The similarities and differences of different plane solitons controlled by (3 + 1) - Dimensional coupled variable coefficient system.

In this paper, a system with controllable parameters for describing the evolution of polarization modes in nonlinear fibers is studied. Using the Horita's method, the coupled nonlinear Schrödinger equations are transformed into the bilinear equations, and the one- and two- bright soliton solutions of system (3) are obtained. Then, the influencing factors on velocity and intensity in the process of soliton transmission are analyzed. The fusion, splitting and deformation of the solitons caused by their interactions are discussed. Finally, a method for adjusting the inconsistencies of sine-wave soliton transmission is given. The conclusions of this paper may be helpful for the related research of wavelength division multiplexing systems.

Introduction

In fiber optics, some studies have been conducted on the traditional optical pulse transmission model [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. With the further study of fiber optics, scientists have extended the study of the traditional optical pulse transmission model nonlinear Schrödinger equation (NLSE) in optical fiber to multi-dimensional NLSE, coupled NLSE (CNLSE) in birefringent fiber, N-coupled NLSE in wavelength division multiplexing system and variable coefficient NLSE in non-uniform fiber [11], [12], [13], [14], [15], [16], [17]. As one of the basic theoretical models for describing nonlinear phenomena, the CNLSEs are widely used in such fields as biophysics, condensed matter physics and nonlinear optics [18], [19], [20], [21]. The classic CNLSE is:where and represent slowly varying amplitudes of two fiber modes, they are complex functions with respect to scale distance and time [22], [23], [24], [25]. The System (1) includes both self-phase modulation and cross-phase modulation, and are cross-phase modulation coefficients, and are the dispersion coefficients of the two wave packets, respectively. For System (1), its exact solutions and soliton transmission characteristics have been studied. By introducing Hirota’s method, the bright soliton and dark soliton solutions of System (1) have been obtained under the conditions of and [26]. The periodic solutions of the systems extended to the N-components have been expressed, and the inelastic interactions caused by intensity redistribution and separation distance have been analyzed [27].

The soliton solution of the high-dimensional CNLEs are more complicated in structure, so that they can produce more abundant new physical phenomena. Therefore, the (1 + 1)-dimensional CNLSEs have been extended to the (2 + 1)-dimensional CNLSEs [28].

System (2) controls the existence and stability of the space vector solitons, and the solutions of System (2) are derived under the condition of parameters, and the elastic and inelastic interactions between two parallel bright solitons have been analyzed [28]. In reference [29], -components (2 + 1)-dimensional CNLSEs have been discussed, which describe the evolution of polarization modes in nonlinear fibers. However, in the process of practical application, some special phenomena such as local defects and damages cannot be explained by constant coefficient system model in optical fiber, which always have an important impact on the optical soliton transmissions and dynamic behavior [30]. Therefore, the variable coefficient CNLSEs have much practical significance and research value. When and develop into and respectively, the bright and dark analytic soliton solutions of the changed System (2) and their related properties have been reported [30], [31].

Further, the higher the dimension of the nonlinear equation, the more accurately the equation can describe the actual physical phenomenon, so that the CNLSE is extended from (2 + 1) dimension to (3 + 1) dimension [32]. Not only that, finding the exact solutions of the variable coefficient CNLES, especially the soliton solutions, has always been a topic of great interest to mathematicians and physicists. Consider the above factors, we will focus on the following (3 + 1)-dimensional variable coefficient system model [32], [33], [34], [35],where and are all perturbed real functions. When they are all constants, the bright soliton solutions of the constant coefficient -dimensional CNLSE has been solved in Ref. [33]. Subsequently, the dark soliton solutions have been derived under the constraints of and in Ref. [34]. The variable-coefficient dark solitons of the system (3) with the constraints , and their different transmission structures have recently been reported [35]. However, after investigation, we found that the bright solitons and the effect of perturbation functions on the soliton transmission process controlled by this variable coefficient (3 + 1)-dimensional CNLSEs have not been studied.

The composition of this paper is divided into the following sections: The derivation of the bilinear forms and the bright analytical solutions of System (3) will be presented in the second part. In the third part, the intensity, velocity and phase during the soliton transmission process on the planes in different directions are analyzed. Further, the influences of perturbation variable parameters on the soliton transmission process and the special phenomena will be explored. Finally, in the fourth part, the final conclusion is drawn.

Material and methods

The bilinear forms of system (3)

It is difficult to directly solve nonlinear equations, so that the following rational transformations are introduced to convert the above System (3) into the bilinear forms:

And then substituting the transformations (4) into System (3), we can get the following expressions:here is a real function, while and are both complex with the variables of and . represents the conjugate symbol. And the operator knowns as the bilinear derivative operator in the above, which is defined as follows [36], [37]:

By setting ( is a positive constant) we can obtain:

To balance the dispersion terms and nonlinear terms, we have the constraints and . Since the denominator cannot be 0, we can get:

From the above process, the bilinear forms of system (3) are:

The One-soliton solutions of System (3)

Next, the bright one-soliton solutions of System (3) will be derived according to the expansions of and with respect to formal parameter .when deriving the one-soliton solutions, the above expansions need to be truncated into and . Making assumptions that , and substituting the assumptions and the truncated expansions into the bilinear Eq. (7), the following relationships can be yielded:

For convenience, make the assumption that , so the one-soliton solutions of System (3) can be written in the following forms:

The two-soliton solutions of System (3)

When deriving the two-soliton solutions, the expansions (7) should be truncated to and . Then, and are set to and , respectively. Here, . Taking the above assumptions into the bilinear equations (7), we can acquire the following results:where

,

Without loss of generality, assuming, then the expressions of the bright two-soliton solutions are as follows:

Results discussion

To explore the traits of the velocity and intensity in solitons transmission process controlled by this model, for intuitive analysis, the above-mentioned one-soliton solutions (9) are transformed as follows:where and represent the real and imaginary parts of , respectively. The characteristic-line equation (12) is introduced in the soliton transmission process to convey the expression of transmission speed [38].

Assuming , , , are real constants and , then substituting them into Eq. (12), the following relationship is obtained:

Differentiate on both sides of Eq. (13), therefore, the soliton transmission velocity in the , , and planes are inferred:

It is shown that the transmission speed of the soliton is affected by wave numbers and disturbance coefficient. What’s more, under the same parameter conditions, the larger the real value of the wave numbers of each plane, the smaller the velocity of the plane. As can be seen from Fig. 1(a) and (b), in the plane, the soliton transmission velocity does not increase or decrease for the changes about the values of and , but its transmission position is shifted to the right. It is because the values of and will affect the initial phase of the soliton in the plane transmission. On the other hand, comparing the soliton transmission volecity on different planes from Fig. 1(a), (b) and (c), as the real part values of , and are , and , respectively, we can see that the speed of Fig. 1 (a) is the largest, and Fig. 1 (c) is the smallest, which confirms the expressions of and from the image aspect.

Fig. 1

The velocity comparison on different planes of one-soliton solitons, corresponding parameters are:

Next, we continue to discuss some special phenomena caused by the effects of perturbation parameters on soliton transmission. When takes a constant, the solitons are linear on the corresponding plane in Fig. 1, but once takes different functions, it will have different shapes on the corresponding plane. For instance, in the plane, when takes or , the solitons appear parabolic in Fig. 2(a) and (b). But if we suppose , there will be a periodic parabolic soliton with different energies in Fig. 2(c) and (d). Not only that, the purpose of changing the period and span of the parabolic solitons can be achieved by adjusting the parameters and . can take various functions, when is taken as , , , , respectively, cubic (Fig. 2(e)), sine(Fig. 2(f)), hyperbolic sine (Fig. 2(g)) and periodic increased amplitude (Fig. 2(h)) solitons are obtained.

Fig. 2

The different shapes of solitons generate on the plane by : (a) ; (b) ; (c) (d) (e) (f) ; (g) ; (h) .

According to Eq. (11), the intensities of and are as follows:

Because , there is

The above equations show that the intensity of the soliton is not related to the constraint parameter , but is related to , the phase constant and , and the parameter . Further, when increases, the intensity of increases but decreases.

Next, we will concentrate on discussing the interactions of the two-solitons in System (3). From Eq. (11), we know that the difference between and is only proportional to the energy, so the following discussion about the soliton’s interactions is only for . As we can see, under certain parameters values, by adjusting the wave number parameters , and , solitons appear to merge, split and deform in the process of interaction. In Fig. 3 (a), the two solitons are fused into a single soliton with greater intensity and wider wave width. However, when the parameters values become , the two solitons do not merge. Instead, one of the solitons absorbed the energy of the other soliton, and the intensity and wave width increased, on the other hand, the energy and wave width of the other soliton are reduced in Fig. 3 (b). The energy and waveform of the solitons have changed after the interaction, which is an inelastic interaction caused by energy redistribution. Further, by adjusting the values of and , the two-solitons are split, and side wave appear. A new soliton is formed between the two solitons, and its energy is greater than that of the two solitons in Fig. 3 (c). Fig. 3 (d) is the cases where the two-solitons split into four waves. This kind of interaction that will generate new solitons may be beneficial to quickly improve the efficiency of optical communications. In addition to fusion and splitting, the two- solitons of System (3) will undergo severe deformation in the area of interaction in Fig. 3(e) and (f). This phenomenon will reduce the accuracy of information transmission and is also a problem that must be solved to improve the transmission efficiency of optical fibers.

Fig. 3

Two-soliton interactions with different constraint coefficients: (a) , (b) , (c) , (d) , (e) (f) .

Finally, parameters and can also modulate the synchronization of soliton transmissions. The propagation of optical soliton in a dispersion-graded fiber is similar to a sinusoidal curve. Therefore, is taken as a sine function to simulate the transmission process of a soliton in a dispersion graded fiber. As can be seen in Fig. 4 (a), the two solitons are sinusoidal waves under the action of , and the vibration directions of the two solitons are opposite. However, with different values of and , the vibration directions of the two solitons become synchronized in Fig. 4 (b). From the previous analysis in Fig. 1(a) and (b), it is known that only the transmission positions of the solitons are different on the different planes in the same direction. Therefore, it can be known from Fig. 4 that the inconsistencies of the sine-wave soliton can be achieved by adjusting parameters and . So that the wave number parameters can not only manage the shape and energy of the solitons themselves, but also modulate the coordination of the two-solitons during the transmissions. At the same time, in Fig. 4, the two solitons only locally deform in the interaction range, and after the interaction, the shape does not change. Thus, the interactions are elastic interactions which has less impact on information transmission during the fiber transmission process.

Fig. 4

Two-soliton interactions with different constraint coefficients: (a) (b) .

Conclusion

In this paper, we have investigated a variable coefficient (3 + 1)-dimensional CNLSE (3) describing circularly polarized waves. The Horita’s method have been used to transform Eq. (3) into the bilinear forms, and the bright one- and two-soliton solutions have been derived. After some derivations, the expressions of soliton transmission velocity and intensity have been obtained. It can be known from the expressions of velocity that in addition to the parameters , , and , the transmission volecity has been controlled by the disturbance coefficient . Moreover, when has took different functions, soliton transmission paths of different shapes have appeared on the corresponding plane. On the other hand, the intensity of the solitons has been affected by the parameter , , , and . Since the parameters , and affect the speed and intensity of the solitons, it is inevitable that the interactions of the solitons would be affected by them in the transmissions. Constantly adjusting the parameters and , it was found that the two solitons had fused, split and deformed. And under certain conditions, the energy of one soliton would be absorbed by the other soliton. In the process of soliton fusion and splitting, both belong to inelastic interactions caused by energy redistribution. Finally, we have found that during the sinusoidal two-soliton transmission, the parameters and can adjust the vibrations synchronization of the two-solitons. This shows that the transmission path and state of the soliton can be controlled by controlling the adjustable parameters.

Compliance with ethics requirements

This article does not contain any studies with human or animal subjects.

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.