We consider the evolution of nonlinear optical pulses in inhomogeneous optical media wherein the pulse propagation is governed by the nonlinear Schrödinger equation with varying dispersion. We adopt the Painlevé analysis in order to obtain the condition for the soliton pulse propagation. We find that there exist two dispersion profiles satisfying this criterion, namely, the constant dispersion and exponentially dispersion decreasing profiles. In the exponentially varying dispersion media, we explain the existence and the formation of chirped optical soliton through the equation for the chirp parameter that results from variational analysis. For further elucidation, we provide the phase-plane diagram in terms of the normalized chirp and intensity (peak power), which explains the physical mechanism of linearly chirped soliton pulse compression in the exponentially dispersion decreasing media. In addition, we discuss the generation of exact chirped higher order solitons using the Bäcklund transformation method. As a special case, we find an oscillatory two-soliton breather pulse, which, at decreasing intervals, evolves into a familiar hyperbolic secant shape whose amplitude is twice that of a fundamental soliton. We highlight on the point that the crux of this work in realizing a compact pulse compressor could possibly result in a myriad of applications in modern optical fiber communications systems.