Some mathematical problems in a neoclassical theory of electric charges

We study here a number of mathematical problems related to our recentlyintroduced neoclassical theory for the electromagnetic phenomena in whichcharges are represented by complex valued wave functions as in the Schrodingerwave mechanics. Dynamics of charges in the non-relativistic case is governed bya system of nonlinear Schrodinger equations coupled with the electromagneticfields, and we prove for it that the centers of wave functions converge inmacroscopic regimes to trajectories of points governed by the Newton'sequations with the Lorentz forces provided the wave functions remain localized.Exact solutions in the form of localized accelerating solitons are found. Ourstudies of a class of time multiharmonic solutions of the same field equationsshow that they satisfy Planck-Einstein relation and that the energy levels ofthe nonlinear eigenvalue problem for the hydrogen atom converge to thewell-known energy levels of the linear Schrodinger operator when the freecharge size is much larger than the Bohr radius.