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 β 1 ( z ) , the group velocity dispersion coefficient β 2 ( z ) and nonlinear coefficient γ ( z ) . Two-soliton solutions for the NLS equation are obtained through the Hirota method. According to the solutions obtained, β 1 ( z ) and γ ( z ) 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.

Femtosecond optical pulses have applications in optical communication, astronomical frequency combs and laser spectroscopy. Here, a hybrid mode-locked erbium-doped fiber (EDF) laser with topological insulator (TI) is proposed, for the first time to our best knowledge. The pulsed laser deposition (PLD) method is employed to fabricate the fiber-taper TI saturable absorber (TISA). By virtue of the fiber-taper TISA, the hybrid EDF laser is passively mode-locked using the nonlinear polarization evolution (NPE) and emits 70 fs pulses at 1542 nm, whose 3 dB spectral width is 63 nm with a repetition rate and transfer efficiency of 95.4 MHz and 14.12%, respectively. Our experiments indicate that the proposed hybrid mode-locked EDF lasers have better performance to achieve shorter pulses with higher power and lower mode-locking threshold in the future.