  An extended analytical model for particle dynamics in fields of a
highly-nonlinear plasma wake field (the bubble or blow out regime) is derived.
A recently proposed piecewise model (Kostyukov et al., New J. Phys., {\bf 12},
045009 (2010)) is generalized to include a time dependent bubble radius and
full field solution in the acceleration direction. Incorporation of the cavity
dynamics in the model is required to simulate the particle trapping properly.
On the other hand, it is shown that the previously reported piecewise model
does not reproduce the formation of a mono energetic peak in the particle
spectrum. The mono energetic electron beams are recovered only when the full
longitudinal field gradient is included in the model.
