# Modulation Instability, Breathers, and Bound Solitons in an Erbium-Doped Fiber System with Higher-Order Effects

Authors: Rui Guo; Hui-Qin Hao; Xiao-Song Gu
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We mainly investigate the generalized nonlinear Schrödinger-Maxwell-Bloch system which governs the propagation of optical solitons in nonlinear erbium-doped fibers with higher-order effects. We deduce Lax pair, analyze modulation instability conditions, construct the Darboux transformation, and derive the Akhmediev breathers, Ma-breathers, bound solitons, and two-breather solutions for this system. Considering the influences of higher-order effects, propagation properties of those solitons are discussed.


---

## Body

## 1. Introduction

In recent years, optical solitons have attracted many researchers for their potential applications in optical fiber transmission systems [1–3]. Based on the balance of the self-phase modulation and group velocity dispersion, the propagation of optical solitons in the picosecond regime is usually governed by the nonlinear Schrödinger [4] equation
(1)iqz+qtt+2|q|2q=0,
where q denotes the slowly varying complex envelope of the wave. When considering propagation characters of the ultrashort pulses, (1) cannot describe the corresponding physical mechanism due to the absences of the fourth-order dispersion, higher-order nonlinearities, and self-steepening effects. Owing to the above three factors, the dynamic features of the ultrashort pulses can be depicted by the following generalized nonlinear Schrödinger equation (GNLS) [5, 6]:
(2)iqz+qtt+2|q|2q+τ(qtttt+8|q|2qtt+2q2qtt*+6q*qt2ffffdf+4|qt|2q+6|q|4q)=0,
where τ is a small dimensionless real parameter, and it is usually positive. In addition, (2) can also govern the nonlinear spin excitations in one-dimensional isotropic biquadratic Heisenberg ferromagnetic spin with the octupole-dipole interaction [7, 8].In real optic fibers, the attenuation usually exists, in this context, erbium-doped fibers can minimize the attenuation [9]. The mathematical description of solitons propagating in erbium-doped fibers is the nonlinear Schrödinger-Maxwell-Bloch (NLS-MB) equations [10, 11]:(3a)qz=i(12qtt+|q|2q)+2p,(3b)pt=2iωp+qη,(3c)ηt=-(qp*+pq*),where subscripts z and t denote the partial derivatives with respect to the longitudinal distance and retarded time. ω is the frequency, the asterisk denotes the complex conjugate, p=v1v2*, and η=|v2|2-|v1|2 with v1 and v2 representing the wave functions in a two-level system [12, 13]. Many research achievements about system (3a), (3b), and (3c) have been obtained [14–16].However, when taking the effects such as the fourth-order dispersion, higher-order nonlinearities, and self-steepening effects into account, the propagation of optical solitons in fibers doped with two-level resonant impurities like erbium is usually described by the following generalized nonlinear Schrödinger-Maxwell-Bloch (GNLS-MB) system [9]:(4a)qz=i(qtt+2|q|2q)+iτ(qtttt+8|q|2qtt+2q2qtt*+6q*qt2+4|qt|2q+6|q|4q)+2p,(4b)pt=2iωp+qη,(4c)ηt=-(qp*+pq*).To our knowledge, investigations on system (4a), (4b), and (4c) have not been reported, and the aim of this paper is mainly to investigate the modulation instability conditions, generate the breather and bound solutions, and discuss the dynamic behaviors of those solutions for system (4a), (4b), and (4c).The outline of this paper will be as follows: in Section2, we will derive Lax pair and analyze the modulation instability conditions for system (4a), (4b), and (4c). In Section 3, by using the Darboux transformation, we will construct two types of one-breather solutions: Akhmediev breathers and Ma breathers on the nonzero continuous wave (cw) background. In Section 4, we will discuss analytically the interactions between neighboring bound solitons and two-breather solutions for system (4a), (4b), and (4c). Finally, our conclusions will be addressed in Section 5.


## 2. Lax Pair and Modulation Instability for System(4a), (4b), and (4c)

Employing the Ablowitz-Kaup-Newell-Segur formalism [17], we can derive the Lax pair for system (4a), (4b), and (4c) as(5a)Ψt=UΨ,(5b)Ψz=VΨ,where Ψ=(ψ1,ψ2)T (T denotes the transpose of a matrix) and the matrices U and V have the form(6a)U=-iλU1+U0,(6b)V=λ4V4+λ3V3+λ2V2+λV1+V0+1λ+ωV-1,with λ as a spectral parameter,
(7)U1=(100-1),U0=(0q-q*0),V4=8iτU1,V2=(-2i-4iτ|q|2-4iτqt-4iτqt*2i+4iτ|q|2),V3=-8τU0,V1=(2τq*qt-2τqqt*2q+4τ|q|2q+2τqtt-2q*-4τ|q|2q*-2τqtt*-2τq*qt+2τqqt*),V0=(a11a12a21-a11),V-1=i(η-p-p*-η),a11=i|q|2+3i|q|4-iτ|qt|2+iτq*qtt+iτqqtt*,a12=iqt+6iτ|q|2qt+iτqttt,a21=iqt*+6iτ|q|2qt*+iτqttt*.
Through symbolic computations, one can verify that system (4a), (4b), and (4c) can be concluded through the zero curvature equation Uz-Vt+UV-VU=0.Modulational instability (MI), which results from the interplay between nonlinearity and dispersive effects, refers to a parametric process in which a continuous or quasi-continuous wave undergoes a modulation of its amplitude or phase in the presence of weak perturbations [18–20]. The MI has some applications in condensate physics, plasma physics, hydrodynamics, nonlinear optics, and some other branches of physics [18]. Now by introducing the steady-state cw solutions for system (4a), (4b), and (4c) as q=q0exp⁡2i(κ+q02+3q04τ)z, p=iκq0exp⁡2i(k+q02+3q04τ)z, η=ωκ, here, with p0 being the input power, we will discuss the modulation instability process for system (4a), (4b), and (4c).We examine the MI process of the steady-state solutions by introducing the following perturbed solutions:(8a)q=(q0+A1)exp⁡2i(κ+q02+3q04τ)z,(8b)p=i(κq0+A2)exp⁡2i(κ+q02+3q04τ)z,(8c)η=ωκ+B,where A1, A2, and B are weak perturbations with the assumed general expressions being A1=u1cos⁡(Kz-Ωt)+iv1sin(Kz-Ωt), A2=u2cos⁡(Kz-Ωt)+iv2sin(Kz-Ωt), B=bcos⁡(Kz-Ωt), where u1, v1, u2, v2, and b are real amplitudes of those perturbations, Ω denotes the real frequency of modulation perturbations, and K is the real disturbance wave number. Inserting (8a), (8b), and (8c) into (4b) and (4c), we can obtain the linearized equations of u1, v1, u2, v2, and b, so u2, v2, and b can be solved as follows:(9a)u2=2ωκ(2u1ω-v1Ω)4q02+4ω2-Ω2,(9b)v2=2κ(2q02v1+2v1ω2-u1ωΩ)4q02+4ω2-Ω2,(9c)b=2κq0(v1Ω-2u1ω)4q02+4ω2-Ω2.Substituting (8a), (8b), (9a), and (9b) into (4a) and (4b), collecting the linear terms, we can derive two linear homogeneous equations for the perturbed unknown functions u1 and v1:(10a)(4q02-Ω2)y1u1+y2v1=0,(10b)y2u1-Ω2y1v1=0,with
(11)y1=2κ-(4q02+4ω2-Ω2)(1+6q02τ-Ω2τ),y2=4κωΩ+K(4q02+4ω2-Ω2).Equations (10a) and (10b) have nontrivial solutions if and only if the following determinant formed by the coefficients matrix vanishes, that is,
(12)|(4q02-Ω2)y1y2y2-Ω2y1|=0.
Equation (12) leads to the dispersion relation of K and Ω which determines the modulation instability process of the steady-state cw solution as
(13)K=4κωΩ+|Ωy1|Ω2-4q02Ω2-4q02-4ω2.
From (13), one can conclude that if Ω2-4q02<0, the value of K will be complex; then the modulation instability will take place with |Im⁡K| as the instability growth rate. So Ω2-4q02<0 is the condition of the modulation instability for system (4a), (4b), and (4c).


## 3. One-Breather Solutions for System (4a), (4b), and (4c)

Based on Lax pair (5a) and (5b), we can construct the following Darboux transformation for system (4a), (4b), and (4c) as(14a)q′=q-2i(λ1-λ1*)ϕ1ϕ2*|ϕ1|2+|ϕ2|2,(14b)p′=pχ12χ2(|ϕ1|2+|ϕ2|2)2-p*(λ1-λ1*)ϕ12ϕ2*2χ2(|ϕ1|2+|ϕ2|2)2+2ηχ1(λ1-λ1*)ϕ1ϕ2*χ2(|ϕ1|2+|ϕ2|2)2,(14c)η′=η(1+2(λ1-λ1*)|ϕ1|2|ϕ2|2χ2(|ϕ1|2+|ϕ2|2)2)+p(λ1-λ1*)χ1ϕ2ϕ1*χ2(|ϕ1|2+|ϕ2|2)2-p*(λ1-λ1*)χ1*ϕ1ϕ2*χ2(|ϕ1|2+|ϕ2|2)2,with(15)χ1=(λ1+ω)|ϕ1|2+(λ1*+ω)|ϕ2|2,χ2=(λ1+ω)(λ1*+ω).
Considering the nonzero continuous wave background, we can take q=qcexp⁡i(κcz+ωct), p=ipcexp⁡i(κcz+ωct), and η=ηc as the initial seeds, where qc, pc, ηc, κc, and ωc are all real parameters. System (4a), (4b), and (4c) requires the following nonlinear dispersion relation:(16a)pc=-12(2qc2-κc-ωc2+6qc4τ-12qc2ωc2τ+ωc4τ),(16b)ηc=-14(2ω-ωc)×(2qc2-κc-ωc2+6qc4τ-12qc2ωc2τ+ωc4τ).By the method of separation of variables and the superposition principle, we can arrive at(17a)ϕ1=(c1exp⁡θ1+c2exp⁡θ2)exp⁡i(κcz+ωct),(17b)ϕ2=c3exp⁡θ1+c4exp⁡θ2,where
(18)θ1=t2[-iωc+-ωc2-4qc2-4λ12-4λ1ωc]+z2[-iκc+-κc2+4a2-4b2-4iaκc],θ2=t2[-iωc--ωc2-4qc2-4λ12-4λ1ωc]+z2[-iκc--κc2+4a2-4b2-4iaκc],
with
(19)a=2iλ12(-1+4λ12τ)-iqc2(-1+4λ12τ-4λ1ωcτ+3ωc2τ)+3iqc4τ+iηcλ1+ω,b=qc(4λ1qc2τ+2λ1-ωc-4λ12ωcτ-2λ1ωc2τllllllllllll+ωc3τ-8λ13τ-6qc2ωcτ)+pcλ1+ω,
and c1, c2, c3, and c4 are complex constants satisfied as
(20)c3=1qc[(-iωc+-ωc2-4qc2-4λ12-4λ1ωc)iωc+iλ1+12llllllllllll×(-iωc+-ωc2-4qc2-4λ12-4λ1ωc)]c1,=(L+iM)c1,c2=qc(1/2)(-iωc+-ωc2-4qc2-4λ12-4λ1ωc)+iωc+iλ1c4=(L+iM)c4.
Suppose that
(21)λ1=12(As+iκs),c4=c1,ζ1+iη1=-ωc2-4qc2-4λ12-4λ1ωc,ζ2+iη2=-κc2+4a2-4b2-4iaκc,
with As, κs, ζ1, η1, ζ2, and η2 being real numbers. Now, substituting (17a) and (17b) into (14a), (14b), and (14c), we can obtain the solutions for system (4a), (4b), and (4c) as(22a)q=G1F1exp⁡i(κcz+ωct),(22b)p=G2F2exp⁡i(κcz+ωct),(22c)η=G3F2,where
(23)G1=qc-2i(λ1-λ1*)[eθ1+eθ2(L+iM)]×[eθ2*+eθ1*(L-iM)],F1=eθ1*[2Leθ2+(1+L2+M2)eθ1]+eθ2*[2Leθ1+(1+L2+M2)eθ2],G2=ipc[μ2+σ12σ2*2(λ1-λ1*)2]+2η0σ1σ2*(λ1-λ1*)μ,F2=(λ1+ω)(λ1*+ω)×[4Lcosh⁡(θ1+θ2*)+2(L2+M2+1)cosh⁡(θ1+θ1*)],G3=ipc(λ1-λ1*)[σ1*σ2μ+σ1σ2*μ*]+ηc[(λ1+ω)(λ1*+ω)(σ1σ1*+σ2σ2*)2lllllllllllll+2σ1σ1*σ2σ2*(λ1-λ1*)2],
with
(24)σ1=eθ1+eθ2(L+iM),σ2=eθ2+eθ1(L+iM),μ=σ1σ1*(λ1+ω)+σ2σ2*(λ1*+ω).Now, we mainly discuss soliton solutions from three different cases.Case 1.
In the case ofqc=0, that is to say, the initial seeds for (4a), (4b), and (4c) are qc=0, pc=0, and ηc=1, solutions (22a), (22b), and (22c) reduce to one-soliton solutions.Case 2.
In the case ofqc≠0, As=ωc=0, and κs2>4qc2, symbolic computation results in the following:
(25)ζ1=-4qc2+κs2,η1=0,ζ2=ζ1h14ω2+κs2,η2=ζ1h24ω2+κs2,
with
(26)h1=2qc2-κc+κs2+6τqc4+2τqc2κs2+τκs4,h2=-2ωκs-4ωτκsqc2-2ωτκs3.
Substituting (25) into (17a) and (17b), we can reduce solutions (22a), (22b), and (22c) to the breathers displayed in Figures 1 and 2.The parameters adopted here areκs=2.3, qc=1, κc=3, ω=1, and τ=0.
(a)
(b)
(c)The parameters adopted here areκs=2.3, qc=1, κc=3, ω=1, and τ=0.1.
(a)
(b)
(c)From Figure1, we can observe that the main features of propagations of those breathers- are periodic in the space coordinate and aperiodic in the time coordinate, so the solitons shown in Figure 1 are Akhmediev breathers [21–23]. In addition, we can observe that Figures 1(a) and 1(c) depict bright breathers, and Figure 1(b) shows the dark one.Comparing Figure2 with Figure 1, one can see that in Figure 2, under the influence of the increasing values of the parameter τ, the number of peaks on the same space interval is increasing when τ goes up from 0 to 0.1. So, we can conclude that the parameter τ controls the period of the Akhmediev breathers.Case 3.
In the case ofqc≠0, As=ωc=0, and κs2<4qc2, through direct computation, one can obtain the following:
(27)ζ1=0,η1=4qc2-κs2,ζ2=η1h1κs-2h2ω4ω2+κs2,η2=η12h1ω+h2κs4ω2+κs2.
Substituting (27) into (17a) and (17b), solutions (22a), (22b), and (22c) become other breathers as displayed in Figure 3.The parameters adopted here areκs=1.3, qc=1, κc=3, ω=1, and τ=0.1.
(a)
(b)
(c)From Figure3, one can observe that those breathers are periodic in the time coordinate and aperiodic in the space coordinate; that is, those are Ma-breather solitons [24–27], and for functions q and η, the solutions are bright solitons while for function p the solution is the dark one. In addition, one can find that the separations between adjacent peaks in Figure 3 gradually increase as κs→2qc and eventually reduce into the rogue waves, the properties of which have been discussed in [28–31].


## 4. Dynamic Features of Two-Soliton Solutions for System (4a), (4b), and (4c)

In this section, we will construct two-soliton solutions for system (4a), (4b), and (4c). Taking the same seeds in Section 2, that is, q=qcexp⁡i(κcz+ωct), p=ipcexp⁡i(κcz+ωct), η=ηc, and iterating the DT twice, one can obtain
(28)(ψ1ψ2)=[λ2(1001)-HΛH-1](φ1φ2),
where
(29)Λ=(λ100λ1*),H=(ϕ1ϕ2*ϕ2-ϕ1*),φ1=d1eθ3+d2e-θ4,φ2=d3eθ3+d4e-θ4,
with
(30)θ3=t2[-iωc+-ωc2-4qc2-4λ22-4λ2ωc]+z2[-iκc+-κc2+4a′2-4b′2-4ia′κc],θ4=t2[-iωc--ωc2-4qc2-4λ22-4λ2ωc]+z2[-iκc--κc2+4a′2-4b′2-4ia′κc],a′=2iλ22(-1+4λ22τ)-iqc2×(-1+4λ22τ-4λ2ωcτ+3ωc2τ)+3iqc4τ+iηcλ2+ω,b′=qc(4λ2qc2τ+2λ2-ωc-4λ22ωcτlllllllllllllll-2λ2ωc2τ+ωc3τ-8λ23τ-6qc2ωcτ)+pcλ2+ω,λ2 is another eigenvalue for Lax pair (5a) and (5b), and d1, d2, d3, and d4 are complex constants satisfied as
(31)d3=1qc[-ωc2-4qc2-4λ22-4λ2ωciωc+iλ2+12llllllllllllll×(-iωc+-ωc2-4qc2-4λ22-4λ2ωc)12]d1,d2=qc(1/2)(-iωc+-ωc2-4qc2-4λ22-4λ2ωc)+iωc+iλ2×d4.Thus, one can derive the expressions of two-soliton solutions for system (4a), (4b), and (4c) as(32a)q(2)=q′-2i(λ2-λ2*)ψ1ψ2*|ψ1|2+|ψ2|2,(32b)p(2)=p′χ1′2χ2′(|ψ1|2+|ψ2|2)2-p′*(λ2-λ2*)ψ12ψ2*2χ2′(|ψ1|2+|ψ2|2)2+2η′χ1′(λ2-λ2*)ψ1ψ2*χ2′(|ψ1|2+|ψ2|2)2,(32c)η(2)=η′(1+2(λ2-λ2*)|ψ1|2|ψ2|2χ2′(|ψ1|2+|ψ2|2)2)+p′(λ2-λ2*)χ1′ψ2ψ1*χ2′(|ψ1|2+|ψ2|2)2-p′*(λ2-λ2*)χ1′*ψ1ψ2*χ2′(|ψ1|2+|ψ2|2)2,with
(33)χ1′=(λ2+ω)|ψ1|2+(λ2*+ω)|ψ2|2,χ2′=(λ2+ω)(λ2*+ω).
Now, we will analyze (32a), (32b), and (32c) under two different cases.


### 4.1. Interaction Characters of Bound Solitons

In this section, we will investigate the interaction between neighboring solitons for system (4a), (4b), and (4c). Taking qc=0, pc=0, ηc=1 and iterating the DT twice, one can generate two-soliton solutions with two spectral parameters λ1=ϑ1+iϱ1 and λ2=ϑ2-iϱ2 for system (4a), (4b), and (4c). Assuming the input launching pulses as q(0,t)=sech(t-t0)+sech(t+t0), here t0 denotes the soliton separation. Supposing that ϑ1=ϑ2=0, we can derive that
(34)ϱ1,2=1+2t0sinh⁡(2t0)±sech⁡t0.
Under the circumstances t0=3 and τ=0, that is, the higher-order dispersive effects are absent, the two-soliton solutions can turn into bound solitons under suitable parameters chosen as shown in Figure 4. We can observe that main features are that the mutual attractions and repulsions repeat periodically when the bound solitons propagate, and this phenomenon may be harmful for optical soliton communication [32–34].The parameters adopted here aret0=3, ω=3, and τ=0.
(a)
(b)
(c)In fact we can suppress the periodical mutual attractions and repulsions through increasing the initial pulse separation, that is, the value oft0. As portrayed in Figure 5, when t0=10, one can find that the mutual attractions and repulsions between two bound solitons disappear, and the two solitons propagate in parallel without any effect on each other even if the propagation distance grows long enough.The parameters adopted here aret0=10, ω=3, and τ=0.
(a)
(b)
(c)Figure6 depicts the effects of the higher-order dispersive terms on the propagations of the solitons. As the pictures show, the periods of the bound solitons can be suppressed when τ decreases.The parameters adopted here aret0=3, ω=3, and τ=5.
(a)
(b)
(c)So we can conclude that the higher-order dispersive terms can control the propagation periods of the bound soliton.


### 4.2. Interaction Characters of Two-Breather Solutions

In this section, we will construct two-breather solutions for system (4a), (4b), and (4c). Taking qc≠0, pc≠0, ηc≠0, ωc=0 and iterating the DT twice, one can get two-breather solutions for system (4a), (4b), and (4c). When |λ1|<|qc| and |λ2|>|qc|, the Akhmediev breathers and Ma-breathers can coexist as portrayed in Figure 7. Main features of the interaction between two breathers in Figure 7 are that they interact perpendicularly and the shapes, amplitudes, and pulse widths of the two breathers all remain invariant, so the interactions are elastic.The parameters adopted here areλ1=i, λ2=4i, qc=3, ωc=2, κc=1, ω=0.5, and τ=0.1.
(a)
(b)
(c)When|λ1|<|qc| and |λ2|<|qc|, the two-breather solutions that evolve from two Ma-breathers come into being as portrayed in Figure 8. And the interactions between these two-breathers in Figure 8 are also elastic.The parameters adopted here areλ1=i, λ2=2i, qc=3, ωc=2, κc=1, ω=0.5, and τ=0.1.
(a)
(b)
(c)When|λ1|>|qc| and |λ2|>|qc|, the two-breather solutions that evolve from two Akhmediev-breathers take place as depicted in Figure 9. And the interactions between these two-breathers in Figure 9 are also elastic.The parameters adopted here areλ1=5i, λ2=4i, qc=3, ωc=2, κc=1, ω=0.5, and τ=0.1.
(a)
(b)
(c)


## 4.1. Interaction Characters of Bound Solitons

In this section, we will investigate the interaction between neighboring solitons for system (4a), (4b), and (4c). Taking qc=0, pc=0, ηc=1 and iterating the DT twice, one can generate two-soliton solutions with two spectral parameters λ1=ϑ1+iϱ1 and λ2=ϑ2-iϱ2 for system (4a), (4b), and (4c). Assuming the input launching pulses as q(0,t)=sech(t-t0)+sech(t+t0), here t0 denotes the soliton separation. Supposing that ϑ1=ϑ2=0, we can derive that
(34)ϱ1,2=1+2t0sinh⁡(2t0)±sech⁡t0.
Under the circumstances t0=3 and τ=0, that is, the higher-order dispersive effects are absent, the two-soliton solutions can turn into bound solitons under suitable parameters chosen as shown in Figure 4. We can observe that main features are that the mutual attractions and repulsions repeat periodically when the bound solitons propagate, and this phenomenon may be harmful for optical soliton communication [32–34].The parameters adopted here aret0=3, ω=3, and τ=0.
(a)
(b)
(c)In fact we can suppress the periodical mutual attractions and repulsions through increasing the initial pulse separation, that is, the value oft0. As portrayed in Figure 5, when t0=10, one can find that the mutual attractions and repulsions between two bound solitons disappear, and the two solitons propagate in parallel without any effect on each other even if the propagation distance grows long enough.The parameters adopted here aret0=10, ω=3, and τ=0.
(a)
(b)
(c)Figure6 depicts the effects of the higher-order dispersive terms on the propagations of the solitons. As the pictures show, the periods of the bound solitons can be suppressed when τ decreases.The parameters adopted here aret0=3, ω=3, and τ=5.
(a)
(b)
(c)So we can conclude that the higher-order dispersive terms can control the propagation periods of the bound soliton.


## 4.2. Interaction Characters of Two-Breather Solutions

In this section, we will construct two-breather solutions for system (4a), (4b), and (4c). Taking qc≠0, pc≠0, ηc≠0, ωc=0 and iterating the DT twice, one can get two-breather solutions for system (4a), (4b), and (4c). When |λ1|<|qc| and |λ2|>|qc|, the Akhmediev breathers and Ma-breathers can coexist as portrayed in Figure 7. Main features of the interaction between two breathers in Figure 7 are that they interact perpendicularly and the shapes, amplitudes, and pulse widths of the two breathers all remain invariant, so the interactions are elastic.The parameters adopted here areλ1=i, λ2=4i, qc=3, ωc=2, κc=1, ω=0.5, and τ=0.1.
(a)
(b)
(c)When|λ1|<|qc| and |λ2|<|qc|, the two-breather solutions that evolve from two Ma-breathers come into being as portrayed in Figure 8. And the interactions between these two-breathers in Figure 8 are also elastic.The parameters adopted here areλ1=i, λ2=2i, qc=3, ωc=2, κc=1, ω=0.5, and τ=0.1.
(a)
(b)
(c)When|λ1|>|qc| and |λ2|>|qc|, the two-breather solutions that evolve from two Akhmediev-breathers take place as depicted in Figure 9. And the interactions between these two-breathers in Figure 9 are also elastic.The parameters adopted here areλ1=5i, λ2=4i, qc=3, ωc=2, κc=1, ω=0.5, and τ=0.1.
(a)
(b)
(c)


## 5. Conclusions

Our main attention has been focused on system (4a), (4b), and (4c) which describes the propagation of optical solitons in nonlinear erbium-doped fibers with higher-order effects. Lax pair and modulation instability conditions for this system have been investigated. Two types of breathers (Akhmediev breathers and Ma-breathers), bound soliton solutions, and two-breather solutions have been constructed by Darboux transformation. Propagation properties of those solitons under the influences of higher-order effects have been discussed.



---

*Source: 185654-2014-04-06.xml*