A generalized higher-order nonlinear-Schrödinger model of the transmission of subpicosend optical pulses in dispersion-decreasing fibers, with variable coefficients of the second- and third-order dispersion, nonlinearity, self-steepening, intra-pulse stimulated Raman scattering, and gain or loss, is considered. Imposing generalized Hirota conditions on the variable coefficients, we obtain exact solutions for a soliton sitting on top of a continuous-wave (CW) background by means of the Darboux transform. In the general form, the same solution provides for an exact description of the development of the modulational instability of a CW state, initiated by an infinitesimal periodic perturbation and leading to formation of a periodic array of solitons with a residual CW background. To obtain a more practically relevant solution for a soliton array without the CW component, we subtract it from the exact solution, and use the result as an initial approximation, to generate solutions in direct simulations. As a result, we obtain robust pulse trains, which are stable against arbitrary perturbations, as well as against violations of the Hirota conditions that were imposed to generate the initial exact solution.