Non uniqueness of non runaway solutions of Abraham–Lorentz–Dirac equation in an external laser pulse

In the paper cite{carati95} it was shown that, for motions on a line under the action of a potential barrier, the third-order ALD equation presents the phenomenon of non uniqueness of non runaway solutions. Namely, at least for a sufficiently steep barrier, the physical solutions of the equation are not determined by the ``mechanical state'' of position and velocity, and knowledge of the initial acceleration too is required. Due to recent experiments, both in course and planned, on the interactions between strong laser pulses and ultra relativistic electrons, it becomes interesting to establish whether such a non uniqueness phenomenon extends to the latter case, and for which ranges of the parameters. In the present work we will consider just the simplest model, i.e., the case of an electromagnetic plane wave, and moreover the ALD equation will be dealt with in the non relativistic approximation. The result we found is that the non uniqueness phenomenon occurs if, at a given frequency of the incoming wave, the field intensity is sufficiently large. An analytic stimate of such a threshold is also given. At the moment it is unclear whether such a phenomenon applies also in the full relativistic case, which is the one of physical interest.