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Born-Infeld nonlinear electrodynamics with point singularities having bothelectric and magnetic charges are considered. Problem of interaction betweenthe associated soliton dyon solutions is investigated. For the case oflong-range interaction at first order by a small field of distant solitons weobtain that the generalized Lorentz force is acted on a dyon underconsideration. Short-range interaction between two dyons having identicalelectric and opposite magnetic charges is investigated for an initialapproximation. We consider the case when the velocities of the dyons have equalmodules and opposite directions on a common line. It is shown that theassociated field configuration has a constant full angular momentum which isindependent of the interdyonic distance and their speed. This property permitsa consideration of this bidyon configuration as an electromagnetic model ofcharged particle with spin. We numerically investigate movement of the dyons inthis configuration for the case when the full electric charge equals theelectron charge and the full angular momentum equals the electron spin. It isshown that for this case the absolute value of relation between electric andmagnetic charges of the dyons equals the fine structure constant. Thecalculation gives that the bidyon may behave as nonlinear oscillator.Associated dependence of frequency on the full energy is obtained for theinitial approximation. In the limits of the electrodynamic model we obtain thatthe quick-oscillating wave packet may behave like massive gravitating particlewhen it move in high background field. We discuss the possible electrodynamicworld with the oscillating bidyons as particles.Comment: 34 pages, 10 postscript figures, LaTeX2e, included jhep.cls; final  version with minor correction

Dyons and interactions in nonlinear (Born-Infeld) electrodynamics