Introduction

Material and methods

Results and discussion

Conclusions

References

Abstract

In nonlinear optics, the soliton transmission in different forms can be described with the use of nonlinear Schrödinger (NLS) equations. Here, the soliton transmission is investigated by solving the NLS equation with the reciprocal of the group velocity b1ðzÞ, the group velocity dispersion coefficient b2ðzÞ and nonlinear coefficient cðzÞ. Two-soliton solutions for the NLS equation are obtained through the Hirota method. According to the solutions obtained, b1ðzÞ and cðzÞ with different function forms are taken to study the characteristics of solitons. The effect of the phase shift on the soliton interaction is discussed, and the non-oscillating soliton amplification, which is transmitted in a bound state, is explored. Parabolic solitons with oscillations are analysed. Moreover, parabolic solitons can be reduced to dromion-like structures. Results indicate that the transmission of solitons can be adjusted with the group velocity dispersion and Kerr nonlinearity coefficients. The phase shift, amplification, oscillation and attenuation of solitons can also be controlled by other related parameters. This work accomplishes the theoretical study of transmission characteristics of optical solitons in spatially dependent inhomogeneous optical fibres. The conclusions of this research have theoretical guidance for the research of optical amplifier, all-optical switches and mode-locked lasers.

Introduction

Solitons have been investigated in such fields as mathematics and physics [1–۶]. They can propagate in a long distance without changes in their waveform, velocity and amplitude [7–۱۰]. The soliton phenomena are closely related to nonlinear evolution equation models [11–۱۷]. Solitons are also applied in particle physics, fluid mechanics, Bose-Einstein condensation and nonlinear optics [18–۲۰]. Some researchers have studied solitons by solving the nonlinear evolution equations and analysing their soliton solutions [21–۲۵]. As one of the classic nonlinear evolution equations, the nonlinear Schrödinger (NLS) equation can be solved to obtain soliton solution, and has been widely investigated by using different methods [26–۳۲]. With the differential quadrature method, the dynamic problems constructed by the NLS equation have been analysed [26]. The existence and stability of the standing wave solutions for the NLS equation in n-dimensional space have been studied [27]. Using the generalized exponential rational function, a new method to solve the exact special solutions of the NLS equation has been proposed [28]. The unified method has been used to acquire optical soliton solutions of the NLS equation [29]. Moreover, the stability of full dimensional KAM tori for the NLS equation has been proved [30]. Local Cauchy theory for the NLS equation has been discussed [31], and nonlinear instability of half-solitons has been analysed [32]. However, the soliton transmission process can be simulated more accurately with the variable coefficient NLS (vcNLS) equation, when the transmission medium or boundary condition is not uniform [33–۴۰]. For the vcNLS equation, dynamics of solitons have been explored [33], and soliton interactions have been discussed [34,35]. In addition, the breather-to-soliton transitions for the vcNLS equation have been found [36], and nonautonomous multipeak solitons have been obtained [37].