  (Abriged) We present analytic models for the formation and evolution of tidal
tails and related structures following impulsive disturbances in galaxy
collisions. Since the epicyclic approximation is not valid for large radial
excursions, we use orbital equations of the form we call p-ellipses. These have
been shown to provide accurate representations of orbits in power-law halo
potentials. In the case of a purely tidal disturbance the resulting tidal tails
have simple structure. Scalings for their maximum lengths and other
characteristics as functions of the tidal amplitude and the exponent of the
power-law potentials are described. The analytic model shows that azimuthal
caustics (orbit crossing zones) are produced generically in these tails at a
fixed azimuth relative to the point of closest approach. Long tails, with high
order caustics at their base are also produced at larger amplitudes. The
analysis is extended to nonlinear disturbances and multiple encounters, which
break the symmetries of tidal perturbations. As the strength of the nonlinear
terms is varied the structure of the resulting forms varies from symmetric
tails to one-armed plumes. Cases with two or more impulse disturbances are also
considered as the simplest analytic models distinguishing between prograde and
retrograde encounters. A specific mechanism for the formation of tidal dwarf
galaxies at the end of tails is suggested as a consequence of resonance effects
in prolonged encounters. Qualitative comparisons to Arp Atlas systems suggest
that the limiting analytic cases are realized in real systems. We identify a
few Arp systems which may have swallowtail caustics, where dissipative gas
streams converge and trigger star formation. UV and optical images reveal
luminous knots of young stars at these 'hinge clump' locations.
