  Based on the already existing analytical theory of the strongly-nonlinear
wakefield (which is called "bubble") in transversely inhomogeneous plasmas, we
study particular behavior of non-loaded (empty) bubbles and bubbles with
accelerated bunches. We obtain an analytical expression for the shape of a
non-loaded bubble in a general case and verify it with particle-in-cell (PIC)
simulations. We derive a method of calculation of the acceleration efficiency
for arbitrary accelerated bunches. The influence of flat-top electron bunches
on the shape of a bubble is studied. It is also shown that it is possible to
achieve acceleration in a homogeneous longitudinal electric field by the
adjustment of the longitudinal density profile of the accelerated electron
bunch. The predictions of the model are verified by 3D PIC simulations and are
in a good agreement with them.
