Summary: In this paper, we investigate a generalized nonautonomous nonlinear equation, which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber. Under certain integrable constraints, bilinear forms, bright one- and two-soliton solutions are obtained. Via certain transformation, we investigate the properties of the solitons with the first-order dispersion parameter \(\sigma_1(x,t)\), second-order dispersion parameter \(\sigma_2(x,t)\), third-order dispersion parameter \(\sigma_3(x,t)\), phase modulation and gain (loss) \(v(x,t)\). Soliton propagation and collision are graphically presented and analyzed: One soliton is shown to maintain its amplitude and width during the propagation. When we choose \(\sigma_1(x,t)\), \(\sigma_2(x,t)\) and \(\sigma_3(x,t)\) differently, travelling direction of the soliton is found to alter. \(v(x,t)\) is observed to affect the amplitude of the soliton. Head-on collision between the two solitons is presented with \(\sigma_1(x,t)\), \(\sigma_2(x,t)\), \(\sigma_3(x,t)\) and \(v(x,t)\) as the constants, and solitons’ amplitudes are the same before and after the collision. When \(\sigma_1(x,t)\), \(\sigma_2(x,t)\) and \(\sigma_3(x,t)\) are chosen as certain functions, the solitons’ traveling directions change during the collision. \(v(x,t)\) can influence the amplitudes of the two solitons.