  Within the framework of numerical simulations, we study the gyrotron dynamics
under conditions of a significant excess of the operating current over the
starting value, when the generation of electromagnetic pulses with anomalously
large amplitudes (``rogue waves'') can be realized. The averaged shape of
high-power pulses is shown to be very close to the celebrated Peregrine
breather. At the same time, we demonstrate that the relation between peak power
and duration of rogue waves is self-similar, but does not reproduce the one
characteristic for Peregrine breathers. Remarkably, the discovered self-similar
relation corresponds to the exponential nonlinearity of an equivalent
Schr\"odinger-like evolution equation. This interpretation can be used as a
theoretical basis for explaining the giant amplitudes of gyrotron rogue waves.
