  The dynamics of linear and nonlinear excitations in a Bose gas in the
Tonks-Girardeau (TG) regime with longitudinal confinement are studied within a
mean field theory of quintic nonlinearity. A reductive perturbation method is
used to demonstrate that the dynamics of shallow dark solitons, in the presence
of an external potential, can effectively be described by a
variable-coefficient Korteweg-de Vries equation. The soliton oscillation
frequency is analytically obtained to be equal to the axial trap frequency, in
agreement with numerical predictions obtained by Busch {\it et al.} [J. Phys. B
{\bf 36}, 2553 (2003)] via the Bose-Fermi mapping. We obtain analytical
expressions for the evolution of both soliton and emitted radiation (sound)
profiles.
