1. Introduction

The possibility of the propagation of solitons in optical fibers was theoretically predicted by Hasegawa and Tappert [1] and was experimentally demonstrated by Mollenauer et al. [2]. As we all know, the propagation of a picosecond optical pulse in a monomode optical fiber (not including optical fiber loss) is described by the nonlinear Schrödinger equation (NLSE) [1]

Depending upon the relative sign between the group velocity dispersion (GVD) and the self phase modulation (SPM) originating from nonlinearity, the NLSE admits two distinct types of solitons, namely, bright (εσ > 0) and dark (εσ < 0) solitons. Generally, bright solitons are well localized structures of light while dark solitons appear as localized intensity dips on a finite carrier wave background and are more robust than bright solitons.

The propagation of light pulses in birefringent fibers, multimode fibers, fiber arrays and also incoherent beam propagation in photorefractive media are more generally modelled by a system of coupled nonlinear Schrödinger (CNLS) type equations. Single mode fibers are actually bimodal because of the presence of birefringence. This birefringence creates two principal transmission axes within the fiber known as the fast and slow axes. When two or more optical fields with different frequencies co-propagate in a fiber, they can interplay through the cross-phase modulation (XPM) mechanism. To describe a two-channel wavelength-division multiplexing (WDM) soliton system with dispersion compensation and lumped amplification, we consider the generalized CNLS equations with distributed coefficients [3,4]

where Ψ₁(z, t) and Ψ₂(z, t) denote two orthogonal components of an electric field, z is the coordinate along the propagation direction of the carrier wave, and t is the retarded time. The second term represents GVD, the third and fourth terms are SPM and XPM for representing the nonlinear effect, and the term in the right stands for the amplification (g > 0) or the attenuation (g < 0). Bright soliton solutions of Eq. (2) have been investigated [4]. However, to our knowledge, similariton solutions of the CNLS equations with variable coefficients have been hardly discussed. Similariton [5–7], which maintains its overall shape but with their parameters, such as amplitudes and widths changing with the modulation of system parameters, has an increasingly vital role in optical fiber communication system.

In this paper, by employing the similarity transformation, Eq. (2) can be transformed into the standard NLSE [Eq. (1)]. Then by making the reverse transformation variables and functions, we obtain the exact bright and dark multi-similariton solutions for Eq. (2). As an application, we will discuss the nonlinear similariton tunneling effect, which is predicted as early as in 1978 by Newell [8]. It was found that in certain circumstances, depending on the ratio of soliton amplitude to barrier height, the soliton can tunnel through the barrier in a lossless manner [8–10]. The review of soliton tunneling principles and research as it currently stands in [11].

2. General similarity transformations

To connect Eq. (2) with Eq. (1), we begin by using the technique of variable transformation

then construct the similarity transformation [5–7]

where both θ and ϑ are constants, the amplitude ρ(z) and the phase ϕ (z, t) are real functions, Φ(Z, T) is a complex function, the effective propagation distance Z(z) and the similarity variable T(z, t) are both to be determined.

The substitution of Eqs. (3) with (4) into Eq. (2) leads to Eq. (1), and after some algebra one gets the following expressions for the pulse:

where D(z) = ∫^z ₀ β(s)ds represents the accumulated dispersion which influences the form of the amplitude, the width w(z), the phase, the chirp and the effective propagation distance, and α = [1 + s ₀ D(z)]⁻¹ is related to the wave front curvature and presents a measure of the phase chirp imposed on the wave. The subscript 0 denotes the initial values of the given functions at distance z = 0. Here free parameters have very clear physical meaning: s ₀ and d ₀ are the initial curvature and position of the wavefront, ρ ₀ and t ₀ are the initial amplitude and position of the pulse center, and w ₀ is related to the initial similariton width. Moreover, we stress that from Eq. (5) the chirp (s ₀ ≠ 0) is an essential feature for similaritons. Thus the difference between soliton and similariton is that pulses are chirped with s ₀ ≠ 0 in the similariton case but pulses are chirp-free with s ₀ = 0 in the soliton case in the analytical solution [Eq. (3)]. As reported, this linear chirp in the similariton leads to efficient compression or amplification and thus are particularly useful in the design of optical fiber amplifiers, and optical pulse compressors.

Note that the existence of such similariton solutions should satisfy the following necessary and sufficient condition between the gain or loss profile and the parameters

which degenerates into the restriction condition for soliton (s ₀ = 0) expressed as Eq. (8) in [4]. Thus our results are more general than them in [4].

3. Bright and dark similaritons

The solution of Eq. (2) can be obtained from the solution of NLSE [Eq. (1)] by exploiting a one-to-one correspondence. Employing the transformations [Eqs. (3), (4)] and Darboux transformation (DT) [12], one can obtain bright multi-similaritons for Eq. (2)

with

where DT times m = 1, …, n, j = 1, 2, complex spectral parameters ${\lambda }_m=\frac{1}{2}({\eta }_m+i{\xi }_m)$ , λ ^* _m is the complex conjugate of λ_m, T,Z,ρ(z) and ϕ (z, t) satisfy Eqs. (5) and (6). (φ _1,1(λ ₁),φ _2,1(λ ₁))^T is the eigenfunction corresponding to λ ₁ for Ψ₀ and ${\phi }_{j,1}=\exp (\frac{{\delta }_j}{2}+i\frac{{\kappa }_j}{2})$ , with

Inserting the zero seeding solution of Eq. (2) as Ψ₀ = 0 into Eq. (8), one can obtain one similariton solution for Eq. (2). Using that one similariton solution as the seed solution in Eq. (8), we can obtain two-similaritons. Thus in recursion, one can generate up to n-similaritons. Here we present bright one- and two-similariton solutions in explicit forms. The one similariton read

where δ ₁ and κ ₁ are given by Eq. (10). The analytical bright similariton pairs read

where $G_1=a_1\cosh {\delta }_2{e}^{i{\kappa }_1}+a_2\cosh {\delta }_1{e}^{i{\kappa }_2}+i{a}_3(\sinh {\delta }_2{e}^{i{\kappa }_1}-\sinh {\delta }_1{e}^{i{\kappa }_2})$ , F ₁ = b ₁ cosh(δ ₁ + δ ₂) + b ₂ cosh(δ ₁ − δ ₂) + b ₃ cos (κ ₂ − κ ₁), $a_j=\frac{{\eta }_j}{2}[{\eta }_j^2-{\eta }_{3-j}^2+{({\xi }_1-{\xi }_2)}^2]$ , $b_j=\frac{1}{4}\{{[{\eta }_1+{\left(-1\right)}^j{\eta }_2]}^2+{({\xi }_1-{\xi }_2)}^2\}$ , a ₃ = η ₁ η ₂(ξ ₁ − ξ ₂) and b ₃ = −η ₁ η ₂,j = 1,2. δ _j and κ _j are given by Eq. (10).

Since a one-to-one correspondence exists between Eq. (2) and the NLSE [Eq. (1)], one can derive dark (gray) multi-similaritons for Eq. (2). For simplicity, here we only present dark (gray) one- and two-similariton solutions in explicit forms. The exact gray similariton pairs read

where $G_2=4\mu ({\omega }_1+{\omega }_2-2\mu )-4i\frac{{\lambda }_1+{\lambda }_2}{{\eta }_1+{\eta }_2}\rho$ , $F_2=4{\mu }^2+{\left(\frac{{\lambda }_1+{\lambda }_2}{{\eta }_1+{\eta }_2}\right)}^2\rho$ , ρ = (ω ₁ − μ) (ω ₂ − μ), ω_j = (ξ_j − iη_j) [ξ_j + iη_j tanh(δ_j)]/μ, δ_j = η_j[T − T _j0 − 2(Ω + ξ_j)εZ],, $\phi (Z,T)=-2\epsilon ({\mu }^2+\frac{{\Omega }^2}{2})Z-\Omega T-{\phi }_0$ , λ_j = ξ_j + iη_j and μ = ∣λ_j∣, j = 1,2. And the exact dark one similariton read

From the expression of δ_j one can clearly see that velocities of each similariton in bright and dark similariton pairs are determined by ξ_jα(z)D(z)/w ² ₀ and (Ω + ξ_j)α(z)D(z)/w ² ₀, which are both related to the parameter ξ_j and GVD function β(z). Therefore, we can trap the velocity of each similariton to control the interaction between two-similaritons by designing appropriate system parameters. The initial position and the initial phase are related to the parameters δ _j0 and κ _j0 for bright simillariton pairs and the parameters T _j0 and φ ₀ for dark simillariton pairs, respectively. Moreover, the spectral parameters η_j and ξ_j control separating or interacting evolutional behavior of similaritons with the suitable initial separation (see Figs. 2 and 3).

4. Nonlinear similariton tunneling effect

To investigate the unique properties of the optical similaritons propagating through dispersion and nonlinear barriers, we consider two particular examples. The first one is a dispersion barrier (DB) or dispersion well (DW) [9]

and the second one is a nonlinear barrier (NB) or nonlinear well (NW) [10]

 

Fig. 1. Evolutions of nonlinear tunneling of bright similariton (s ₀ = 0.03) and soliton (s ₀ = 0) in the (a) and (b) dispersion barrier with h ₁ = 2, (c) dispersion well with h ₁ = −0.5 and (d) nonlinear barrier with h ₂ = 2. Other parameters: k = 2,z ₀₁ = z ₀₂ = 7, σ = ε = t ₀ = d ₀ = δ ₁₀ = w ₀ = 1, θ = η ₁ = ξ ₁ = 0.5, γ ₀ = β ₀ = −1.

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Fig. 2. The nonlinear tunneling of separating and interacting bright similariton pairs in the dispersion barrier with (a) and (b) h ₁ = 2,z ₀₁ = 7, (c) h ₁ = 2,z ₀₁ = 3 and (d) dispersion well with h ₁ = −0.9, k = 0.6,z ₀₁ = 5.2. The parameters are (a)η ₁ = ξ ₂ = 1.5, η ₂ = ξ ₁ = 1.4, θ ₁₀ = −θ ₂₀ = 2,ϕ ₁₀ = ϕ ₂₀ = 1, and (b)–(d)η ₁ = η ₂ = 1,ξ ₁ = −ξ ₂ = 1.5, θ ₁₀ = −θ ₂₀ = 5, ϕ ₁₀ = ϕ ₂₀ = 0. Other parameters are the same as that in Fig. 1.

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where h ₁ and h ₂ denote the heights of the DB (DW) or NB (NW), respectively. k is related to the barrier (or well) width, and z ₀₁ and z ₀₂ represent the longitudinal coordinates indicating the locations of barriers (or wells). The process of similariton amplification is nontrivial, because the potential jump β(z) (15) or γ(z) [Eq. (16)] necessitates the corresponding jump g from Eq. (7).

Figure 1 displays the dynamical evolution scenarios I = ∣Ψ₁∣² of the nonlinear tunneling of the single bright similariton with s ₀ ≠ 0 or soliton with s ₀ = 0 [Eq. (11)] through both the DB or DW [Eq. (15)] and the NB or NW [Eq. (16)]. As β(z) > 0, we assume h ₁ > −1, where h ₁ > 0 indicates the DB, and −1 < h ₁ < 0 represents the DW. Similarly, h ₂ > 0 and −1 < h ₂ < 0 denote the NB and NW, respectively. As shown in Figs. 1(a) and 1(b), it can be seen that when the similariton and soliton pass through the DB, the pulses are amplified and form the peaks, then attenuates and recovers original shape, respectively. When the similaritons pass through the DW and the NB, the pulses’ amplitudes diminish. The pulses form the dips, then attenuates and increases their amplitudes, as shown in Figs. 1(c) and 1(d), respectively. When the similariton goes through the NW, the dynamical behavior is similar to the case in Fig. 1(a) except increasing its amplitudes after passing across the well, and we omit it. From Figs. 1(a) and 1(d), the DB and NB have different effects on the propagation of the similaritons, i.e., one forms a peak, the other a dip.

In the high bit rate and long-distance optical communication systems, there are always multiple pulses [3]. Therefore, it is necessary to study the multi-similaritons transmission. From Fig. 2(a), similariton pairs pass through the DB and do not interact with each other along the optical fibers. That is what we want in the ultralarge capacity transmission systems and can enhance the capacity of the systems. Although not strictly valid, a two-similariton solution may be interpreted as the nonlinear superposition of two interacting one-similariton solutions approaching each other from infinity. Two interacting similaritons are exhibited in Fig. 2(b). From Figs. 2(c) and 2(d), one can make use of the barrier and well to regulate the influence of interaction by controlling their heights, widths and locations. If the locations of the barrier or well coincide with that of interaction, the barrier in Fig. 2(c) increases the level of interaction, while the well in Fig. 2(d) lessens the degree of interaction.

 

Fig. 3. Evolutions of nonlinear tunneling of dark similaritons in the (a) and (d) nonlinear barrier, and (b) and (c) dispersion barrier. The parameters are σ = −ε = γ ₀ = β ₀ = 1 with (a)η ₁ = ξ ₁ = 0.5,Ω = 1, (b)η ₁ = ξ ₂ = 1.5, η ₂ = ξ ₁ = 1.4,T ₁₀ = −T ₂₀ = 1, and (c) and (d) η ₁ = −ξ ₂ = 1.5, η ₂ = ξ ₁ = 1.4,T ₁₀ = −T ₂₀ = 10,z ₀₂ = 8. Other parameters are the same as that in Fig. 1.

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From Fig. 3(a) and 3(b), when the optical wave passes through the NB and DB, the dark similaritons forms a channel in the background near z = z ₀₂ and a peak near z = z ₀₁. The peak and channel originating from the NB and DB also influence the level of interaction, as shown in Figs. 3(c) and 3(d). If the locations of the barriers coincides with that of interaction, the DB in Fig. 3(c) can partially counteract the influence of interaction, and the shape in the location of interaction becomes from concave to convex. While the NB in Fig. 3(d) increases the degree of interaction and form a deeper dip in the location of interaction.

It was found that in certain circumstances, depending on the ratio of similariton amplitude to barrier height, the similariton can tunnel through the barrier in a lossless manner. Next we consider a dispersion barrier or well on an exponential background in the form

Figure 4 illustrates the evolution of the bright similariton through the dispersion barrier for the decaying or increasing parameter g ₀. When g ₀ = 0.055, the amplitude of the wave is almost unchanged. When g ₀ > 0.055 and g ₀ < 0.055, the amplitudes gradually increases and decreases along z, respectively. This implies that the parameter g ₀ can control the amplitude of similariton after passing through the barrier.

The dynamic behaviors of component Ψ₂ are similar to that of Ψ₁, and here we omit it.

5. Conclusions

In summary, we have analytically obtained bright and dark similaritons for the generalized CNLSE with distributed coefficients in the birefringent fiber. Under the parameter condition, we discuss the nonlinear similariton tunneling effect. The results show that the effect of dispersion barrier and well on the similaritons equals that of nonlinear well and barrier, respectively. One can make use of the barrier and well to regulate the influence of interaction between similaritons. The decaying or increasing parameter g ₀ can control the amplitude of similariton after passing through the barrier. We expect that these results for similariton tunneling would inaugurate a new and exciting area in the application of optical similaritons.

 

Fig. 4. Bright similariton through the dispersion barrier [Eq. (17)] for the different system parameter g ₀: (a) g ₀ = 0.1, (b) g ₀ = 0.055 and (c) g ₀ = −0.1 with r = 1. Other parameters are the same as that in Fig. 1.

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