The laser-ion acceleration from the ultra-short and ultra-intense laser-matter interactionsattracts more and more interest nowadays. When a laser pulse interacts with a target,relativistic electrons are generated in a period of few femtoseconds and driven away by the ponderomotiveforce, then a huge charge-separation field forms. In general cases, the ion accelerationis determined by this charge-separation field and the scale length of the plasma density. A generaltime-dependent solution is obtained to describe laser-plasma isothermal expansions into a vacuum,which is the fundamental theory of the laser-ion acceleration. It is adequate for non-quasi-neutralplasmas and different types of the scale length of the density gradient. The previous solutionsare some special cases of our general solution. It is found that there exist both a compressionlayer of the ion velocity distribution and a potential well for some initial conditions. However,many unaccounted idiographic solutions, which may be used to reveal new mechanisms of ionacceleration, may be deduced from our general solutions.
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