Parametric instability of the real scalar pulsons

We investigate stability of localized time-periodic states (pulsons) of a real scalar field satisfying the Klein-Gordon equation with the logarithmic nonlinearity. It is shown that the pulsons of any amplitudes, remaining well-localized objects, lose their coherence with time. We thus conclude that, strictly speaking, all pulsons of the model considered are unstable. Nevertheless, for some amplitudes the rate of the coherence breaking is found to be very small, so that the corresponding pulsons are long-lived.