  A star destroyed by a supermassive black hole (SMBH) in a tidal disruption
event (TDE) is transformed into a filamentary structure known as a tidally
disrupted stellar debris stream. We show that when ideal gas pressure dominates
the thermodynamics of the stream, there is an exact solution to the
hydrodynamics equations that describes the stream evolution and accounts for
self-gravity, pressure, the dynamical expansion of the gas, and the transverse
structure of the stream. We analyze the stability of this solution to
cylindrically symmetric perturbations, and show that there is a critical stream
density below which the stream is unstable and is not self-gravitating; this
critical density is a factor of at least 40-50 smaller than the stream density
in a TDE. Above this critical density the stream is overstable, self-gravity
confines the stream, the oscillation period is exponentially long, and the
growth rate of the overstability scales as $t^{1/6}$. The power-law growth and
small power-law index of the overstability implies that the stream is
effectively stable to cylindrically symmetric perturbations. We also use this
solution to analyze the effects of hydrogen recombination, and suggest that
even though recombination substantially increases the gas entropy, it is likely
incapable of completely destroying the influence of self-gravity. We also show
that the transient produced by recombination is far less luminous than previous
estimates.
