1. Introduction

Propagation of wave packets in nonlinear media with periodically varying dispersion or refractive index is one of the fundamental problems of modern photonics. Such remarkable applications as optical pulse transmission in dispersion managed (DM) fiber links [1], [2], a stretched pulse generation in mode-locked laser systems and recirculating fiber loops [3], [4], evolution of soliton-like beam in a periodically modulated nonlinear waveguide [5], [6] should be mentioned as examples.

An impressive progress was achieved in this area from the theoretical point of view during the last years. Such effective methods as the guiding-center concept, different variational approaches, the multiscale theory, and the numerical averaging method [1], [7] have been developed. Dark, gray and antisymmetric solitons were recently found inDMstructures [8], [9], [10]. Sequences of DM solitons can be effectively used for data transmission because of enhanced robustness. Moreover, interaction between DM solitons can be strongly suppressed under appropriate conditions [11], [12].

Recently, basic properties of doubly-periodical nonlinear waves (cn-, dn- and sn-type) in dispersion-managed systems were considered [13]. It was shown, that in the strong localization limit the energy of breathing elliptic wave is well above that of classical one with a constant average dispersion. Possibility of stabilization of elliptic waves in DM fibers was also demonstrated.

In this paper our attention is focused on the cases of weak and moderate localization that have essentially new features in comparison with soliton propagation. The analysis is based on the concept of the truncated elliptical wave that means its representation by a finite sum of harmonics with specially adjusted coefficients. Similar approach was used for the study of modulation instability [14], [15]. For weak localization this procedure offers quite acceptable approximation, which always can be revised by increasing number of harmonics. Fixing the average dispersion, the dispersion difference and increasing the map period we show how the regular propagation dynamics similar to that in fiber with average dispersion is replaced by the chaotic behavior. Growth of the map period leads to the appearance of regions with chaotic behavior in the phase space coexisting with the regions of quasi-periodic behavior. Close vicinity of a mapping point corresponding to the truncated cnoidal wave can still remain neutrally stable, but small perturbations move the solution into the chaotic region. Long-scale DM (when the dispersion map period is comparable with the longitudinal period of the cnoidal wave) leads to the deterministic KAM chaos even in the case of lowest dimensionality of the phase space [16]. Note, that the scenario of transition to chaos has much in common with the polarization chaos in nonlinear birefringent resonators [17].

The paper is organized as follows. First, we discuss the mathematical formulation of the problem and approximation of elliptic waves with a finite number of harmonics. Then we examine in detail the case of lowest dimensionality in the spectral domain that offers the opportunity of mapping on the Poincaré sphere. It seems to be very special case, nevertheless, it is capable to model cn, sn and dn-type waves. It also demonstrates such important features as possibility to find self repeating (breathing) solutions and transition to chaos. Finally, we discuss the expansion of the method to bigger number of harmonics, physical meaning and limitations of the solutions obtained.

2. Mathematical model and system parameters

We use the (1+1)D Schrödinger type equation for the lossless nonlinear optical fiber modified to include longitudinally varying group velocity dispersion d(ξ):

In this equation q(η,ξ)=(L_dis /L_spm )^1/2 $I_0^{-1/2}$ A(η,ξ) is the normalized complex amplitude; A(η,ξ) is the slowly varying envelope; I ₀ is the peak input intensity; η=(t-z/u_gr )/τ ₀ is the running time; τ ₀ is the characteristic time scale; $u_{\mathit{gr}}={\left(\frac{\mathit{dk}}{d\omega }\right)}_{\omega ={\omega }_0}^{-1}$ is the group velocity; k ₀=k(ω ₀) is the wave number; ω ₀ is the carrying frequency; ξ=z/L_dis is the normalized propagation distance; L_dis =${\tau }_0^2$/|β ₂| is the dispersion length; coefficient ${\beta }_2={\left(\frac{d^{2}k}{d{\omega }^2}\right)}_{\omega ={\omega }_0}$ is defined by the group velocity dispersion (GVD) for a standard communication fiber; L_spm =2c/(ω ₀ n ₂ I ₀) is the self-phase modulation length; n ₂ is the coefficient of nonlinearity.

The normalized dispersion coefficient in Eq.(1) for one period of the two-step symmetric dispersion map is introduced by the relations

Here L ₀=(2a+b)L is the period of the dispersion map, d ₀>0 is one half of the dispersion difference, a,b are positive parameters; the average dispersion is given by d_av =(2a-b)/(2a+b)d ₀, and L is the characteristic length. Note, that the first segment with anomalous group velocity dispersion (focusing) is followed by a segment with a normal GVD (defocussing) and terminated with the segment with the anomalous GVD.

Two specific doubly-periodic solutions of Eq.(1) are known for the constant anomalous GVD d(ξ)=d_av >0 in a form of elliptic dn- and cn- waves.

and one stationary periodic solution for normal constant dispersion d(ξ)=d_av <0:

Here cn(η,ξ),sn(η,ξ),dn(η,ξ) are Jacobi elliptic functions; 0≤m≤1 is the modulus of the elliptic function that describes the degree of localization of the wave field energy; κ>0 is the arbitrary form-factor; ψ ₀ is the constant phase. The transverse period of the dn-wave equals l_dn =2K(m)/κ, where K(m) is the elliptic integral of the first kind, whereas transverse periods of cn- and sn- waves are equal to to l_cn =l_sn =4K(m)/κ. It is worth mentioning that the analytical solutions given by Eqs. (3–5) are a good initial guess for calculation of the profile of the true breathing elliptic wave by the numeric averaging method [4,7]. Trigonometric series for elliptic functions are well known:

$sn(\eta ;m)=8\pi l_{\mathit{sn}}^{-1}\sum _{n=1}^{\infty }{\rho }^{n-\frac{1}{2}}{\left(1-{\rho }^{2n-1}\right)}^{-1}\sin \left[\frac{2\pi \left(2n-1\right)\eta }{l_{\mathit{sn}}}\right],$ and $\rho =\exp \left[\frac{-\pi K\left(\sqrt{1-m^2}\right)}{K\left(m\right)}\right]$ . In the limit of weak localization m→0, K(m)→π/2, $K\left(\sqrt{1-m^2}\right)\to \infty$ , and only few terms are sufficient for adequate representation of corresponding elliptic functions.

In general complex form the truncated elliptic wave can be written as:

where S_n (ξ) are the amplitudes of harmonics, the sum is taken over appropriate frequencies given by Eqs. (6). For our normalization, the smallest nonzero Ω_n is of an order of unity. By substitution of Eq. (7) into Eq. (1), it is possible to obtain the system of ordinary differential equations in the Hamiltonian form:

and the Hamiltonian is given by:

where each index l takes any possible value of n.

Intuitively it is clear that the solution of the truncated set of equations is adequate if the contribution of higher harmonics in the solution of the complete equation remains small upon propagation. This is true for the stationary cnoidal waves, where amplitudes of harmonics remain the same, thus higher harmonics do not grow. In fact, the solutions of truncated equations which approximate cnoidal waves can be found quite easily. They correspond to the Hamiltonian minimum on the sphere of constant intensity I=∑S_n S*_n=const. If we restrict the minimization procedure to real amplitudes, the resulting values coincide quite well with the Fourier coefficients of the cnoidal wave expansion Eq. (6) with the same period and the same intensity, if the number of harmonics is sufficient to represent well the cnoidal wave with these parameters.We will call these solutions truncated elliptic waves. Dn-type truncated elliptic waves are obtained if the number of harmonics is odd, cn- and sn- waves, if the number of harmonics is even, depending on the symmetry/antisymmetry of coefficients. Truncated cnoidal waves are neutrally stable, small variations of initial amplitudes produce a small jitter around the trajectory of the cnoidal wave in the phase space. This is the consequence of the complete integrability of the nonlinear Schrödinger equation - the solution moves to the nearby invariant torus.

3. Truncated elliptic waves in a dispersion-managed fiber

The dispersion management takes the situation away from the complete integrability. According to the KAM theorem for weak perturbation most invariant tori survive, but degenerate trajectories break into chaotic regions. For bigger perturbation the volume occupied by chaotic trajectories grows and finally all the phase space becomes chaotic. First of all we will illustrate this process when only two harmonics are involved (weak localization, m≪1). Note, that both dn- and cn- type solutions can be approximated, if we take into account the symmetry of coefficients S_n =S_-n . Sn-type solutions are obtained with an even number of harmonics and antisymmetric amplitudes.

 

Fig. 1. Phase portraits of the dn-type wave on the Poincaré sphere for 3 harmonics taken into account. B=2^3/2 Im(S _Ω S*₀), C=2^3/2 Re(S _Ω S*₀), Ω=2.The dispersion management period is L ₀=1.2L. a - L=0.1. The center of the pattern B=C=0,A=1 corresponds to a constant intensity. It is unstable stationary point of the map. Two stable periodic points in upper and lower parts correspond to truncated cnoidal waves, the upper is marked with an arrow. b - L=0.75, c - L=0.95, d - L=1.25.

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In the case of two harmonics the mapping on the Poincaré sphere proves to be very effective tool for analysis of the dynamic behavior in general and transition to chaos in particular. The Stokes parameters are introduced instead of the spectral amplitudes by following relations:

The intensity conservation leads to the identity:

thus the trajectories A(ξ),B(ξ),C(ξ) lie on a sphere. Their shape can be determined by rewriting Hamiltonian in terms of Stokes parameters, the trajectories are intersections of a sphere with parabolic cylinders for the case of dn-wave, and with elliptic cylinders in a case of cnwaves (see [18] for details). To illustrate the propagation in a dispersion-managed fiber we calculate numerically and plot on the Poincaré sphere mapping points corresponding to the solution parameters after consecutive dispersion map periods. Typically, 150 points per trajectory are taken. The sum of intensities of harmonics (the sum of amplitude squares) is taken equal to 1. The Stokes parameters were scaled to be plotted on the sphere with a radius 1 both for cn-and dn- cases.

The dn-type wave evolution is presented in Fig. 1. We have chosen the dispersion map given by the Eq. (2) with d ₀=1,a=0.5,b=0.2. The length L from Eq. (2) served as a variable parameter (the period of the map is then L ₀=1.2L). For the completely integrable case (uniform fiber with positive dispersion, no DM) the mapping points form the closed lines corresponding to the solution trajectories.

If L is small in comparison with a typical longitudinal period of the cnoidal waves (which is physically of an order of dispersion length in our case), the trajectory pattern is very similar to that one with a dispersion equal the average d_av =2/3 (see Fig. 1-a). When the map period grows, chaotic trajectories appear close to the unstable periodic points of the map. To trace the chaos development one could change the nonlinearity strength or the temporal period instead. The length changes were chosen because the transition there is clearly seen.

For the dn-type solution, the point in the center of the circle in Fig. 1 corresponds to uniform intensity distribution (only zeroth harmonic exists), and it is unstable because of the modulation instability for anomalous GVD and positive nonlinearity. The trajectory passing through this point becomes chaotic first. For L=0.75 (Fig. 1(b)), the chaotic region is quite small, but it increases rapidly with L, (Figs. 1(c) and (d)). The process is characterized by a formation of a typical homoclinic tangle near the unstable periodic points. The point in the phase space corresponding the breathing dn-type cnoidal wave is quite close to this chaotic region. While the nearest vicinity of cnoidal wave remains stable with growing dispersion map period L, the region of the neutral stability diminishes, and finally even the trajectories very close to the cnoidal wave point become chaotic. For moderate dispersion management there is also a big number of chains with alternating stable and unstable periodic points which is typical for KAM chaos (Figs. 1(c) and (d)).

We will now briefly discuss the key features of KAM chaos [16]. For integrable Hamiltonian systems, trajectories in the 2N-dimensional phase space lie on the N-dimensional tori. The action-angle variables can be introduced, and each trajectory has N characteristic frequencies. The general Kolmogorov-Arnold Moser theorem states that for small perturbations in integrable Hamiltonian systems most of invariant tori survive, but some trajectories with degenerate frequencies start to break into chaotic regions. In our case the Poincaré sphere is the two-dimensional subspace of the four-dimensional phase space, and the closed line of the non-perturbed problem (without DM) are intersections of the sphere with the invariant tori. For dn-wave, the degenerate torus corresponds to the stationary point B=C=0 on the sphere. One stable and one unstable manifold start in the vicinity of this stationary point. They are formed by points which either approximate the stationary point with any iteration moving forward or backwards in time. The manifolds produce the eight-figure, the point which starts on the unstable manifold in the close vicinity of the stationary point first moves away and then approximates it on the stable manifold.

For dispersion management case the picture is different - the manifolds are arranged in very complicated loops called homoclinic tangle [16]. The center of the eight-figure becomes diffuse, as it can be seen in Fig. 1(b). The trajectories which have periodic points under perturbation first break into chains with alternating hyperbolic and elliptic periodic points, and similar homoclinic tangles are formed in the vicinity of hyperbolic ones. Some of such chains can be seen in Figs. 1(c) and (d), and this behavior is quite typical. The numerical example of the homoclinic tangle formation with more detailed description related to the polarization dynamics in nonlinear resonators can be found in Ref. [17].

For the cn-case, the topology of the trajectories is differnt. Here the unstable trajectory does not exist for the integrable case (Fig. 2(a)). Thus, with the growth of L the chains of periodic points of the map appear, and then the chaos develops close to unstable periodic points of these chains. Thus, for the cn-wave for the same period and intensity as the dn-wave, the chaotic region develops for bigger dispersion management length, and starts quite far away from the the region of truncated cnoidal wave in the phase space. Consequently, cn-waves are more stable with respect to a long-scale dispersion management. Note, that for the weakly localized sn-wave patterns on the sphere are qualitatively similar to that for cn-wave, so we will not go into details.

 

Fig. 2. The same, as in Fig. 1, but for the cn-type wave with 4 harmonics taken into account. A=2(S _1/2Ω S*_1/2Ω-S _3/2Ω S*_3/2Ω), C=2Re(S _3/2Ω S*_1/2Ω), Ω=2. The dispersion management periods are L=0.1, 1.4, 1.8, and 2.8 for pictures a–d.

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When the number of harmonics taken into account grows, the main features of the presented scenario are still valid. The truncated cnoidal wave exists, and for weak dispersion management there is a solution which corresponds to a wave with dispersion averaged over the period. The long-scale dispersion management leads to the appearance of chaotic regions in a phase space. The projection on the Poincaré sphere in multi-harmonics case is less informative, because even for the integrable case the projections do not form closed lines. Some amount of jitter appears due to the growth of harmonics number. To illustrate the influence of additional harmonics on the solution we have calculated the propagation of the cn-type wave using 4, 8 and 10 harmonics truncation (Fig. 3). It is seen that for small L the solution retains its structure. The threshold value of L producing developed chaotic behavior diminishes if additional harmonics are involved. If the perturbation is not symmetric, the cnoidal wave can start to move, but generally retains its structure (Fig. 3(e)).

The chaotic behavior for KAM chaos is quite subtle and it does not mean the sharp onset of the chaos in all the phase space, but rather the appearance of small chaotic regions first. The volume occupied by these regions increases with a “degree of non-integrability”, which depends on nonlinearity strength and the perturbation parameters (length of the dispersion management period in our case). When the phase space is many-dimensional, visualizing the corresponding processes is quite complicated in itself. For our simulation we have chosen the parameters of the wave for which the dynamics for no dispersion management is quite well described by only two independent amplitudes. For the Poincaré sphere representation this means that the trajectory remains in the region close to the point B=C=0 on the sphere, where higher harmonics are small in comparison with central ones (see Figs. 1(a) and 2(a)). The KAM theorem means that then the dynamics is still well described by two amplitudes for small dispersion management. This is confirmed by the calculation, because for small dispersion management addition of new harmonics in the truncation scheme does not change drastically the dynamics (Figs. 3(a) and (b)). As the chaos first develops in the region of phase space well described by two-amplitude dynamics (e.g., Fig. 1(b)), the truncation is generally adequate to describe the chaos onset. For big dispersion management, when the trajectories on the Poincaré sphere spread to the bigger region, the truncated dynamics with only two independent amplitudes is no more adequate. Thus, our simulations model rather the onset of the chaotic behavior, than the developed chaos for strong nonlinearity and dispersion management.

 

Fig. 3. Influence of higher harmonics on the cnoidal wave. The intensity distributions over 200 periods in a wave which is close to the cn-wave and has initial parameters S _1/2Ω=cos(0.29)/√2, S _3/2Ω=sin(0.29)/√2 are shown. a - 4 harmonics, L=0.1, b - 10 harmonics, L=0.1 c - 4 harmonics, L=3, d - 8 harmonics, L=2.4, e - same as b, but initial conditions are not symmetric S _-3/2Ω=(1+0.2i)S _3/2Ω, S _-1/2Ω=(1+0.2i)S_1/2Ω

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4. Concluding remarks

Concluding the paper we would like to mention that truncated elliptic waves demonstrate very rich and interesting propagation dynamics in dispersion-managed fibers including neutrally stable propagation, quasi-periodic behavior and transition to Kolmogorov-Arnold -Moser chaos. Some specific features of this nontrivial dynamics appear at the level of the lowest dimensionality and are related rather to the discreteness of the truncated elliptic waves in the spectral domain and the integrability of corresponding governing equations, than with the number of spectral degrees of freedom. The key parameters that define the propagation scenario are the degree of nonlinearity proportional to light intensity, nonlinearity constant and temporal period, and the degree of deviation from integrable task given in our case by the dispersion map period. Both factors are necessary for the chaotic behavior development.