<p>We investigate the possibility of forming self-similar solitary waves characterized by a nonlinear chirp within an inhomogeneous waveguide material exhibiting variations in second-order dispersion, cubic nonlinearity, quintic nonlinearity, and gain/loss parameters along the transmission direction of the waveguide. We find that the optical system supports nonlinearly chirped localized waves with bright, dark, kink, and anti-kink shapes, which propagate self-similarly on a continuous-wave background. More significantly, we demonstrate that self-similar waves possessing a linear chirp of the governing generalized cubic-quintic nonlinear Schrödinger equation are special limited cases of the derived self-similar solitary pulses with a nonlinear chirp. As a practical example, we analyze the evolutionary behavior of the obtained chirped self-similar solitary waves in a specified soliton control system, and present a series of fascinating properties of nonlinear waves.</p>