Wave-corpuscle mechanics for elementary charges
  It is well known that a concept of point charge interacting with electromagnetic (EM) field has a problem. To address that problem we introduce a concept of wave-corpuscle to describe spinless elementary charges interacting with the classical EM field. Every charge interacts only with the EM field and is described by a complex valued wave function over 4-dimensional space time continuum. A system of many charges interacting with the EM field is defined by a local, gauge and Lorentz invariant Lagrangian with a key ingredient - a nonlinear self-interaction term providing for a cohesive force assigned to every charge. An ideal wave-corpuscle is an exact solution to the Euler-Lagrange equations describing both free or accelerated motion. It carries explicitly features of a point charge and the de Broglie wave. A system of well separated charges moving with nonrelativistic velocities are represented accurately as wave-corpuscles governed by the Newton motion equations for point charges interacting with the Lorentz forces. In this regime the nonlinearities are "stealthy" and don't show explicitly anywhere, but they provide for binding forces that keep localized every individual charge. 
Anatoli Babin and Alexander Figotin