Analytical solution of the classical Dirac particle CM/CC dynamical system

Develop analytical solutions for the coupled non-linear second-order ordinary differential equations that govern the time evolution of the center of mass q(t) and the center of charge r(t) of the classical Dirac particle in external electromagnetic fields, where the center-of-mass acceleration is determined by the Lorentz force evaluated at the center-of-charge and the center-of-charge acceleration depends on the relative variables and velocities, specifically for the circularly polarized electromagnetic plane wave traveling along the OY axis with fields E_x(t,y) = k E sin(omega (t − y/c) + sigma), E_z(t,y) = E cos(omega (t − y/c) + sigma), B_x(t,y) = B cos(omega (t − y/c) + sigma), and B_z(t,y) = −k B sin(omega (t − y/c) + sigma).

Background

The paper models a classical Dirac particle by two distinguished points: the center of charge r and the center of mass q. Their dynamics in external electromagnetic fields is described by a coupled system of non-linear second-order ordinary differential equations: the center-of-mass acceleration depends on the Lorentz force computed at the center-of-charge, while the center-of-charge acceleration involves the relative separation and velocities subject to the constraints |u| = c and |v| < c.

For the circularly polarized plane-wave considered, the authors provide explicit component-wise equations for dv_x/dt, dv_y/dt, and dv_z/dt in terms of the wave parameters (frequency nu, polarization k, phase sigma) and the interaction strength a. Despite extensive numerical experimentation, they report that they could not find analytical solutions and restrict their paper to numerical integration, highlighting the need for an analytical treatment of the system.