Periodic and solitary waves generating in optical fiber amplifiers and fiber lasers with distributed parameters

We study self-similar dynamics of picosecond light pulses generating in optical fiber amplifiers and fiber lasers with distributed parameters. A rich variety of periodic and solitary wave solutions are derived for the governing generalized nonlinear Schr\"{o}dinger equation with varying coefficients in the presence of gain effect. The constraint on distributed optical fiber parameters for the existence of these wave solutions is presented. The dynamical behaviour of those self-similar waves is discussed in a periodic distributed amplification system. The stability of periodic and solitary wave solutions is also studied numerically by adding white noise. It is proved by using the numerical split-step Fourier method that the profile of these nonlinear self-similar waves remains unchanged during evolution.