  To clarify the relationship between phase-stable acceleration (PSA) and
unlimited relativistic acceleration (URA) (Phys. Rev. Lett. 104, 135003
(2010)), an analytical relativistic model is proposed in the interactions of
the ultra-intense laser and nanometer foils, based on hydrodynamic equations.
The dependence of the ion momentum on time is consistent with the previous
results and checked by PIC simulations. Depending on the initial ion momentum,
relativistic RPA contains two acceleration processes: phase-stable limited
relativistic acceleration (PS-LRA) and URA. In PS-LRA, the potential is a deep
well trapping the ions. The ion front, i.e., the bottom, separates it into two
parts: the left half region is PSA region; the right half region is PSD region,
where the ions climb up and are decelerated to return back. In PS-LRA, the
maximum ion energy is limited. If the initial ion momentum large enough, ions
will experience a potential downhill and drop into a bottomless abyss, which is
called phase-lock-like position. URA is not phase-stable any more. At the
phase-lock-like position the ions can obtain unlimited energy gain and the ion
density is non-zero. You cannot have both PSA and URA.
