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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.
[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. 15 240
Google Scholar
[3] Bagchi D 2021 Phys. Rev. E 104 054108
Google Scholar
[4] Friesecke G, Mikikits-Leitner A 2015 J. Dyn. Differ. Equ. 27 627
Google Scholar
[5] Pan Q, Yin H M, Chow K W 2021 J. Mar. Sci. Eng. 9 577
Google Scholar
[6] Van Simaeys G, Emplit P, Haelterman M 2002 J. Opt. Soc. Am. B 19 477
Google Scholar
[7] Conforti M, Mussot A, Kudlinski A, Trillo S, Akhmediev N 2020 Phys. Rev. A 101 023843
Google 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 2019348118
Google Scholar
[9] Chen S C, Liu C, Yao X, Zhao L C, Akhmediev N 2021 Phys. Rev. E 104 024215
Google Scholar
[10] Akhmediev N, Korneev V 1986 Theor. Math. Phys. 69 1089
Google Scholar
[11] Ablowitz M J, Herbst B M 1990 Siam J. Appl. Math. 50 339
Google Scholar
[12] Wabnitz S, Wetzel B 2014 Phys. Lett. A 378 2750
Google Scholar
[13] Wang X, Dong Z, Deng Z 2021 Results Phys. 29 104715
Google Scholar
[14] Chabchoub A, Hoffmann N, Tobisch E, Waseda T, Akhmediev N 2019 Wave Motion 90 168
Google Scholar
[15] Yao X, Yang Z Y, Yang W L 2021 Nonlinear Dyn. 103 1035
Google Scholar
[16] Kuznetsov E A 2017 JETP Lett. 105 125
Google Scholar
[17] Erkintalo M, Genty G, Wetzel B, Dudley J M 2011 Phys. Lett. A 375 2029
Google Scholar
[18] Liu C, Chen S C, Yao X, Akhmediev N 2022 Physica D 433 133192
Google Scholar
[19] Che W J, Chen S C, Liu C, Zhao L C, Akhmediev N 2022 Phys. Rev. A 105 043526
Google Scholar
[20] Liu C, Wu Y H, Chen S C, Yao X, Akhmediev N 2021 Phys. Rev. Lett. 127 094102
Google Scholar
[21] Yao X, Liu C, Yang Z Y, Yang W L 2022 Phys. Rev. Res. 4 013246
Google Scholar
[22] Akhmediev N N 2001 Nature 413 267
Google Scholar
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Google Scholar
[24] Grinevich P, Santini P 2018 Phys. Lett. A 382 973
Google Scholar
[25] Wabnitz S, Akhmediev N 2010 Opt. Commun. 283 1152
Google 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 99
Google Scholar
[27] Erkintalo M, Hammani K, Kibler B, Finot C, Akhmediev N, Dudley J M, Genty G 2011 Phys. Rev. Lett. 107 253901
Google Scholar
[28] Soto-Crespo J M, Ankiewicz A, Devine N, Akhmediev N 2012 J. Opt. Soc. Am. B 29 1930
Google Scholar
[29] Mussot A, Kudlinski A, Droques M, Szriftgiser P, Akhmediev N 2014 Phys. Rev. X 4 011054
Google Scholar
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Google 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 1
Google Scholar
[32] Naveau C, Szriftgiser P, Kudlinski A, Conforti M, Trillo S, Mussot A 2019 Opt. Lett. 44 5426
Google Scholar
[33] Naveau C, Szriftgiser P, Kudlinski A, Conforti M, Trillo S, Mussot A 2019 Opt. Lett. 44 763
Google 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 041017
Google Scholar
[35] Goossens J W, Hafermann H, Jaouen Y 2019 Sci. Rep. 9 1
Google Scholar
[36] Vanderhaegen G, Szriftgiser P, Kudlinski A, Conforti M, Trillo S, Droques M, Mussot A 2020 Opt. Express 28 17773
Google Scholar
[37] Bao C, Jaramillo-Villegas J A, Xuan Y, Leaird D E, Qi M, Weiner A M 2016 Phys. Rev. Lett. 117 163901
Google 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 133033
Google Scholar
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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}} $ Figure 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}} $ Figure 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
Figure 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,
$\varOmega = {\varOmega _{\max }} = 5\sqrt {10} \;{\text{rad/ps}}$ Figure 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}} $ Figure 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
Figure 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
Figure 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
Figure 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.
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[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. 15 240
Google Scholar
[3] Bagchi D 2021 Phys. Rev. E 104 054108
Google Scholar
[4] Friesecke G, Mikikits-Leitner A 2015 J. Dyn. Differ. Equ. 27 627
Google Scholar
[5] Pan Q, Yin H M, Chow K W 2021 J. Mar. Sci. Eng. 9 577
Google Scholar
[6] Van Simaeys G, Emplit P, Haelterman M 2002 J. Opt. Soc. Am. B 19 477
Google Scholar
[7] Conforti M, Mussot A, Kudlinski A, Trillo S, Akhmediev N 2020 Phys. Rev. A 101 023843
Google 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 2019348118
Google Scholar
[9] Chen S C, Liu C, Yao X, Zhao L C, Akhmediev N 2021 Phys. Rev. E 104 024215
Google Scholar
[10] Akhmediev N, Korneev V 1986 Theor. Math. Phys. 69 1089
Google Scholar
[11] Ablowitz M J, Herbst B M 1990 Siam J. Appl. Math. 50 339
Google Scholar
[12] Wabnitz S, Wetzel B 2014 Phys. Lett. A 378 2750
Google Scholar
[13] Wang X, Dong Z, Deng Z 2021 Results Phys. 29 104715
Google Scholar
[14] Chabchoub A, Hoffmann N, Tobisch E, Waseda T, Akhmediev N 2019 Wave Motion 90 168
Google Scholar
[15] Yao X, Yang Z Y, Yang W L 2021 Nonlinear Dyn. 103 1035
Google Scholar
[16] Kuznetsov E A 2017 JETP Lett. 105 125
Google Scholar
[17] Erkintalo M, Genty G, Wetzel B, Dudley J M 2011 Phys. Lett. A 375 2029
Google Scholar
[18] Liu C, Chen S C, Yao X, Akhmediev N 2022 Physica D 433 133192
Google Scholar
[19] Che W J, Chen S C, Liu C, Zhao L C, Akhmediev N 2022 Phys. Rev. A 105 043526
Google Scholar
[20] Liu C, Wu Y H, Chen S C, Yao X, Akhmediev N 2021 Phys. Rev. Lett. 127 094102
Google Scholar
[21] Yao X, Liu C, Yang Z Y, Yang W L 2022 Phys. Rev. Res. 4 013246
Google Scholar
[22] Akhmediev N N 2001 Nature 413 267
Google Scholar
[23] Chin S A, Ashour O A, Belić M R 2015 Phys. Rev. E 92 063202
Google Scholar
[24] Grinevich P, Santini P 2018 Phys. Lett. A 382 973
Google Scholar
[25] Wabnitz S, Akhmediev N 2010 Opt. Commun. 283 1152
Google 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 99
Google Scholar
[27] Erkintalo M, Hammani K, Kibler B, Finot C, Akhmediev N, Dudley J M, Genty G 2011 Phys. Rev. Lett. 107 253901
Google Scholar
[28] Soto-Crespo J M, Ankiewicz A, Devine N, Akhmediev N 2012 J. Opt. Soc. Am. B 29 1930
Google Scholar
[29] Mussot A, Kudlinski A, Droques M, Szriftgiser P, Akhmediev N 2014 Phys. Rev. X 4 011054
Google Scholar
[30] Mussot A, Naveau C, Conforti M, Kudlinski A, Copie F, Szriftgiser P, Trillo S 2018 Nat. Photonics 12 303
Google 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 1
Google Scholar
[32] Naveau C, Szriftgiser P, Kudlinski A, Conforti M, Trillo S, Mussot A 2019 Opt. Lett. 44 5426
Google Scholar
[33] Naveau C, Szriftgiser P, Kudlinski A, Conforti M, Trillo S, Mussot A 2019 Opt. Lett. 44 763
Google 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 041017
Google Scholar
[35] Goossens J W, Hafermann H, Jaouen Y 2019 Sci. Rep. 9 1
Google Scholar
[36] Vanderhaegen G, Szriftgiser P, Kudlinski A, Conforti M, Trillo S, Droques M, Mussot A 2020 Opt. Express 28 17773
Google Scholar
[37] Bao C, Jaramillo-Villegas J A, Xuan Y, Leaird D E, Qi M, Weiner A M 2016 Phys. Rev. Lett. 117 163901
Google 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 133033
Google Scholar
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