  We investigate stability of both localized time-periodic coherent states
(pulsons) and uniformly distributed coherent states (oscillating condensate) of
a real scalar field satisfying the Klein-Gordon equation with a logarithmic
nonlinearity. The linear analysis of time-dependent parts of perturbations
leads to the Hill equation with a singular coefficient. To evaluate the
characteristic exponent we extend the Lindemann-Stieltjes method, usually
applied to the Mathieu and Lame equations, to the case that the periodic
coefficient in the general Hill equation is an unbounded function of time. As a
result, we derive the formula for the characteristic exponent and calculate the
stability-instability chart. Then we analyze the spatial structure of the
perturbations. Using these results we show that the pulsons of any amplitudes,
remaining well-localized objects, lose their coherence with time. This means
that, strictly speaking, all pulsons of the model considered are unstable.
Nevertheless, for the nodeless pulsons the rate of the coherence breaking in
narrow ranges of amplitudes is found to be very small, so that such pulsons can
be long-lived. Further, we use the obtaned stability-instability chart to
examine the Affleck-Dine type condensate. We conclude the oscillating
condensate can decay into an ensemble of the nodeless pulsons.
