  A model for the excitation of a non-linear ion-wake mode by a train of plasma
electron oscillations in the non-linear time-asymmetric regime is developed
using analytical theory and particle-in-cell based computational solutions. The
ion-wake is shown to be a driven non-linear ion-acoustic wave in the form of a
cylindrical ion-soliton. The near-void and radially-outwards propagating
ion-wake channel of a few plasma skin-depth radius, is explored for application
to "Crunch-in" regime of positron acceleration. The coupling from the electron
wakefield mode to the ion-mode dictates the long-term evolution of the plasma
and the time for its relaxation back to an equilibrium, limiting the
repetition-rate of a plasma accelerator. Using an analytical model it is shown
that it is the time asymmetric phases of the oscillating radial electric fields
of the nearly-stationary electron bubble that excite time-averaged inertial ion
motion radially. The electron compression in the back of the bubble sucks-in
the ions whereas the space-charge within the bubble cavity expels them, driving
a cylindrical ion-soliton structure with on-axis and bubble-edge
density-spikes. Once formed, the channel-edge density-spike is sustained over
the length of the plasma and driven radially outwards by the thermal pressure
of the wake energy in electrons. Its channel-like structure is independent of
the energy-source, electromagnetic wave or particle beam, driving the bubble
electron wake. Particle-In-Cell simulations are used to study the ion-wake
soliton structure, its driven propagation and its use for positron acceleration
in the "Crunch-in" regime.
