1. Introduction

Solitons, the special wave envelopes that maintain their overall shapes, sizes and speeds during propagation and after collision [1], have been widely investigated in many fields of physics, especially in the contexts of nonlinear optics and Bose-Einstein condensates, where the governing equation is the famous standard nonlinear Schrödinger equation (NLSE). It is well-known that the standard NLSE admits sech (tanh) type soliton solution when the dispersion (or diffraction) and nonlinearity parameters are of the same (different) sign. Recently, a more general class of shape-preserving waves, which maintain their overall shapes but with their parameters such as amplitudes and widths changing with the propagation distance, have been found in nonlinear optical systems described by the nonautonomous NLSE with varying coefficients [2, 3, 4, 5, 6, 7, 8, 9], and in nonlinear optical systems described by the autonomous NLSE in the presence of amplification and spatial inhomogeneity [10, 11, 12, 13]. In contrast to the concept of solitons, these self-similar waves were called similaritons, including the exact similaritons and the asymptotic similaritons. The exact similaritons are described by the sech and tanh functions (correspond to the bright and dark similaritons) [3, 5, 8, 9, 12]. Note that the exact dark similaritons are embedded in the infinite-width continuous wave background. The asymptotic similaritons are mainly described by the Gaussian-Hermite functions and the compact parabolic functions (correspond to the Gaussian-Hermite and parabolic similaritons) [2, 4, 6, 7, 10, 11, 13]. They appear at the asymptotic limit, for example, the parabolic similaritons appear in nonlinear optical fiber amplifier when the the relative strength of nonlinearity is much larger than that of dispersion (diffraction). Compared to the exact similaritons where their generation need delicate balances between system parameters, the asymptotic similaritons can be easily generated (see [14, 15] for the generation of asymptotic compact parabolic pulses in dispersion-decreasing fibers).

In nonlinear fiber optics, dispersion-managed optical fibers are of fundamental interest in optical soliton transmission, since exact optical similariton can be formed in lossy fibers with exponentially-decreasing dispersion [16]. Although the exact multiple optical similaritons can be obtained by the inverse scattering technique [3], the initial conditions described by the multiple similariton solutions are difficult to realize in real applications. In fact, the simplest initial pulses are the direct linearly add (for bright solitons) or multiplication (for dark solitons, see Fig. 1) of many single solitons, with their separations larger than their widths. Due to the non-linearity, the interaction between these similaritons are inevitable. The interaction of bright similaritons were well studied, and it was shown that the interaction between two exact bright similaritons could lead to the formation of molecule-like bound state [9]. Also, the interaction of more than two bright similaritons were studied numerically in loss- and dispersion-managed fibers [17]. However,to the best of our knowledge, less work has been paid to the interaction dynamics of dark similaritons, especially when they are embedded in the varying and finite-width backgrounds.

To make the dark optical similaritons applicable in real applications, one should investigate how does the varying and finite-width background and the interaction between dark similaritons affect their propagations. In this paper, we shall consider these problems in the dispersion-managed fiber, where the governing equation for the slowly-varying pulse envelop u(z,t) is the following normalized NLSE:

Here β(z) represents the group velocity dispersion (GVD) parameter, and γ is the nonlinearity parameter. Fibers with varying GVD can be fabricated by changing their core diameters during the drawing process [18], or by using comb-like dispersion profiles [19]. Both approaches have been recently successfully adopted to the normal dispersion regime in order to experimentally generate compact parabolic self-similar pulses [14,20]. Higher-order dispersion is not included in Eq. (1), since it is possible to manufacture dispersion-managed fiber with reduced third-order dispersion [14, 21]. The fiber loss is ignored, but the method presented below can be easily applied to the nonlinear optical systems with dispersion, nonlinearity and amplification managements. Finally, for the existence of dark similariton, and without loss of generality, we hereafter set β(z) to be positive (normal dispersion regime) with β(0) = 1 and γ=1.

2. Self-similar transformation

In this section, we revist the self-similar propagation of the background for the propagation of dark similaritons. Firstly, we assume that the width of the background is proportional to W(z), then its amplitude is proportional to $1/\sqrt{W\left(z\right)},$, since the pulse energy P = ∫_-∞ ^∞∣u∣² dt in Eq. (1) is conserved. Secondly, as the width changes, the background acquires a local expansion velocity W_zt/W for the self-similar evolution, which equals −β(u ^* u_t − uu_t ^*)/2i∣u∣² = −β ϕ_t, such that the pulse phase contains a quadratic term ϕ = −W_zt ²/2βW [13]. Finally, inspired by the fact that for exact similariton there exists a one-to-one correspondence between the nonautonomous NLSE and standard NLSE [9], we anticipate that a NLSE with a trapping potential could be derived due to the requirement of a finite-width background. Introducing the following self-similar transformation [2, 22, 23]:

we cast Eq. (1) into the following NLSE with a harmonic trapping potential term:

by requiring that Z = −∫₀ ^z/W(z′)dz′, T = t/W with W(0) = 1, and ν(z) = β(z)/W(z). Here K(> 0) is a constant, and W(z) is determined by the following second-order differential equation:

Thus once the background function U(Z,T) and W_z(0) are known, the evolution of slowly-varying pulse envelope u(z,t) can be completely determined. It must be emphasized that the coefficient ν in Eq. (3) is generally not a constant, which is different from the self-similar transformations used in [2, 24] that reduce the nonautonomous NLSE into Eq. (3) with constant ν. Note that reducing the nonautonomous NLSE into Eq. (3) with constant ν, one certainly add constraints to the function form of the fiber parameters. Let us take Eq. (1) for an example. From the expression of ν and Eq. (4), we know that only when β(z) =exp[a(z+z ₀)] and W_z =β(z)/ν with a and z ₀ being arbitrary constant can we reduce Eq. (1) to the standard NLSE (the fact that ν is constant implies that β is proportional to W. Since β(0) = W(0) = 1, we have ν = 1). Similarly, if one reduce Eq. (1) to Eq. (3) with constant ν and constant K(≠ 0) [2, 24], from Eq. (4) it immediately follows that the GVD parameter must satisfy the following form

where $\delta =\sqrt{C^2+K}$[2]. However, if we reduce Eq. (1) to Eq. (3) with varying ν and constant K, the constraint in the function form of GVD can be weakened. As shown below, we only require the GVD to be the decreasing function of the propagation distance for the generation of parabolic similaritons [14, 15]. Therefore, when considering the pulse propagating in fibers with dispersion, nonlinearity and amplification managements, the advantages of our method is obvious, because the strict balance between fiber control parameters can be weakened a lot, or even eliminated.

Now we consider the finite-width background function U(Z, T). Note that in Eq. (3), the pulse energy is still P. When it is small, the nonlinearity is negligible, and the dark solitons cannot be generated. On the contrary, when the pulse energy is high and ν is small, according to the Thomas-Fermi approximation, the finite-width background can be well described by the following compact parabolic function:

at $\mid T\mid \le \sqrt{2A^2/K}=T_{max},$, and U_b = 0 otherwise, where the positive constant K is related to the energy and amplitude of the pulse as $P=\sqrt{32A^3/9K}.$.

The requirement of small ν can be easily satisfied. Specifically, when the GVD takes the form described by Eq. (5), we have ν = 1; otherwise, ν is a function of z. In real experiments, in order to obtain the parabolic intensity profile, β is always a decreasing function of the propagation distance z [14, 15], hence W is an increasing function of z, such that ν is less than 1. Therefore, if we let the transformed propagation distance Z varies from 0 to negative (corresponding to z varying from 0 to positive), then the transformed GVD parameter ν is less than 1 in the framework of Eq. (3).

Note that the center of the compact parabolic background given by Eqs. (2) and (6) is quiescent, according to the Galilei transformation: t → t − t_c, u → u exp[ib ₀(t − t_c) − ib ₀ ² D/2], where t_c =t_c(0) − b ₀ D with b ₀ and D being the arbitrary constant and ∫₀ ^z β(z′)dz′, respectively, the moveable compact parabolic background can be obtained [6]. Also note that the compact parabolic background is linearly chirped, which can be incorporated with highly efficient pulse compression [11].

3. The approximate dark similaritons

Since the GVD and nonlinearity parameters are of different sign, dark solitons can be formed on the compact parabolic background. We firstly consider the propagation of dark solitons in the framework of Eq. (3), since the corresponding background (6) does not vary with respect to the transformed propagation distance Z. Then we will take into account the variable transformations used in the self-similar transformation (2) and the Galilei transformation, and discuss the propagation of dark similaritons in the original coordinates z and t.

We recall that, when ν is a constant, the approximate dark soliton solution of Eq. (3), which is embedded in the compact parabolic background (6), can be written as follows [24]:

where Q = q√ν (< T_max) and A√ν sin ϕ are respectively the center-of-mass and velocity of the dark soliton. When the dark soliton does not initially rest on the coordinate origin or it has an initial velocity, i.e., Q(0) ≠ 0 or sin ϕ(0) ≠ 0, it will oscillate periodically, and the center-of-mass motion of the dark soliton can be approximately captured by Q_zz + KvQ/2 = 0 and Q_z = A√ν sinϕ, when its oscillation amplitude is small (Q ≪ T_max) and the pulse energy P is large enough [24, 25]. In this case, the dark soliton behaviors like a quasiparticle of mass 2/ν, oscillating harmonically in the trap, and its energy E = Q_z ²/ν + KQ ²/2 is a conserved quantity. Note that here the energy is composed of the kinetic and potential energy of the quasiparticle, instead of the pulse energy P defined before. Also note that the oscillation frequency of the center-of-mass of the dark soliton is $\sqrt{\mathrm{Kv}/2};$; as its center-of-mass oscillates, the dark soliton periodically changes its degree of darkness, and this periodic transformation of the black soliton (sinϕ = 0) to the gray one (sinϕ ≠ 0) and vice versa causes the oscillation frequency of dark soliton smaller than that of the linear oscillator described by the Schrödiger equation [26].

When ν varies smoothly and slowly, the dark soliton gradually sheds parts of its energy that appears in the form of dark solitons with larger speeds. When the emission of dark solitons (radiation) is not severe, it is reasonable to assume that the main dark solitons can be approximately described by Eq. (7), with the varying mass 2/ν and energy E = Q_z ²/ν + KQ ²/2. Considering the fact that ν is a function of the transformed propagation distance Z, we obtain the following governing equation for the center-of-mass motion of the main dark soliton

after differentiating the energy with respect to Z; and the parameter ϕ, which is related to the darkness of dark solitons, can be determined through the equation Q_z = A√ν sin ϕ, as that for dark soliton when ν is constant.

We then estimate how slowly should ν be. From Eq. (7), we find that the coefficient of T is A sinϕ/ν, which is inversely proportional to the effective width of dark soliton. When ν is constant, as ϕ periodically changes, the dark soliton can maintain its dynamical stability, i.e., the dark soliton can change periodically from black to gray. In this case, the fastest change rate of its inverse width is proportional to $\sqrt{K/2}$. When v varies with respect to Z, the change rate of the inverse width of the main dark soliton, which is caused by the variation of ν, is nearly proportional to ∣v_z/2ν∣. Therefore, the main dark can maintain its stability when ∣v_z/ν∣ ≤ $\sqrt{2K.}$. This is just the validity criteria for Eq. (8) to hold. From the expression for ν and the relation between z and Z, one could find that ∣v_z/ν∣ = β_zW/β − W_z. As demonstrated before, generally, we need the GVD to be a decreasing function of the propagation distance and hence W is a increasing function of z. Therefore, at the limit z → ∞, the right hand term of Eq. (4) can be ignored; and hence we find that W_z = β. Thus, for exponentially decreasing GVD, i.e., β = exp(−λ_z), we have ∣v_z/ν∣ = β_zW/β − W_z = λ. That is, Eq. (8) is valid when $\lambda \le \sqrt{2K}.$. Similarly, we find numerically that as soon as ${\beta }_z/\beta \le \sqrt{2K,}$, the condition $\mid v_Z/v\mid \le \sqrt{2K}$ is always satisfied.

Taking into account the self-similar and Galilei transformations, we find that the function form of the dark similariton of Eq. (1)can be written as follows:

It is clear that the center-of-mass of the dark similariton is t_d = t_c + t_r, where t_r = WQ is the relative displacement of the dark similariton to the center of the compact parabolic background t_c. The second derivative of tr with respect to the propagation distance z reads:

where t_r(0) = Q(0) − t_c(0). This equation shows that the dark similariton experiences a force F(t_r,t_rz), which is not only related to its position and velocity, but also to the GVD parameter, the width of the compact parabolic background and their derivatives. Specifically, when the GVD takes the special form (5), the above equation can be greatly simplified, which reads

Finally, according to the discussion in Eq. (8), we have

Up to now, we have proposed an analytical expression to describe the propagation of single dark similariton on the finite-width self-similar background. We then consider the interaction of N-dark similaritons on the finite-width self-similar background. Similarly, we will firstly consider the interaction of N-dark solitons on the compact parabolic background (6) in the framework of Eq. (3).

When the neighbor dark solitons do not overlap, that is, ∣Q_i − Q _i±1∣ > 1/A, it is reasonable to assume that the multiplication of N single dark solitons

is an approximate solution of Eq. (3). Under the small amplitude oscillation approximation of dark solitons, we have found that the repulsive interaction force between i-th and j-th dark soliton can be described by [27]:

when ν is a constant, where Q_ij = Q_i − Q_j is the relative separation between i-th and j-th dark solitons. The force increases as Q_ij decreases before the two dark solitons begin to overlap, and then decreases to zero as Q_ij approaches zero. The latter is not physical since the repulsive interaction force should be a monotone decreasing function of the the relative separation when the dark solitons are described by Eq. (12). However, in numerical simulations, we have found that the above expression could give a very good description of the dark solitons’ trajectories, provided that arbitrary two dark solitons are initially well separated. We have also compared our results with that given in Ref [28], a extreme good agreements were found.

 

Fig. 1. Interaction of two dark similaritons on the self-similar compact parabolic background. The initial parameters for the first dark soliton (left) are Q ₁(0) = −5,ϕ ₁(0) = 0, that of the second (right) are Q ₂(0) =5,ϕ ₂(0) =π/8. Other parameters are A=2, K = 0.01, C = −0.1, t_c(0) = 1 and b = 0.1. Red and blue lines represents the theoretical predictions and numerical simulations, respectively.

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Fig. 2. Evolution of three dark similaritons on the self-similar compact parabolic background (contour plot of ∣u∣²). The red, blue and green lines represent the trajectories of dark similaritons for Q ₁ (0) = 5,ϕ ₁(0) = π/8, Q ₂(0) = 0,ϕ ₂(0) = − π/8, and Q ₃(0) = −5,ϕ ₃(0) = 0, respectively. Other parameters are the same with that in Fig. 1. Inset: the comparison between the numerical/theoretical (blue/red lines) results.

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When ν varies smoothly and slowly, we assume that the interaction force between dark solitons can still be described by Eq. (13). Then, following the same procedure done for single dark similaritons, the center-of-mass of each dark similariton is t_di = t_c + t_ri (i = 1,2, …, N is the index of dark similaritons) with t_ri = WQ_i, which is determined by

 

Fig. 3. Evolution of the dark similaritons, which initially locate at t = −1, with ϕ(0) = 0 in (a) and ϕ(0) = −π/5 in (b). From tom to bottom, z =30, 27, 24, 21 and 18. The black and color lines are the results of theoretical predictions and numerical simulations, respectively. The parameters used are A = 2,T_max = 30, W_z(0) = 0, t_c(0) = b = 0, and β(z) = exp(−0.01z).

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Fig. 4. Evolution of two dark similaritons on the self-similar compact parabolic background (contour plot of ∣u∣²). The red and blue lines represent the trajectories of dark similaritons for t _r1 (0) = 5,ϕ ₁(0) = π/10 and t _r2(0) = −5,ϕ ₂(0) = −π/10, respectively. The background parameter are the same with that in Fig. 3, and β = 1 − 0.01z. Inset: the comparison between the numerical/theoretical(blue/red lines) results.

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where Δ = A(t_ri −t_rj)/W√ν, t_ri(0) = Q_i(0) − T_c(0), and sinϕ_i, is determined by (t_ri W_z/W −t_riz)/A√ν. Then, Eq. (14), together with self-similar and Galilei transformations and Eq. (12), determines the interaction dynamics of dark similaritons on the finite-width, self-similar parabolic background.

4. Numerical simulations

The theoretical predictions have been compared to the direct numerical simulation results of Eq. (1), where good agreements, not only for the centers but also for the depths of each dark similaritons, are shown in Fig. 1 and Fig. 2 for the interaction of multiple dark similaritons on the self-similar compact parabolic background. Note that the background itself is moveable. Also note that when the GVD takes the special form (5), the parameter ν in the reduced NLSE (3) is 1, therefore, there are almost no emission of dark solitons. However, when the GVD takes other forms, the emission of dark solitons (radiation) increase [see Fig. 3(a) for exponentially-decreasing GVD], which could even badly distort the parabolic profile of the background [29] [see Fig. 3(b)]. In this case, it is hard to exactly determine the center-of-mass motion of the dark similaritons, but we found that, through numerical simulations, the analytical expressions give a close estimation for the evolution of the dark similaritons.

Further numerical simulations show that the theoretical expressions give a good description for the evolution of dark similaritons when they are initially well separated [see Fig. 4 for linearly-decreasing GVD]. Note that the power profile at z = 8 does not agree with that of numerical simulations, this is because dark similaritons are too close such that the linear superposition (12) breaks down.

5. Conclusions

In summary, we have presented an analytical method to describe the interaction dynamics of multiple dark similaritons embedded on the finite-width, moveable, self-similar parabolic background. Two kind of interactions are considered. The first is the interaction between dark similariton and the finite-width and varying background, and the other is the interaction between dark similaritons. It was found that, if the nonautonomous NLSE can be casted into a standard NLSE with an additional harmonic trapping term, the evolution of dark similaritons on the parabolic background is stable and exactly predictable; otherwise, the propagation of dark similaritons is accompanied with dark wave emission, which will severely destroy the parabolic background when the velocity of dark similariton is large. However, even in this case, our analytic expressions could give a close predictions for the evolution of dark similaritons.

Since the generation of compact parabolic background in dispersion-decreasing fibers are feasible, we believed that the validity of our analytical expressions for the interaction of dark similaritons can be easily checked by the experiments. Further, we also believed that our method can be extended to study the propagation and interaction of dark similaritons in fibers with dispersion, nonlinearity and amplification managements. As mentioned in Section 2, our method can weaken the strict balance between the fiber control parameters such as dispersion, nonlinearity and amplification for the propagation of dark solitons, which will present an easy way for the dark soliton management.