Exact chirped self-similar solutions of the generalized nonlinear Schrödinger equation with varying dispersion, nonlinearity, gain or absorption, and nonlinear gain have been found. The stability of these nonlinearly chirped solutions is then demonstrated numerically by adding Gaussian white noise and by evolving from an initial chirped Gaussian pulse, respectively. It is reported that the pulse position of these chirped pulses can be precisely piloted by tailoring the dispersion profile, and that the sech-shaped solitary waves can propagate stably in the regime of beta(z)gamma(z) > 0 as well as the regime of beta(z)gamma(z) < 0 , according to the magnitude of the nonlinear chirp parameter. Our theoretical predictions are in excellent agreement with the numerical simulations.