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In the paper \cite{carati95} it was shown that, for motions on a line underthe action of a potential barrier, the third-order Abraham-Lorentz-Diracequation presents the phenomenon of nonuniqueness of nonrunaway solutions.Namely, at least for a sufficiently steep barrier, the physical solutions ofthe equation are not determined by the "mechanical state" of position andvelocity, and knowledge of the initial acceleration too is required. Due torecent experiments, both in course and planned, on the interactions betweenstrong laser pulses and ultra relativistic electrons, it becomes interesting toestablish whether such a nonuniqueness phenomenon extends to the latter case,and for which ranges of the parameters. In the present work we will considerjust the simplest model, i.e., the case of an electromagnetic plane wave, andmoreover for the Abraham-Lorentz-Dirac equation dealt with in thenonrelativistic approximation. The result we found is that the nonuniquenessphenomenon occurs if, at a given frequency of the incoming wave, the fieldintensity is sufficiently large. An analytic estimate of such a threshold isalso given. At the moment it is unclear whether such a phenomenon applies alsoin the full relativistic case, which is the one of physical interest.

Nonuniqueness of nonrunaway solutions of Abraham-Lorentz-Dirac equation in an external laser pulse