Yun-Hong Zhang(張?jiān)萍t) and Chong Liu(劉沖),3,?

1School of Physics,Northwest University,Xi’an 710127,China

2Shaanxi Key Laboratory for Theoretical Physics Frontiers,Xi’an 710127,China

3Peng Huanwu Center for Fundamental Theory,Xi’an 710127,China

Keywords: modulation instability,parametric resonance breather,three-mode truncation,energy exchange

1.Introduction

Modulation instability (MI)[1,2]is a fundamental physical process that exists in a variety of nonlinear systems.[3-6]It is characterized by the dynamical growth and evolution of periodic modulation on a continuous wave background.The MI process can be regarded as a degenerate four-wave mixing process,[7]in which the energy exchange between the central mode (pump) and the spectral sidebands is very intense.This dynamics is closely related to the celebrated Fermi-Pata-Ulam(FPU)recurrence.[8-12]In general,the full evolution of MI in integrable systems can be accurately described by the Akhmediev breathers.[13-17]Performing the Fourier spectral analysis for the Akhmediev breathers, one can obtain an accurate mathematical expression of FPU recurrence.[13,15-17]In contrast, exact description of full evolution of MI in nonintegrable systems remains an open issue.It is recently suggested that, the mode truncation method[18-20]can be used to provide an analytical description of the nonlinear stage of MI dynamics in non-integrable systems with parametric resonance(PR).[20-23]It is demonstrated in the periodic nonlinear Schr¨odinger equation(NLSE)that such parametric instability yields PR breathers in the nonlinear stage,which exhibits periodic oscillation in the evolution direction.[20]These results enrich greatly the types of nonlinear excitations induced by the parametric instability.

In this paper,we consider the parametric instability in the periodic pure-quartic NLSE where the fourth-order dispersion is modulated periodically.It should be pointed out that significant progress has been made on the pure-quartic NLSE in experiments,[24-30]where pure quartic solitons have been observed in photonic crystal waveguides.[24]Such experiments in turn stimulate intense theoretical studies,[31-41]in which nontrivial heteroclinic-structure transition of the pure-quartic MI has been revealed.[35]Thus, we expect more complex properties of parametric instability in the periodic pure-quartic NLSE.By using the Floquet analysis and the three-mode truncation method,we revealed in this paper,the complex nonlinear stage of parametric instability exhibits PR breathers with internal oscillation.These result could be useful for experimental observation of complex PR nonlinear excitations in pure-quartic NLSE optical systems.

2.Physical model and floquet analysis

The periodic pure-quartic NLSE is given by

wherezandtare the propagation distance and the retarded time,respectively,and thez-dependent fourth-order dispersion is given by

where ˉβ4denotes the average andgΛ(z)=cos(kgz) has periodΛ=2π/kg.When the parametric driving is absent, i.e.,βm=0, equation (1) reduces to pure-quartic NLSE,[24]and the MI exists only when ˉβ4＜0.In particular, the nonlinear stage of MI has be achieved recently by the three-mode truncation.[35]

However,in the case whenβm?=0,the MI becomes different.Namely, (i) the MI regime obtained by the Floquet analysis switches to ˉβ4＞0; (ii)the unique nonlinear dynamics of MI should be studied by the method of three-mode truncation.

Our aim is to obtain the Floquet map,which is defined by matrixΦ.The latter satisfies

We then consider the case whenβm?= 0.Parametric instability occurs whenk(Ωp) =pπ/Λ=pkg/2 (p=±1,±2,±3,...).The latter is the so-called parametric resonance condition.The instability frequencyΩpcorrespond to the tips of the Arnold tongues[20]

To do so, we follow the idea proposed in Ref.[20].Namely,a fiber with periodic dispersion parameters can be thought as two fiber segmentsaandbwith different dispersion parameters.The Floquet map is given by the product of two matrices describing each uniform segment

Figure 1 shows the gain spectra of the periodic pure quartic-NLSE on the (Ω,βm) plane when ˉβ4= 1.As can be seen, the Floquet analysis shows the MI bands which are presently known as Arnold tongues[20,22][Fig.1(a)].Such the MI exhibits narrow band around the tongue tip frequenciesΩp.Moreover,the profiles of the gain spectra whenβm=2 shown in Fig.1(b).Clearly, differentΩpare generally incommensurate, which greatly reduces the possibility that the harmonics of a probed frequency experience exponential amplification due to higher-order bands.Thus, the nonlinear stage of such MI can be studied by the method of three-modes truncation.

Fig.1.Results of the linear Floquet analysis for Eq.(1).(a)False color plot showing the first three Arnold tongues in the plane(Ω,βm).The black dotted line represents the Ωp, p=1,2,3.(b)The gain curve GF(Ω)at βm=2.In this paper,we focus on the wave dynamics in the first PR band.

3.The three-mode truncation method

Three-mode model allows us to unveil the nonlinear stage of parametric instability for the periodic pure-quartic NLSE.We consider the harmonically perturbed plane wave(HPPW)as follows:ˉβ4Ω4/12+2P=pkg, this value corresponds to the quasiphase matching relation of the parametric resonance.

For the periodic pure-quartic NLSE,Hexists four groups of stationary points (Δφe,ηe) (the solution of ˙η= ˙Δφ=0).Namely,

Figure 2 shows the bifurcation diagram and the level set ofHversus frequency Ω.Firstly, in the frequency range of Arnold tongue,the heteroclinic separatrix(ηe=0)divides the phase plane into inner and outer orbits [see Fig.2(b)].Secondly, once the frequency fall outside the Arnold tongue, the topological structure of HamiltonianHwill suddenly change,and the new heteroclinic separatrix divides the phase plane into three different domains[see Figs.2(c)and 2(d)].On the bifurcation diagram[see Fig.2(a)],the red and green lines intersect.It is worth noting that by comparing Figs.2(c) and 2(d), the heteroclinic separatrix will change toηe=1 fromηe=0 at the intersection of the two lines.

Fig.2.Bifurcation diagram and the level set of H.(a) Normalized sideband fraction η for the nonlinear eigenmodes as a function of normalized frequency Ω;the red lines correspond to Δφ =0,stable;the green lines corresponding to Δφ =π/2, unstable; the blue lines is the gain GF(Ω) when βm =2.Panels (b) and (d) show the Hamiltonian contours when Ω =2.8 inside PR gain bandwidth and Ω=2.87,Ω=2.9,beyond the PR gain bandwidth,respectively.

To get a deeper understanding of the dynamics, we analyzed the stability of the eigenmodes.It is checked by the given perturbation solution (Δφ,η) = (Δφe+δφ,ηe+δη).After linearizing the eigenmodes,the Jacobian matrix is given by

4.Heteroclinic structures

The structure illustrated in Fig.2 can characterize the nonlinear stage of parametric instability of the pure-quartic NLSE.To illustrate this point, we numerically integrate Eq.(1).The initial input condition is given by

whereη0?1,and Δφ0=θ0+φp/2.

Our discussion begins with the near-separation line dynamics of the system.As shown in Figs.3(a) and 3(b), nonlinear excitations induced by the PR are observed.Such nonlinear excitations are the PR breathers.The spectra exhibit FPU-like recurrence[see Figs.3(c)and 3(d)].We found that the nonlinear pattern with initial phase Δφ0=π/2 is similar to that with the phase Δφ0=0.However, such the two patterns correspond to different phase trajectories[see Figs.3(e)and 3(f)].When Δφ0=0, the trajectory is inside the separatrix, and the whole trajectory rotates to become a homoclinic cycles[see Fig.3(e)].When Δφ0=π/2, the trajectory spans the entire phase plane(-π,π)[see Fig.3(f)].Due to the rapid oscillations in the evolutionary direction,the effect of the initial phase on the evolution pattern cannot be directly revealed from the space-time evolution.This is an important characteristic of the hidden heteroclinic structure of PR in the periodic NLSE.By comparing to the numerical results of Eq.(16)(the red curve)with Eq.(1)(the blue scatter curve),there is a slight deviation between the numerical results of Eqs.(16) and (1),This is due to the fact that the three-mode truncation ignores the higher-order sidebands.

Fig.3.Two types of quasiperiodic recurrences from numerical integration of Eq.(1).(a)The intensity|A|2 when Δφ0=0.(b)The intensity|A|2 when Δφ0 =π/2.(c), (d)Fourier modes of central pump|A0|2 (black lines)and sideband power |A1|2 (red lines) versus z when Δφ0 =0 and Δφ0 =π/2,respectively.(e), (f) Projections of the quasiperiodic recurrences in phase plane with Δφ0 =0 and Δφ0 =π/2,respectively.Here βm =2,η0 =0.02,and Ω =2.8.

Whenη=0.001,Δφ0=0.298639π,the heteroclinic separatrix in the phase plane is obtained.The nonlinear evolution is shown in Fig.4(a).

Another interesting result is the internal oscillation structure of the PR breathers as shown in Fig.4(b).To illustrate clearly this point, we extract the evolution of the separatrix with differentβm.Moreover,by defining the parametric resonance periodD1(the distance between the two peaks formed by the parameter resonance)and the internal oscillation periodD2(the number of internal oscillation peaks contained in one resonance period), we analyzed the variation of the two periods in Fig.4(d)versusβm.As can be seen from the figure,D1keeps unchanged once the modulation frequencykgis fixed.However,D2varies asβmincreases.Specifically,withβmincreasing from 0, the oscillating structure began to appear inside the parametric resonance.Then,D2gradually decreases.Whenβm=0.9,the internal oscillation period reaches saturation.Asβm=2.6,D2increases again,and the internal oscillation decreases.

Fig.4.The evolution of PR breathers through numerical integration of Eq.(1).(a) The PR breathers of the intensity |A|2 correspond to the separatrix.Here βm =2, Ω =2.8, η0 =0.001, and the initial phase Δφ0 =0.298639π.(b)Enlarged view of the structure of the region inside the dotted line in panel (a).(c) Fourier modes of the central pump |A0|2 and the first-order sidebands|A1|2 versus z.(d)The variation of two periods D1 and D2 with βm.The red circle corresponds to D2 in panel(b).

5.The energy exchange between spectrum sidebands and the pump

Linear stability analysis suggests that the sideband growth rate is the largest at the frequency corresponding to the maximum gainGF, i.e., the conversion efficiency of the central pump and sideband is the largest.However,as shown in Fig.2(a),the intersection of the two lines on the bifurcation diagram falls outside the Arnold tongue.Thus,we guess that the strongest nonlinear transition occurs at frequencies outside the PR bandwidth.

Figure 5(a)shows the variation of the maximum achievable sideband as a function of frequency.It can be seen that the sideband fraction almost increases until it reaches the maximum at the frequencyΩc.Beyond the critical frequencyΩc,the energy contained in the sideband becomes lower, proving that the transition suddenly drops.Obviously, the result of nonlinear analysis deviates from that of Floquet (linear)analysis.There is a strong contrast between the frequency conversion dynamics before and after the frequencyΩc[see Figs.5(b) and 5(c)].Their evolutionary trajectories are located in different regions of the phase plane [Figs.5(d) and 5(e)].The critical frequency corresponding to the evolution along the heteroclinic orbit in Fig.2(c) can be calculated as the implicit solution of equation

This indicates that the largest achievable sideband occurs atΩ=Ωc.

Fig.5.(a)The output sideband fraction η(z=10)versus Ω from numerical integration for η0=0.03 and Δφ0=0(solid black curve);the red curve gives the maximum achievable conversion from Eq.(22),with superimposed small-signal PR gain GF (blue curve).(b), (c) Pump and sideband mode evolutions with the same initial condition around Ωc,they are Ω =2.87 and Ω =2.9,respectively.(d),(e)Corresponding evolution trajectories of panels(b)and(c)on the phase plane.Here βm =2, η0 =0.02, and the initial phase Δφ0=0.

6.Conclusion

In conclusion, we study the parametric instability in the periodic pure-quartic NLSE by using the Floquet analysis and the three-mode truncation method.We obtain the PR breathers with internal oscillation.Within the frequency range of Arnold tongue, the phase plane is divided into internal and external orbits by the heteroclinic separatrix.Once the modulation frequency exceeds the MI range, the heteroclinic separatrix will separate the phase plane into three different regions.

Moreover, we demonstrated that the maximum energy exchange between the spectrum sidebands and pump occurs outside the gain bandwidth.Our results reveal the richness of nonlinear dynamics in periodic pure-quartic NLSE, which could be useful for experimental observation of complex PR nonlinear excitations in pure-quartic NLSE optical systems.

Acknowledgements

Project supported by the National Natural Science Foundation of China (Grant Nos.12175178 and 12247103), the Natural Science Basic Research Program of Shaanxi Province,China (Grant No.2022KJXX-71), and the Shaanxi Fundamental Science Research Project for Mathematics and Physics(Grant No.22JSY016).