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We obtain exact self-similar solutions to an inhomogeneous nonlinear Schrödinger equation, describing propagation of optical pulses in fiber amplifiers with distributed dispersion and gain. We show that there exists a one-to-one correspondence between such self-similar waves and solitons of the standard, homogeneous, nonlinear Schrödinger equation if a certain compatibility condition is satisfied. As this correspondence guarantees the stability of the novel self-similar waves, we refer to them as {\it similaritons}. We demonstrate that, the character of similariton interactions crucially depends on the sign of the similariton phase chirp. In particular, we show that the similariton interactions can under certain conditions lead to the formation of molecule-like bound states of two similaritons.

Interactions of chirped and chirp-free similaritons in optical fiber amplifiers