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Stability plays a significant role in successfully observing Fermi-Pasta-Ulam-Tsingou (FPUT) phenomenon in experiment. However, there are few relevant studies in the literature. The primary object of this work is to study the stability of FPUT phenomenon in the nonlinear fibers numerically. In this study, we take the sinusoidally perturbed continuous waves (CWs) with white noise as the imposed initial condition, which can be readily realized in real experiments. We find that both the perturbation amplitude and phase difference between the perturbation and pump can drastically affect the stability of the resulting FPUT phenomenon. Firstly, as the perturbation amplitude increases, the FPUT phenomenon becomes much more stable. When the perturbation amplitude reaches a critical value, the most stable FPUT phenomenon can be observed. With the further increase of the perturbation amplitude, the stability of the resulting FPUT phenomenon weakens. Secondly, the phase difference between the perturbation and pump takes distinct effects on the stability of FPUT phenomenon for perturbation frequency located inside and outside the conventional modulation instability (MI) band. For the perturbation frequency located inside the conventional MI band, as the phase difference between the perturbation and pump increases from zero, the corresponding FPUT phenomenon first is less stable, and then becomes most instable at a critical phase difference; after that, the stability of the FPUT phenomenon is enhanced again. For the perturbation frequency located outside the conventional MI band, the stability of FPUT phenomenon is enhanced monotonically as the phase difference increases from 0 to π/2. In order to observe a much more stable FPUT phenomenon, as shown in the above results, the perturbation amplitude should be moderately large, and the phase difference between the perturbation and the pump should be appropriate to avoid the most instable FPUT phenomenon. 1. Introduction
In the early 1950s, Italian physicists Fermi, Pasta, Ulam and their student Tsingou simulated the evolution of a nonlinear system composed of 64 harmonic oscillators on the first generation of large computers, hoping to prove the principle of energy equipartition in statistical physics. At the initial moment, all the energy of these harmonic oscillators is concentrated on one oscillator, and the initial energy of other oscillators is zero. According to the principle of energy equipartition, the system should finally reach an equilibrium state, that is, the energy should be equally divided. [ 1 ] The discovery of this phenomenon has greatly promoted the development of nonlinear physics. The early research on FPUT phenomenon was mainly concentrated in the field of fluid mechanics. Because of the high cost of experimental observation requiring large ships and other experimental equipment, the research on FPUT phenomenon was gradually shelved. With the development of optical precision instruments and the progress of experimental technology, the observation of FPUT phenomenon in optical fibers was first realized in the laboratory in 2001. [ 2 ] Since then, the nonlinear physics community has set off a wave of research on the FPUT phenomenon. [ 3 – 15 ] .
The formation of FPUT phenomenon is related to the double periodic wave state and modulation instability of nonlinear system. [ 3 – 15 ] Modulation instability describes the exponential growth of a small disturbance applied to the plane wave of a system with the increase of propagation distance, which can effectively explain the initial growth process of the amplitude of the first-order disturbance signal with the increase of propagation distance in FPUT phenomenon. [ 16 – 18 ] , the growth of the amplitude of higher order perturbation components can be explained by the cascading modulation instability [ 6 , 19 , 20 ] When the amplitude of multiple perturbations increases to its maximum, the amplitude decays rapidly, but the reason has not yet been effectively explained. In theory, the relationship between different types of FPUT phenomena and modulation instability has also been studied. [ 21 ] The influence of high-order effect on FPUT and the frequency spectrum characteristics of FPUT under high-order effect are discussed. [ 22 , 23 ] The analysis of FPUT phenomenon by three-wave truncation method is explored. [ 13 ] In recent years, it has also been found that the FPUT phenomenon can be excited not only in the modulation instability region (called conventional modulation instability), but also outside the modulation instability region (called unconventional modulation instability). [ 24 – 26 ] This may be related to the fact that the nonlinear effect of perturbation is neglected in the calculation of modulation instability. In addition, the study of FPUT phenomenon has been extended to two coupled nonlinear systems. Based on the Manakov model equation, it is found that the two coupled systems have the characteristics of asymmetric spectrum. The main results include the analytical calculation of the related asymmetric spectrum. [ 27 ] Non-degenerate Akhmediev breather solution [ 28 ] And complex higher-order modulation instability evolution [ 29 ] In recent years, different techniques have been used to observe the FPUT phenomenon in optical fibers, such as the destructive "cutback" measurement. [ 30 ] All-fiber measurement method based on OTDR system [ 31 – 33 ] Cyclic fiber loop method based on periodic amplification [ 16 , 34 ] And other different technical means.
Stable nonlinear wave is the key to experimental observation. However, as far as we know, there are very few studies on the stability of FPUT phenomenon. The relevant studies mainly discuss the stability of FPUT phenomenon by analyzing the stability of a specific double-periodic wave. [ 26 , 35 ] The analysis results are limited to a specific analytical two-periodic wave solution, and the stability of FPUT phenomenon can not be directly analyzed from the actual experimental conditions. In this paper, the actual experimental conditions are considered, and the stability of FPUT phenomenon in optical fiber is analyzed by numerical simulation method with the sinusoidal disturbance plane wave with a certain noise as the incident condition. In the study, we are not limited to a specific value of the "disturbance" amplitude and "disturbance" frequency (many related literatures in the study.
2. Model equation
The governing equation describing the propagation of a slowly varying electric field envelope in a nonlinear single-mode fiber is the classical nonlinear Schrodinger equation, namely。
i∂a∂z±12β2∂2a∂t2+γ|a|2a=0, (1) Among a Represents the slowly varying envelope amplitude of the electric field; z Represents the transmission distance; t Represents the time coordinate in the group velocity reference frame; β 2 Is the group velocity dispersion parameter, which can be positive or negative ( β 2 0 stands for normal dispersion, β 2 < 0 stands for anomalous dispersion); γ Nonlinear coefficient representing the Kerr effect of an optical fiber.
Without loss of generality, for the convenience of analysis, we can put the model equation ( 1 ) is normalized to。
i∂A∂Z±12sgn(β2)∂2A∂T2+|A|2A=0, (2) Among。
A=√γLda,Z=zLd,T=zT0,Ld=T20|β2|,sgn(β2)=±1. (3) ( 3 ) In the formula L d Is the dispersion length, T 0 Is the initial pulse width.
The analysis later in this paper will be based on the normalization equation ( 2 ) to expand the discussion. Equation ( 2 The plane wave solution for) is。
A=A0exp(iA20Z), (4) Among A 0 Represents the amplitude of the incident plane wave. According to the linear stability analysis method, in the case of normal dispersion, that is, when β 2 0 (or sgn ( β 2 ) = 1), the fiber has no modulational instability, and in the case of anomalous dispersion, that is, when β 2 0 (or sgn ( β 2 ) = – 1), the fiber has modulation instability, based on the normalized model equation ( 2 The fiber gain spectrum of。
g=|Im(K)|=12|Ω|√4A20−Ω2,Ω<Ωc=2A0, (5) Among K Is a sinusoidal disturbance beam; Ω Is the sinusoidal perturbation frequency; Im ( K ) representative K The imaginary part of the optimal gain is。
gmax (6) We adopt in this paper initial conditions that are relatively easy to implement in experiments, namely。
\begin{split} A\left( {Z = 0,T} \right) =\;& A_0^{}\big[ 1 + \sqrt \delta \cdot {{\mathrm{e}}^{{\mathrm{i}}\varOmega t + {\mathrm{i}}{\varphi _0}}} \\ \;& + \sqrt \delta \cdot {{\mathrm{e}}^{ - {\mathrm{i}}\varOmega t + {\mathrm{i}}{\varphi _0}}} + {\mathrm{Noise}} \big] , \end{split} (7) Where the equation ( 7 ) The second and third terms on the right side represent two sinusoidal perturbation signals applied on either side of the pump light frequency ( \pm \varOmega Frequency components) whose amplitudes are all \sqrt \delta , φ 0 Represents the phase difference between the sinusoidal signal and the pump light, and the third term represents the inevitable white noise under the experimental conditions. In the simulation, let。
{\mathrm{Noise}} = 0.01 \times \left( {1 - 2 \cdot {\text{rand}}} \right) , (8) Where rand represents a series of random numbers uniformly distributed in the range [0,1], and the amplitude of the noise is 1% of the amplitude of the pump light.
3. 研究结果
本文采用的模拟方法为经典的谱方法与自适应步长的龙格库塔方法相结合的模拟方法. 在本文的模拟结果中, 不失一般性, 令平面波振幅A0 = 1(对应的最佳扰动频率为 {\varOmega _{\text{m}}} = \sqrt 2 , 临界扰动频率为 {\varOmega _{\text{c}}} = 2 ), 为使结果具有普适性, 同时不激发高阶谐波, 在我们的研究中, 正弦扰动频率Ω > Ωc/2 = 1, 其值不仅可以位于调制不稳定性区域内(Ω < 2), 也可以位于调制不稳定性区域外(Ω > 2); 正弦扰动振幅δ不局限于无穷小量, 其值任意.
3.1 扰动振幅对FPUT现象的影响
通过大量的数值模拟, 可以发现对一给定扰动频率, 扰动振幅可以极大地影响FPUT现象的稳定性. 图1(a)和图1(b)分别反应当扰动频率Ω = 1.5, φ0 = 0.5π, 扰动振幅δ = 0.01时扰动平面波随传输距离的时空和频率谱演化, 可看出在该参数条件下, 扰动平面波振幅可周期性的回归到初始态(z = 0处), 此即FPUT现象, 稳定的FPUT现象可传输至约z = 15处, 当z > 15, FPUT现象逐渐消失, 波形很快变得杂乱无章; 随着扰动振幅δ的增加, 发现FPUT现象的稳定性逐渐增强, 当增加至δ = 0.1, FPUT现象的稳定性增至最强, 如图1(c)和图1(d)所示(δ = 0.1), 稳定的FPUT现象可传输至约z = 22.5处; 随着扰动振幅的进一步增加, FPUT现象稳定性逐步减弱, 如图1(e)和图1(f)所示, 其中 δ = 0.25, 这时稳定的FPUT现象只可传输至约z = 15.
图 1 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(1%噪声), 参数为Ω = 1.5, φ0 = 0.5π (a), (b) δ = 0.01; (c), (d) δ = 0.1; (e), (f) δ = 0.25Fig. 1. The evolution of perturbed plane waves in nonlinear fibers with the propagation distance in the temporal and spectral domain (1% noise) with Ω = 1.5, φ0 = 0.5π: (a), (b) δ = 0.01; (c), (d) δ = 0.1; (e), (f) δ = 0.25.由图1可看出, 随着传输距离的增加, 规则的双周波演化(也即FPUT现象)最终都变得杂乱无章, 这应该与调制不稳定性及级联调制不稳定性相关. 由级联调制不稳定性效应所出现的高阶频率谱成分也具有调制不稳定性增益, 当这些高频成分的振幅长到一定值, 彼此之间及与低频率成分之间就产生相互作用, 从而改变波形演化的动力学特征; 另一方面, 噪声在实验中不可避免, 而噪声中也存在众多具有非零增益的频率成分, 这些频率成分的振幅随传输距离增加而快速增长, 当其增加到一定程度, 彼此之间都会产生相互作用. 故图1中规则的双周期波随着传输距离的增加都会变得杂乱无章. 这里的论述可用于解释下文中所有相关图中规则双周波演化最终都变得杂乱无章的原因.
通过比较图1可看出: 第一, 在给定的扰动频率下, 随着扰动振幅的增加, FPUT现象的稳定性先增强再减弱; 第二, 在给定的扰动频率下, 存在一个临界的扰动振幅δcr, 当δ < δcr时, 随着δ的增加, FPUT现象的稳定性逐渐增强; 当δ = δcr时, FPUT现象稳定性增至最强; 当δ > δcr时, 随着δ的增加, FPUT现象稳定性逐渐减弱.
上述结论也适用于扰动频率位于调制不稳定性带外的情况, 如图2所示(所用参数为Ω = 2.2 > Ωc = 2, φ0 = 0.5π, 扰动振幅从上到下依次是δ = 0.01, 0.25, 0.5). 当δ = 0.01时, 稳定的FPUT现象只可传输至距离约z = 9处, 当δ = 0.25时, 稳定的FPUT现象可传输距离增至约z = 22处, 当δ = 0.5时, 稳定的FPUT现象可传输距离减至约z = 13处.
图 2 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(1%噪声), 参数为Ω = 2.2, φ0 = 0.5π (a), (b) δ = 0.01; (c), (d) δ = 0.25; (e), (f) δ = 0.5Fig. 2. The evolution of perturbed plane waves in nonlinear fibers with the propagation distance in the temporal and spectral domain (1% noise) with Ω = 2.2, φ0 = 0.5π: (a), (b) δ = 0.01; (c), (d) δ = 0.25; (e), (f) δ = 0.5.临界扰动振幅δcr值与扰动频率密切相关, 其值随着扰动频率的增加而增加, 如图3所示. 此结论不仅适用于扰动信号与泵浦光之间相位差φ = 0.5π的情况, 还适用于其他相位差的情况, 主要区别是随着φ的减小, 较小扰动频率所需的临界扰动振幅δcr相对较小(比较图3中Ω < 1.7时φ = 0.5π和φ =0.3π对应的δcr值曲线), 反之则比较大(比较图3中Ω > 1.7时φ = 0.5π和φ =0.3π对应的δcr值曲线).
图 3 临界扰动振幅δcr与扰动频率的依赖关系(1%噪声)Fig. 3. The dependence of the critical perturbation amplitude (δcr) on the perturbation frequency (1% noise).3.2 扰动信号与泵浦光之间相位差对FPUT现象稳定性的影响
扰动信号与泵浦光之间相位差(φ0)是影响FPUT现象演化规律的另一个非常重要的物理 参量, 在图1和图2中, φ0 = 0.5π, 下面将探讨扰动振幅一定时, 相位差φ0从0逐渐增加至0.5π时所引起的对FPUT现象稳定性的影响, 为突出相位差φ0的影响, 我们在该部分的研究中令扰动振幅保持不变.
图4展示了当扰动频率位于调制不稳定性带内时相位差φ0对FPUT现象稳定性的影响. 如图4(a)所示, 当φ0 = 0.1π时, FPUT现象可比较稳定的传输到z = 25处; 当相位差φ0增至0.3π时, FPUT现象只可以较稳定的传输到z = 15处(图4(b)); 当相位差φ0继续增至0.5π时, 稳定的FPUT现象可传输距离又增至z = 20处(图4(c)). 由此可见, 相位差φ0可极大程度地影响FPUT现象的稳定性, 通过大量的数值模拟, 我们发现当φ0从0开始逐渐增加时, FPUT稳定性先逐渐减弱, 当φ0增加至某临界值φ0cr, FPUT稳定性减至最弱, 当φ0继续增加时, FPUT稳定性又逐步增强. 上述结论适用于调制不稳定性带内的其他扰动频率值.
图 4 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(1%噪声), 参数为Ω = 1.5, δ = 0.1 (a), (b) φ0 = 0.1π; (c), (d) φ0 = 0.3π; (e), (f) φ0 = 0.5πFig. 4. The evolution of perturbed plane waves with the propagation distance in the temporal and spectral domain (1% noise) with parameters Ω = 1.5, δ = 0.1: (a), (b) φ0 = 0.1π; (c), (d) φ0= 0.3π; (e), (f) φ0 = 0.5π.图5显示了当扰动振幅分别为δ = 0.1 和δ =0.01时, 临界值φ0cr随扰动频率的变化关系. 由图5可见, 临界值φ0cr随扰动频率的增加而快速减小, 且在较大的频率范围内, 临界值φ0cr随扰动振幅δ的增加而增加. 在此值得注意的是当扰动频率接近截止频率时, 即当Ω → 2时, φ0cr → 0, 也就是说当扰动频率接近截止频率, 且当扰动信号和泵浦光同相时, FPUT现象的稳定性最差.
图 5 扰动信号与泵浦光之间的临界相位差与扰动频率的依赖关系(1%噪声)Fig. 5. The dependence of the critical phase difference between the perturbation and the pump on the perturbation frequency (1% noise).当扰动频率位于调制不稳定性带外时, 相位差对FPUT现象稳定性的影响与扰动频率位于带内的情况截然不同, 随着相位差φ0的增加, FPUT现象稳定性呈现单调增强的趋势, 如图6所示, 当φ0 = 0.1π时(图6(a)), FPUT只可稳定的传输至约z = 5; 当φ0 = 0.3π时(图6(b)), FPUT可稳定的传输至约z = 15; 当φ0 = 0.5π时(图6(c)), FPUT可稳定传输距离增至约z = 20.
图 6 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(1%噪声), 参数为Ω = 2.2, δ = 0.25 (a), (b) φ0 = 0.1π; (c), (d) φ0 = 0.3π; (e), (f) φ0 = 0.5πFig. 6. The evolution of perturbed plane waves with the propagation distance in the temporal and spectral domain (1% noise) with parameters Ω = 2.2, δ = 0.25: (a), (b) φ0 = 0.1π; (c), (d) φ0= 0.3π; (e), (f) φ0 = 0.5π.3.3 噪声对FPUT现象稳定性的影响
噪声在实验中是不可避免的, 同时, 在数值模拟中, 也会因数值方法的不稳定性引起, 故在研究工作中必须考虑噪声对FPUT现象稳定性的影响. 图7展示了当(8)式中噪声系数为0.1时, 即当噪声的振幅为泵浦光振幅的10%时, 扰动平面波随传输距离的时空和相应的频谱演化, 其他参数与图1(噪声系数为0.01)完全相同. 比较图7与图1可看到噪声越大, 所观察到的FPUT现象循环数越少, FPUT现象稳定性越差, 该结论与Wabnitz和Wetzel研究的[5]结论一致. 物理上这是可以完全理解的, 噪声中有众多不稳定性增益频率成分, 其振幅随着传输距离的增加而增加, 当振幅达到一定量值, 彼此之间以及与一阶扰动频率成分之间均会产生相互作用, 从而破坏FPUT现象的稳定性演化. 当噪声系数增加时, 噪声中所有频率成分的初始振幅都将增加, 这时所有具有非零增益的频率成分的振幅都会随着传输距离的增加而极速增加, 故在较短的传输距离内就能彼此之间以及与一阶及级联的高阶扰动频率成分之间发生强相互作用, 从而快速破坏FPUT现象的稳定性传输.
图 7 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(10%噪声), 参数为Ω = 1.5, φ0 = 0.5π (a), (b) δ = 0.01; (c), (d) δ = 0.1; (e), (f) δ = 0.25Fig. 7. The evolution of perturbed plane waves in nonlinear fibers with the propagation distance in the temporal and spectral domain (10% noise) with Ω = 1.5, φ0 = 0.5π: (a), (b) δ = 0.01; (c), (d) δ = 0.1; (e), (f) δ = 0.25.上述扰动振幅及扰动信号与泵浦光之间相位差对FPUT现象的稳定性影响, 不能由调制不稳定性及级联调制不稳定性来解释, 它们只能用来解释一阶及高阶扰动振幅的初始增长, 由于能量守恒, 其振幅并不会无限的增加, 当振幅增加到一定值, 就会出现衰减至初始态的现象, 衰减的物理原因至今尚未清楚, 我们认为扰动振幅对FPUT现象的稳定性影响可能主要与其关联的严格双周期波的稳定性紧密相关. 经典的非线性薛定谔模型方程(1)是可积系统, 理论上讲, 具有非常多的严格双周期波解, 其中一些解在一定条件下可退化到(7)式的初始态, 该些解的稳定性可能均与(7)式中的扰动振幅δ和相位差φ0紧密相关, 而且与本文上述结论一致.
4. 总结与讨论
FPUT现象的稳定性是影响其实验观测的一个重要特性, 本文结合实验中的实际条件, 通过大量的仿真模拟研究了扰动振幅及扰动信号与泵浦光间相位差对FPUT现象稳定性的影响, 研究发现二者均可以极大程度地影响FPUT现象的稳定性.
当扰动振幅从无穷小量逐渐增加时, FPUT现象的稳定性逐渐增强; 当扰动振幅值达到临界扰动振幅值, FPUT现象的稳定性增至最强; 当扰动振幅继续增加时, FPUT现象的稳定性又逐渐减弱, 该临界扰动振幅值随扰动频率的增加而增加.
扰动信号与泵浦光间相位差对调制不稳定性带内和带外的FPUT现象稳定性影响不同. 当扰动频率位于调制不稳定性带内时, 随着相位差的增加, FPUT现象的稳定性逐步减弱; 当相位差增至临界值, FPUT现象的稳定性减至最弱; 随着相位差的继续增加, FPUT现象的稳定性又开始逐步增强. 临界相位差值随扰动频率的增加而减弱. 当扰动频率位于调制不稳定性带外时, FPUT现象的稳定性随着相位差的增加而增强.
噪声的强度并不影响上述主要结论, 但是噪声的存在将降低FPUT现象的稳定性, 噪声振幅越大, FPUT现象的稳定性越差.
利用上述主要结论, 可以用来指导FPUT现象的实验观测. FPUT现象并不能无限的周期性传输, 它的周期性总是在一定传输距离后被破坏掉, 由上述主要结论, 我们得知在实验中设置的扰动振幅要适当大, 不能受到调制不稳定性的局限性而把扰动振幅总设置在非常小的范围值内, 同时, 扰动信号与泵浦光的相位差要适当, 当扰动频率位于传统调制不稳定性区域内时, 该相位差要尽量避免中间值, 即要么使其为0, 要么为0.5π, 但是当扰动频率位于传统调制不稳定性区域外时, 该相位差要位于0.5π为最佳.
综上, 我们在研究中并不局限于某一特定值的扰动振幅和扰动频率, 其中扰动振幅任意, 扰动频率横跨了调制不稳定性区域内外的频率空间, 故我们的研究结果具有广泛性、可推广性、可为光纤中FPUT现象的理论和实验研究提供有益的帮助.
数据可用性声明
支撑本研究成果的数据集可在科学数据银行https://doi.org/10.57760/sciencedb.j00213.00066中访问获取.
[1] Fermi E, Pasta P, Ulam S, Tsingou M 1955 Studies of the Nonlinear Problems Los Alamos, May 1, 1955 pLA-1940
[2] Van Simaeys G, Emplit G, Haelterman M 2001 Phys. Rev. Lett. 87 033902
Google Scholar
[3] Akhmediev N N 2001 Nature 413 267
Google Scholar
[4] Devine N, Ankiewicz A, Genty G, Dudley J M, Akhmediev N 2011 Phys. Lett. A 375 4158
Google Scholar
[5] Wabnitz S, Wetzel B 2014 Phys. Lett. A 378 2750
Google Scholar
[6] Chin S A, Ashour O A, Blic M R 2015 Phys. Rev. E 92 063202
Google Scholar
[7] Bao C, Jaramillo-Villegas J A, Xuan Y, Leaird D E, Qi M, Weiner A M 2016 Phys. Rev. Lett. 117 163901
Google Scholar
[8] Kimmoun O, Hsu H C, Branger H, Li M S, Chen Y Y, Kharif C, Onorato M, Kelleher E J R, Kibler B, Akhmediev N, Chabchoub A 2016 Sci. Rep. 6 28516
Google Scholar
[9] Deng G, Li S, Biondini G, Trillo S 2017 Phys. Rev. E 96 052213
Google Scholar
[10] 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
[11] Naveau C, Szriftgiser P, Kudlinski A, Conforti M, Trillo S, Mussot A 2019 Opt. Lett. 44 763
Google Scholar
[12] Vanderhaegen G, Szriftgiser P, Kudlinski A, Conforti M, Trillo S, Droques M, Mussot A 2020 Opt. Express 28 17773
Google Scholar
[13] Sheveleva A, Andral U, Kibler B, Colman P, Dudley J M, Finot C 2022 Optica 9 656
Google Scholar
[14] Chen S C, Liu C 2022 Physica D 438 133364
Google Scholar
[15] Sinthuja N, Rajasekar S, Senthilvelan M 2023 Nonlinear Dyn. 111 16497
Google Scholar
[16] Kraych A E, Agafontsev D, Randoux S, Suret P 2019 Phys. Rev. Lett. 123 093902
Google Scholar
[17] Chowdury A, Ankiewicz A, Akhmediev N, Chang W 2018 Chaos 28 123116
Google Scholar
[18] Copie F, Suret P, Randoux S 2022 Opt. Lett. 47 3560
Google Scholar
[19] Kimmoun O, Hsu H C, Kibler B, Chabchoub A 2017 Phys. Rev. E 96 022219
Google Scholar
[20] Yin H M, Chow K W 2021 Physica D 428 133033
Google Scholar
[21] Mussot A, Naveau C, Conforti M, Kudlinski A, Copie F, Szriftgiser P, Trillo S 2018 Nat. Photonics 12 303
Google Scholar
[22] Liu C, Wu Y H, Chen S C, Yao X, Akhmediev N 2021 Phys. Rev. Lett. 127 094102
Google Scholar
[23] Yao X K, Liu C, Yang Z Y, Yang W L 2022 Phys. Rev. Res. 4 013246
Google Scholar
[24] Conforti M, Mussot A, Kudlinski A, Trillo S, Akhmediev N 2020 Phys. Rev. A 101 023843
Google Scholar
[25] 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. U.S.A. 118 e2019348118
Google Scholar
[26] Cheung V Y Y, Yin H M, Li J H, Chow K W 2023 Phys. Lett. A 476 128877
Google Scholar
[27] Chen S C, Liu C, Yao X, Zhao L C, Akhmediev N 2021 Phys. Rev. E 104 024215
Google Scholar
[28] Liu C, Chen S C, Yao X K, Akhmediev N 2022 Chin. Phys. Lett. 39 094201
Google Scholar
[29] Chen S C, Liu C, Akhmediev N 2023 Phys. Rev. A 107 063507
Google Scholar
[30] Hammani K, Wetzel B, Kibler B, Fatome J, Finot C, Millot G, Akhmediev N, Dudley J M 2011 Opt. Lett. 36 2140
Google Scholar
[31] Naveau C, Vanderhaegen G, Szriftgiser P, Martinelli G, Droques M, Kudlinski A, Conforti M, Trillo S, Akhmediev N, Mussot A 2021 Front. Phys. 9 637812
Google Scholar
[32] Hu X Y, Chen W, Lu Y, Yu Z J, Chen M, Meng Z 2018 IEEE Photon. Technol. Lett. 30 47
Google Scholar
[33] Vanderhaegen G, Szriftgiser P, Kudlinski A, Armaroli A, Conforti M, Mussot A, Trillo S 2023 Phys. Rev. A 108 033507
Google Scholar
[34] Goossens J W, Hafermann H, Jaouën Y 2019 Sci. Rep. 9 18467
Google Scholar
[35] Yin H M, Li J H, Zheng Z, Chiang K S, Chow K W 2024 Chaos 34 013120
Google Scholar
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图 1 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(1%噪声), 参数为Ω = 1.5, φ0 = 0.5π (a), (b) δ = 0.01; (c), (d) δ = 0.1; (e), (f) δ = 0.25
Figure 1. The evolution of perturbed plane waves in nonlinear fibers with the propagation distance in the temporal and spectral domain (1% noise) with Ω = 1.5, φ0 = 0.5π: (a), (b) δ = 0.01; (c), (d) δ = 0.1; (e), (f) δ = 0.25.
图 2 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(1%噪声), 参数为Ω = 2.2, φ0 = 0.5π (a), (b) δ = 0.01; (c), (d) δ = 0.25; (e), (f) δ = 0.5
Figure 2. The evolution of perturbed plane waves in nonlinear fibers with the propagation distance in the temporal and spectral domain (1% noise) with Ω = 2.2, φ0 = 0.5π: (a), (b) δ = 0.01; (c), (d) δ = 0.25; (e), (f) δ = 0.5.
图 3 临界扰动振幅δcr与扰动频率的依赖关系(1%噪声)
Figure 3. The dependence of the critical perturbation amplitude (δcr) on the perturbation frequency (1% noise).
图 4 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(1%噪声), 参数为Ω = 1.5, δ = 0.1 (a), (b) φ0 = 0.1π; (c), (d) φ0 = 0.3π; (e), (f) φ0 = 0.5π
Figure 4. The evolution of perturbed plane waves with the propagation distance in the temporal and spectral domain (1% noise) with parameters Ω = 1.5, δ = 0.1: (a), (b) φ0 = 0.1π; (c), (d) φ0= 0.3π; (e), (f) φ0 = 0.5π.
图 5 扰动信号与泵浦光之间的临界相位差与扰动频率的依赖关系(1%噪声)
Figure 5. The dependence of the critical phase difference between the perturbation and the pump on the perturbation frequency (1% noise).
图 6 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(1%噪声), 参数为Ω = 2.2, δ = 0.25 (a), (b) φ0 = 0.1π; (c), (d) φ0 = 0.3π; (e), (f) φ0 = 0.5π
Figure 6. The evolution of perturbed plane waves with the propagation distance in the temporal and spectral domain (1% noise) with parameters Ω = 2.2, δ = 0.25: (a), (b) φ0 = 0.1π; (c), (d) φ0= 0.3π; (e), (f) φ0 = 0.5π.
图 7 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(10%噪声), 参数为Ω = 1.5, φ0 = 0.5π (a), (b) δ = 0.01; (c), (d) δ = 0.1; (e), (f) δ = 0.25
Figure 7. The evolution of perturbed plane waves in nonlinear fibers with the propagation distance in the temporal and spectral domain (10% noise) with Ω = 1.5, φ0 = 0.5π: (a), (b) δ = 0.01; (c), (d) δ = 0.1; (e), (f) δ = 0.25.
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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] Van Simaeys G, Emplit G, Haelterman M 2001 Phys. Rev. Lett. 87 033902
Google Scholar
[3] Akhmediev N N 2001 Nature 413 267
Google Scholar
[4] Devine N, Ankiewicz A, Genty G, Dudley J M, Akhmediev N 2011 Phys. Lett. A 375 4158
Google Scholar
[5] Wabnitz S, Wetzel B 2014 Phys. Lett. A 378 2750
Google Scholar
[6] Chin S A, Ashour O A, Blic M R 2015 Phys. Rev. E 92 063202
Google Scholar
[7] Bao C, Jaramillo-Villegas J A, Xuan Y, Leaird D E, Qi M, Weiner A M 2016 Phys. Rev. Lett. 117 163901
Google Scholar
[8] Kimmoun O, Hsu H C, Branger H, Li M S, Chen Y Y, Kharif C, Onorato M, Kelleher E J R, Kibler B, Akhmediev N, Chabchoub A 2016 Sci. Rep. 6 28516
Google Scholar
[9] Deng G, Li S, Biondini G, Trillo S 2017 Phys. Rev. E 96 052213
Google Scholar
[10] 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
[11] Naveau C, Szriftgiser P, Kudlinski A, Conforti M, Trillo S, Mussot A 2019 Opt. Lett. 44 763
Google Scholar
[12] Vanderhaegen G, Szriftgiser P, Kudlinski A, Conforti M, Trillo S, Droques M, Mussot A 2020 Opt. Express 28 17773
Google Scholar
[13] Sheveleva A, Andral U, Kibler B, Colman P, Dudley J M, Finot C 2022 Optica 9 656
Google Scholar
[14] Chen S C, Liu C 2022 Physica D 438 133364
Google Scholar
[15] Sinthuja N, Rajasekar S, Senthilvelan M 2023 Nonlinear Dyn. 111 16497
Google Scholar
[16] Kraych A E, Agafontsev D, Randoux S, Suret P 2019 Phys. Rev. Lett. 123 093902
Google Scholar
[17] Chowdury A, Ankiewicz A, Akhmediev N, Chang W 2018 Chaos 28 123116
Google Scholar
[18] Copie F, Suret P, Randoux S 2022 Opt. Lett. 47 3560
Google Scholar
[19] Kimmoun O, Hsu H C, Kibler B, Chabchoub A 2017 Phys. Rev. E 96 022219
Google Scholar
[20] Yin H M, Chow K W 2021 Physica D 428 133033
Google Scholar
[21] Mussot A, Naveau C, Conforti M, Kudlinski A, Copie F, Szriftgiser P, Trillo S 2018 Nat. Photonics 12 303
Google Scholar
[22] Liu C, Wu Y H, Chen S C, Yao X, Akhmediev N 2021 Phys. Rev. Lett. 127 094102
Google Scholar
[23] Yao X K, Liu C, Yang Z Y, Yang W L 2022 Phys. Rev. Res. 4 013246
Google Scholar
[24] Conforti M, Mussot A, Kudlinski A, Trillo S, Akhmediev N 2020 Phys. Rev. A 101 023843
Google Scholar
[25] 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. U.S.A. 118 e2019348118
Google Scholar
[26] Cheung V Y Y, Yin H M, Li J H, Chow K W 2023 Phys. Lett. A 476 128877
Google Scholar
[27] Chen S C, Liu C, Yao X, Zhao L C, Akhmediev N 2021 Phys. Rev. E 104 024215
Google Scholar
[28] Liu C, Chen S C, Yao X K, Akhmediev N 2022 Chin. Phys. Lett. 39 094201
Google Scholar
[29] Chen S C, Liu C, Akhmediev N 2023 Phys. Rev. A 107 063507
Google Scholar
[30] Hammani K, Wetzel B, Kibler B, Fatome J, Finot C, Millot G, Akhmediev N, Dudley J M 2011 Opt. Lett. 36 2140
Google Scholar
[31] Naveau C, Vanderhaegen G, Szriftgiser P, Martinelli G, Droques M, Kudlinski A, Conforti M, Trillo S, Akhmediev N, Mussot A 2021 Front. Phys. 9 637812
Google Scholar
[32] Hu X Y, Chen W, Lu Y, Yu Z J, Chen M, Meng Z 2018 IEEE Photon. Technol. Lett. 30 47
Google Scholar
[33] Vanderhaegen G, Szriftgiser P, Kudlinski A, Armaroli A, Conforti M, Mussot A, Trillo S 2023 Phys. Rev. A 108 033507
Google Scholar
[34] Goossens J W, Hafermann H, Jaouën Y 2019 Sci. Rep. 9 18467
Google Scholar
[35] Yin H M, Li J H, Zheng Z, Chiang K S, Chow K W 2024 Chaos 34 013120
Google Scholar
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