  Conditions for the existence, uniqueness and stability of self-consistent
bunch steady states are considered. For the existence and uniqueness problems,
simple algebraic criteria are derived for both the action and Hamiltonian
domain distributions. For the stability problem, van Kampen theory is used. The
onset of a discrete van Kampen mode means the emergence of a coherent mode
without any Landau damping; thus, even a tiny couple-bunch or multi-turn wake
is sufficient to drive the instability. The method presented here assumes an
arbitrary impedance, RF shape, and beam distribution function. Available areas
on the intensity-emittance plane are shown for resistive wall wake and single
harmonic, bunch shortening and bunch lengthening RF configurations. Thresholds
calculated for the Tevatron parameters and impedance model are in agreement
with the observations. These thresholds are found to be extremely sensitive to
the small-argument behaviour of the bunch distribution function. Accordingly, a
method to increase the LLD threshold is suggested. This article summarizes and
extends recent author's publications.
