  The laser-ion acceleration in the ultra-short and ultra-intense laser-matter
interactions attracts more and more interest nowadays. Since electrons gain
relativistic energy from laser pulse in a period of several femtoseconds and
driven away by the ponderomotive force of laser pulse, a huge charge-separation
field pulse is generated. In general cases, the ion acceleration is determined
by this charge-separation field. A novel general time-dependent solution for
laser-plasma isothermal expansions into a vacuum with different types of the
scale length of the density gradient which correspond to different charge
separation forms is obtained. The previous solutions are some special cases of
our general solution. A series of new solutions have been proposed and may be
used to predict new mechanisms of ion acceleration. However, many unaccounted
idiographic solutions that may be used to reveal new acceleration mode of ions
such as shock wave acceleration, may be deduced from our general solutions.
