Propagation and interaction between special fractional soliton and soliton molecules in the inhomogeneous fiber.

INTRODUCTION: Fractional nonlinear models have been widely used in the research of nonlinear science. A fractional nonlinear Schrödinger equation with distributed coefficients is considered to describe the propagation of pi-second pulses in inhomogeneous fiber systems. However, soliton molecules based on the fractional nonlinear Schrödinger equation are hardly reported although many fractional soliton structures have been studied.
OBJECTIVES: This paper discusses the propagation and interaction between special fractional soliton and soliton molecules based on analytical solutions of a fractional nonlinear Schrödinger equation.
METHODS: Two analytical methods, including the variable-coefficient fractional mapping method and Hirota method with the modified Riemann-Liouville fractional derivative rule, are used to obtain analytical non-travelling wave solutions and multi-soliton approximate solutions.
RESULTS: Analytical non-travelling wave solutions and multi-soliton approximate solutions are derived. The form conditions of soliton molecules are given, and the dynamical characteristics and interactions between special fractional solitons, multi-solitons and soliton molecules are discussed in the periodic inhomogeneous fiber and the exponential dispersion decreasing fiber.
CONCLUSION: Analytical chirp-free and chirped non-traveling wave solutions and multi-soliton approximate solutions including soliton molecules are obtained. Based on these solutions, dynamical characteristics and interactions between special fractional solitons, multi-solitons and soliton molecules are discussed. These theoretical studies are of great help to understand the propagation of optical pulses in fibers.

Introduction

A soliton is known as a self-reinforcing wave packet that keeps its shape and propagating velocity. Soliton exhibits its rich structures including optical soliton [1], plane soliton [2], soliton molecules [3], rogue waves [4], etc., and helps develop some breakthrough branches of physical sciences [5], [6], [7] such as optics, condense physics, fluid and plasma [8], [9], [10], [11].

Soliton molecules mean robust multi-soliton bound states [12], and their dynamics have become hot topics in several contexts, including optical systems [13], [14] and Bose–Einstein condensates [15]. The formation of optical soliton molecules originates from the existence of attractors of a nonlinear dynamical system. Once formed, soliton molecules will stably travel around a mode-locked laser cavity [14]. For a soliton molecule, temporal separations among solitons are the most relevant degrees of freedom [13]. The real-time internal dynamics of two-soliton and three-soliton molecules were experimentally studied [14], [16]. Optical soliton molecular complexes have been experimentally observed in a passively mode-locked fiber laser [17]. The breathing soliton molecules were also experimentally found in a mode-locked fiber laser [18].

These studies above focused on experimental observations [12], [13], [14], [15], [16], [17], [18]. However, theoretical investigation on soliton molecules was less carried out. Until fairly recently, the formation mechanism of soliton molecules was theoretically proposed [3], [19], [20], [21]. Soliton molecules based on fractional nonlinear models(FNMs) are hardly reported although many fractional soliton structures have been studied [22], [23].

In recent years, FNMs have been widely used in the research of nonlinear science [24], [25], [26]. At the same time, FNMs with distributed coefficients have certain representative significance. Many phenomena can be described successfully by using FNMs such as plasma, nonlinear optics and chaotic oscillations [24]. Many researchers have already succeeded in the study of fractional models [27], [29]. Effective methods for solving fractional nonlinear Schrödinger (FNLS) equation have been achieved [30], such as the fractional F-expansion method [28], fractional Riccati method [26] and fractional bi-function method [31]. At the same time, some numerical methods have also been successfully used to solve the fractional nonlinear Schrodinger equation[32], and numerical solutions were derived by using Riesz-Feller Derivative and non-standard discretization[33], [34]. These numerical methods have been validated by the researchers [35].

The novelty of this paper lies in presenting a new strategy to get analytical fractional non-travelling wave solution and multi-soliton solutions of a FNLS equation by altogether utilizing fractional mapping method and Hirota method with the modified Riemann–Liouville(RL) fractional derivative rule. Another novelty is to study the dynamics of special fractional soliton and soliton molecules, and discuss the formation mechanism of soliton molecules. The stability of the special fractional soliton and soliton molecule is analyzed through a series of numerical studies. These conclusions possess theoretical guidance for the related experimental study in all-optical switches, optical amplifier and mode-locked lasers.

Material and methods

To our knowledge, most practical nonlinear physical models have distributed coefficients. For example, modern communication systems use variable dispersion fiber. The NLS equation with distributed coefficients can describe the propagation of pi-second pulses in inhomogeneous fiber systems [38], [39]. Recently, many scholars believe that the evolution of the development function can be well described by FNMs. When describing practical problems, compared with the integer model, FNMs are more satisfactory [36]. The FNLS equation with distributed coefficients was introduced as [37]

where the complex envelope and its derivatives

with the fractional orders and , the delay time t and longitudinal propagation distance x. Functionsandare coefficients of the Kerr nonlinearity and dispersion, and function is adiabatic amplification (loss) for or gain for . If , Eq. (1) is the variable-coefficient NLS equation [38]. When functionsandare constant and , Eq. (1) describe solitons in the homogeneous fiber [39]. Here the modified RL fractional derivative is defined as [40]

they has the following properties [41], [42]with the following inequality [42]

In Eq. (3), the fractal index is usually calculated from a gamma function. Abdel-Salam et al. found that for the Mittag-Leffler function as , the value of the fractal index is equal to one [42].

Analytical non-travelling wave solutions of FNLS equation (1)

We suppose that Eq. (1) has the form of the following solutionwhere the amplitude , chirped phaselinear phase and phase shift are functions of × , making , separating the imaginary and real parts, thus

We suppose that Eqs. (6), (7) have following solutionwhere are functions of × , is a function of × and t,with group velocity and pulse width are functions of × , by the leading term analysis in Eq. (7) can get . Here satisfies the fractional mapping equation [28]with arbitrary constants . Eq. (9) has different forms of solutions listed in Table 1.

Table 1

Some solutions of Eq. (9).

Case	a0	a2	a4	Y
1	1	−2	1	Y=tanh(ψ,λ)
2	0	1	−1	Y=sech(ψ,λ)
3	1/4	−1/2	1/4	Y=tanh(ψ,λ)1±sech(ψ,λ),Y=coth(ψ,λ)+csch(ψ,λ),Y=tanh(ψ,λ)+isech(ψ,λ)
4	0	1	1	Y=csch(ψ,λ)
5	1	1	0	Y=sinh(ψ,λ)
6	−1	1	0	Y=cosh(ψ,λ)
7	1	−1	0	Y=sin(ψ,λ),Y=cos(ψ,λ)
8	0	−1	1	Y=csc(ψ,λ),Y=sec(ψ,λ)
9	1	2	1	Y=cot(ψ,λ),Y=tan(ψ,λ)

In Table 1, the extended hyperbolic and trigonometric functions satisfy the following definitions

Substituting ansatz (8) with and (9) into Eqs. (6), (7), considering the approximation of the generalized binomial theorem, and making the coefficients of and zero getwhere with .

Solving Eqs. (11), using Eqs. (5), (8) with solutions in Table 1, we can get several families of solutions of Eq. (1). Due to the limit of length, we only list part of solutions.

Family 1. when where andare functions of and C1~C5 are arbitrary constants.where andare functions of and C1~C5 are arbitrary constants.where andare functions of and C1~C5 are arbitrary constants.where and are functions of and C1~C5 are arbitrary constants.

Fractional chirp-free dark soliton solution

Fractional chirp-free combined soliton solution

Fractional chirped dark soliton solution

Fractional chirped combined soliton solution

Family 2. when where where

Fractional chirp-free bright soliton solution

Fractional chirped bright soliton solution

Multi-soliton approximate solutions of FNLS equation (1)

In order to study the dynamics of multi-soliton solutions, we use the inequality (4) in Eq.(1), thus Eq.(1) is converted into

Because the approximate expression (4) is used here, thus we call these solutions derived in the following as approximate solutions. Similar to the solving procedure in Section 3, we can also derive fractional bright and dark soliton approximate solution with and without chirped phase by using the fractional mapping method. For the limit of length, we do not list them.

Using the Hirota method, we can get a chirp-free bright soliton approximate solution like that derived from the fractional mapping method. However, we can not get chirped bright soliton approximate solution.

Next, we will get chirp-free multi-soliton approximate solution. From the Hirota method [43], we assume that where is complex function, and is real function. We get the bilinear equation of Eq. (18) as followswhere * denotes the complex conjugate, and are Hirota bilinear operator. In order to solve Eqs. (19), (20), we can write and as power series expansions [43], [44]

Two soliton solutions via the Hirota method

In order to get the two soliton solution, we suppose that Eq. (18) has a solution aswherewith Here andare real numbers related to the group velocity, phase velocity, soliton position and phase of the j-th soliton respectively, and are pending functions. Substituting into Eq. (19), collect the coefficient of , we get and

Substituting into Eq. (19), collect the coefficient of , we getwhere , , , is an arbitrary constant.

Substituting and into Eq.(19), collect the coefficient of , we getwhere

Substituting ,and into Eq.(20), collect the coefficient of , we getwhere

We assume that we get the two soliton solutionwhere , , and are given in Eqs. (23), (24)-(26). We list coefficients of Eq. (27) in Appendix A.

Three soliton solutions via the Hirota method

In order to get the three soliton solution, we assume that Eq. (18) has a solution as [43]

using the similar steps in Section 4.1, we can define , we can get the three soliton solution of Eq. (18)where parameters are listed in Appendix A.

Results discussion

Based on analytical solutions (12), (13), (14), (15), (16), (17), (27), (29), some special fractional soliton, multi-soliton and soliton molecules can be constructed. When the order of the fractional soliton equals one, the fractional soliton becomes a traditional integer soliton, especially two soliton and three soliton solutions are similar to those solutions in Refs. [40], [41].

In order to facilitate the study of the dynamic characteristics of the soliton, we select the exponential distributed control system [45]where parameters describe the group velocity dispersion. Particularly, if , system (30) depicts the exponentially dispersion decreasing fiber [46]. If , system (30) depicts the periodic inhomogeneous fiber [47].

Special fractional soliton

The evolution of fractional chirp-free bright-type soliton solution (16) with different values of fractional orders versus × and t is shown in the periodic inhomogeneous fiber system in Fig. 1, Fig. 2.The amplitude and the velocity of the bright soliton depend on and respectively, the phase shift is determined by, and the time shift is related to. We find that the dynamic characteristics of soliton are affected by different values of the fractional order. When the value of fractional orderis closer to 1, solution (16) describes the classic bright soliton in Fig. 1(a), where the amplitude gradually decreases due to . When, the amplitude of the soliton decreases rapidly and then tends to be fixed magnitude in Fig. 1(b). By comparing Fig. 2(a) with Fig. 2(b), we can find that the position of the soliton on the t-axis changes, which means that the soliton has a phase shift. When the value of fractional order is closer to 0.5, the periodicity of the soliton is significantly weakened. It can be clearly seen that the propagation speed of the soliton along the fiber has changed due to the presence of the parameter C1. This shows that the soliton keeps its sech-function shape even if the velocity is changed. This is an important property of solitons.

Fig. 1

Chirp-free bright soliton (16) with the intensity. Parameters are with (a) and (b)

Fig. 2

Density map of chirp-free bright soliton . Parameters are same as Fig. 1.

Multi-soliton and soliton molecules

The interaction of the two-soliton solution (27) in the exponential system is shown in Fig. 3. The two solitons continue to move after elastic collision, and their shape remain unchanged. By changing the parameters of phase velocity and, we can control the phase shift of the soliton. By changing the parameters of soliton position and, we can change the interval between solitons, but the amplitude of solitons has not changed. In addition, when we change the value of integral constant , we can only change the amplitude of solitons. When the values of group velocity parameters and change, the amplitude and phase of soliton will also vary.

Fig. 3

The dynamical interaction of the two-soliton solution: (a) density plot, (b) intensity plot and (c) numerical rerun with 5% white random noise. Parameters are

Besides the interactions between two soliton, we can also discuss the two-soliton bounded state, currently termed “soliton molecules” [11], [12]. Fig. 4 shows a soliton molecule consisting of two solitons. Recently, Lou et al. gave the velocity resonance conditions of soliton molecules consisting of integer-order solitons in fluid [3], [20]. Analyzing two-soliton solution (27), we give the condition of velocity resonance for soliton molecules as which means. By using the condition of velocity resonance, we can obtain two-soliton molecule. The amplitudes of two soliton in the molecule are different, due to, although their speeds are the same. The distance between the two solitons of the molecule depends on the parameters and . If two solitons of the molecule are close enough, the soliton molecule will become an asymmetric soliton, which similar to that in Ref. [48].

Fig. 4

A soliton molecule consisting of two solitons: (a) density plot and (b) intensity plot. Parameters are

In order to analyze the stability of the interaction between two solitons and soliton molecules, we conduct direct numerical rerun for equation (1) using the split-step pulse propagation method. Here the initial field comes from solution (27) with initial 5% white noise. By numerical estimation in Fig. 3(c), we see that two solitons stably propagate a long distance after their interactions. For soliton molecules within Fig. 5, two solitons stably form the bounded state a long-distance against the initial 5% white noise. In both evolutionary processes, two solitons have maintained a steady movement.

Fig. 5

The numerical rerun of the soliton molecule in Fig. 4 with 5% white random noise: (a) intensity plot and (b) density plot. Parameters are

Furthermore, we can also study the three-soliton molecule, which has the similar velocity resonance condition to the two-soliton molecule. It is easy to know that if the solution (29) satisfies the following resonance conditions as namely then we can obtain three-solitons molecule. Similarly, we can deduce that if N-solitons satisfies the following resonance conditions we can obtain N-solitons molecule. In Fig. 6, we can find that the amplitudes of three solitons in the molecule are different owing to, although their velocities are the same.

Fig. 6

A soliton molecule consisting of three solitons: (a) density plot and (b) intensity plot. Parameters are

By adjusting the parameters, we can study the interaction of two-soliton molecules and a single bright soliton. Fig. 7 exhibits the elastic collision between a molecule consisting of two solitons and a single bright soliton. Due to, the amplitudes of two solitons in the soliton molecule are different, although they have the same speed. Along the propagation distance × , the single bright soliton interacts with the big soliton of the molecule and produces a phase shift, and then interacts with the small soliton of molecule and also appear a phase shift. After the collision, the soliton molecule and the single bright soliton maintain their amplitudes, shapes and widths, which is the elastic interaction. It is known that the collision between the soliton and the soliton in the NLSE is elastic, so the soliton molecule has similar properties as the soliton.

Fig. 7

The interaction of soliton molecules and a bright solitons: (a) density plot and (b) intensity plot. Parameters are

By the numerical rerun in Fig. 8, we see that the three-soliton molecule also stably forms the bounded state a long distance, and maintains good stability. The numerical calculation indicates no collapse, that is, its velocity, shape, amplitude and width are nearly unchanged against the initial 5% white noise.

Fig. 8

The numerical rerun of the three-soliton molecule in Fig. 6 with 5% white random noise: (a) intensity plot and (b) density plot. Parameters are

Conclusions

In conclusion, we consider a fractional NLS equation with distributed coefficients, which describes the propagation of pi-second pulses in inhomogeneous fiber systems. We get analytical chirp-free and chirped non-travelling wave solutions and multi-soliton approximate solutions by two analytical methods, namely, the variable-coefficient fractional mapping method and Hirota method. We give the form conditions of soliton molecules, and study the dynamical characteristics of special fractional solitons, multi-solitons and soliton molecules in the periodic inhomogeneous fiber and the exponentially dispersion decreasing fiber. In the fractional order, the soliton still maintains good stability and forms the bound state of the soliton molecule. An elastic collision occurs between the soliton molecule and the single bright soliton, and their amplitudes, shapes and widths are maintained, and thus soliton molecules have the similar properties as the soliton.

In Fig. 1, Fig. 2, when with the integer case and with the fractional case, the motion effect of soliton is similar, however, the periodicity of the fractional soliton is significantly weakened. In Fig. 3, after the two fractional solitons collide, they keep their original shapes and continue to move. This shows that the fractional order does not affect the properties of integer solitons. Thus the fractional order has a certain effect on the movement of the soliton, but does not change the nature of the soliton. These results have theoretical guidance for the related experimental study in all-optical switches, optical amplifier and mode-locked lasers.

Via the simialr analysis and calculation, we can extend our methods in this paper to two-dimensional FNLS equations and coupled FNLS equations. This will be the direction and focus of our next research.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.