  We study the formation of singularity for the isothermal Euler-Poisson system
arising from plasma physics. Contrast to the previous studies yielding only
limited information on the blow-up solutions, for instance, sufficient
conditions for the blow-up and the temporal blow-up rate along the
characteristic curve, we rather give a constructive proof of singularity
formation from smooth initial data. More specifically, employing the stable
blow-up profile of the Burgers equation in the self-similar variables, we
establish the global stability estimate in the self-similar time, which yields
the asymptotic behavior of blow-up solutions near the singularity point. Our
analysis indicates that the smooth solution to the Euler-Poisson system can
develop a cusp-type singularity; it exhibits $C^1$ blow-up in a finite time,
while it belongs to $C^{1/3}$ at the blow-up time, provided that smooth initial
data are sufficiently close to the blow-up profile in some weighted
$C^4$-topology. We also present a similar result for the isentropic case, and
discuss noteworthy differences in the analysis.
