  非线性光纤中, 扰动平面波(a)和相应频谱(b)随传输距离的演化; 平面波演化至z1FPUT = 14.6 m处的波形(c)及频谱(d). 图中P0 = 1 kW, δ = 0.001, Ω = 22.3607 rad/ps

    Fig. 8.  Evolution of perturbed plane wave (a) and corresponding spectra (b) with transmission distance; wave form (c) and corresponding spectra (d) at z1FPUT = 14.6 m in typical single-core fibers for P0 = 1 kW, δ = 0.001, Ω = 22.3607 rad/ps.

  • [1]

    Fermi E, Pasta P, Ulam S, Tsingou M 1955 Studies of the Nonlinear Problems Los Alamos, May 1, 1955 pLA-1940

    [2]

    Zabusky N J, Kruskal M D 1965 Phys. Rev. Lett. 15 240Google Scholar

    [3]

    Bagchi D 2021 Phys. Rev. E 104 054108Google Scholar

    [4]

    Friesecke G, Mikikits-Leitner A 2015 J. Dyn. Differ. Equ. 27 627Google Scholar

    [5]

    Pan Q, Yin H M, Chow K W 2021 J. Mar. Sci. Eng. 9 577Google Scholar

    [6]

    Van Simaeys G, Emplit P, Haelterman M 2002 J. Opt. Soc. Am. B 19 477Google Scholar

    [7]

    Conforti M, Mussot A, Kudlinski A, Trillo S, Akhmediev N 2020 Phys. Rev. A 101 023843Google Scholar

    [8]

    Vanderhaegen G, Naveau C, Szriftgiser P, Kudlinski A, Conforti M, Mussot A, Onorato M, Trillo S, Chabchoub A, Akhmediev N 2021 Proc. Natl. Acad. Sci. USA 118 2019348118Google Scholar

    [9]

    Chen S C, Liu C, Yao X, Zhao L C, Akhmediev N 2021 Phys. Rev. E 104 024215Google Scholar

    [10]

    Akhmediev N, Korneev V 1986 Theor. Math. Phys. 69 1089Google Scholar

    [11]

    Ablowitz M J, Herbst B M 1990 Siam J. Appl. Math. 50 339Google Scholar

    [12]

    Wabnitz S, Wetzel B 2014 Phys. Lett. A 378 2750Google Scholar

    [13]

    Wang X, Dong Z, Deng Z 2021 Results Phys. 29 104715Google Scholar

    [14]

    Chabchoub A, Hoffmann N, Tobisch E, Waseda T, Akhmediev N 2019 Wave Motion 90 168Google Scholar

    [15]

    Yao X, Yang Z Y, Yang W L 2021 Nonlinear Dyn. 103 1035Google Scholar

    [16]

    Kuznetsov E A 2017 JETP Lett. 105 125Google Scholar

    [17]

    Erkintalo M, Genty G, Wetzel B, Dudley J M 2011 Phys. Lett. A 375 2029Google Scholar

    [18]

    Liu C, Chen S C, Yao X, Akhmediev N 2022 Physica D 433 133192Google Scholar

    [19]

    Che W J, Chen S C, Liu C, Zhao L C, Akhmediev N 2022 Phys. Rev. A 105 043526Google Scholar

    [20]

    Liu C, Wu Y H, Chen S C, Yao X, Akhmediev N 2021 Phys. Rev. Lett. 127 094102Google Scholar

    [21]

    Yao X, Liu C, Yang Z Y, Yang W L 2022 Phys. Rev. Res. 4 013246Google Scholar

    [22]

    Akhmediev N N 2001 Nature 413 267Google Scholar

    [23]

    Chin S A, Ashour O A, Belić M R 2015 Phys. Rev. E 92 063202Google Scholar

    [24]

    Grinevich P, Santini P 2018 Phys. Lett. A 382 973Google Scholar

    [25]

    Wabnitz S, Akhmediev N 2010 Opt. Commun. 283 1152Google Scholar

    [26]

    Fatome J, El-Mansouri I, Blanchet J L, Pitois S, Millot G, Trillo S, Wabnitz S 2013 J. Opt. Soc. Am. B 30 99Google Scholar

    [27]

    Erkintalo M, Hammani K, Kibler B, Finot C, Akhmediev N, Dudley J M, Genty G 2011 Phys. Rev. Lett. 107 253901Google Scholar

    [28]

    Soto-Crespo J M, Ankiewicz A, Devine N, Akhmediev N 2012 J. Opt. Soc. Am. B 29 1930Google Scholar

    [29]

    Mussot A, Kudlinski A, Droques M, Szriftgiser P, Akhmediev N 2014 Phys. Rev. X 4 011054Google Scholar

    [30]

    Mussot A, Naveau C, Conforti M, Kudlinski A, Copie F, Szriftgiser P, Trillo S 2018 Nat. Photonics 12 303Google Scholar

    [31]

    Kimmoun O, Hsu H, Branger H, Li M, Chen Y, Kharif C, Onorato M, Kelleher E J, Kibler B, Akhmediev N 2016 SCI. REP-UK 6 1Google Scholar

    [32]

    Naveau C, Szriftgiser P, Kudlinski A, Conforti M, Trillo S, Mussot A 2019 Opt. Lett. 44 5426Google Scholar

    [33]

    Naveau C, Szriftgiser P, Kudlinski A, Conforti M, Trillo S, Mussot A 2019 Opt. Lett. 44 763Google Scholar

    [34]

    Pierangeli D, Flammini M, Zhang L, Marcucci G, Agranat A J, Grinevich P G, Santini P M, Conti C, DelRe E 2018 Phys. Rev. X 8 041017Google Scholar

    [35]

    Goossens J W, Hafermann H, Jaouen Y 2019 Sci. Rep. 9 1Google Scholar

    [36]

    Vanderhaegen G, Szriftgiser P, Kudlinski A, Conforti M, Trillo S, Droques M, Mussot A 2020 Opt. Express 28 17773Google Scholar

    [37]

    Bao C, Jaramillo-Villegas J A, Xuan Y, Leaird D E, Qi M, Weiner A M 2016 Phys. Rev. Lett. 117 163901Google Scholar

    [38]

    Agrawal G P 2000 Nonlinear Science at the Dawn of the 21st Century (Berlin Heidelberg: Springer-Verlag) pp195–211

    [39]

    Yin H M, Chow K W 2021 Physica D 428 133033Google Scholar

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出版历程
  • 收稿日期:  2022-05-12
  • 修回日期:  2022-05-26
  • 上网日期:  2022-09-08
  • 刊出日期:  2022-09-20

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