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

and Applied Analysis 3


Introduction
In recent years, optical solitons have attracted many researchers for their potential applications in optical fiber transmission systems [1][2][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 where 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]: where 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]: where subscripts and denote the partial derivatives with respect to the longitudinal distance and retarded time. is the frequency, the asterisk denotes the complex conjugate, = V 1 V * 2 , and = |V 2 | 2 − |V 1 | 2 with V 1 and V 2 representing the wave functions in a two-level system [12,13]. Many research achievements about system (3a), (3b), and (3c) have been obtained [14][15][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 2 Abstract and Applied Analysis is usually described by the following generalized nonlinear Schrödinger-Maxwell-Bloch (GNLS-MB) system [9]: 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 Section 2, 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 twobreather solutions for system (4a), (4b), and (4c). Finally, our conclusions will be addressed in Section 5.
We examine the MI process of the steady-state solutions by introducing the following perturbed solutions: where 1 , 2 , and are weak perturbations with the assumed general expressions being perturbations, and is the real disturbance wave number. Inserting (8a), (8b), and (8c) into (4b) and (4c), we can obtain the linearized equations of 1 , V 1 , 2 , V 2 , and , so 2 , V 2 , and can be solved as follows: 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 1 and V 1 : Equations (10a) and (10b) have nontrivial solutions if and only if the following determinant formed by the coefficients matrix vanishes, that is, Equation (12) leads to the dispersion relation of and Ω which determines the modulation instability process of the steady-state cw solution as From (13), one can conclude that if Ω 2 − 4 2 0 < 0, the value of will be complex; then the modulation instability will take place with | Im | as the instability growth rate. So Ω 2 − 4 2 0 < 0 is the condition of the modulation instability for system (4a), (4b), and (4c).

Case 2.
In the case of ̸ = 0, = = 0, and 2 > 4 2 , symbolic computation results in the following: Abstract and Applied Analysis with ℎ 1 = 2 2 − + 2 + 6 4 + 2 2 2 + 4 , Substituting (25)  From Figure 1, 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][22][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 Figure 2 with Figure 1, one can see that in Figure 2, under the influence of the increasing values of the parameter , the number of peaks on the same space interval is increasing when goes up from 0 to 0.1. So, we can conclude that the parameter controls the period of the Akhmediev breathers.
From Figure 3, 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][25][26][27], and for functions and , the solutions are bright solitons while for function the solution is the dark one. In addition, one can find that the separations between adjacent peaks in Figure 3 gradually increase as → 2 and eventually reduce into the rogue waves, the properties of which have been discussed in [28][29][30][31].
In fact we can suppress the periodical mutual attractions and repulsions through increasing the initial pulse separation, that is, the value of 0 . As portrayed in Figure 5, when Figure 6 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 decreases.
So we can conclude that the higher-order dispersive terms can control the propagation periods of the bound soliton.

Interaction Characters of Two-Breather Solutions.
In this section, we will construct two-breather solutions for system (4a), (4b), and (4c). Taking ̸ = 0, ̸ = 0, ̸ = 0, = 0 and iterating the DT twice, one can get two-breather solutions for system (4a), (4b), and (4c). When | 1 | < | | and | 2 | > | |, 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.
When | 1 | < | | and | 2 | < | |, 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.
When | 1 | > | | and | 2 | > | |, 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.

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.