ANZIAM  J.  44 (2002), 51-59
Nonlinear electron solutions and their characteristics at infinity

Hilary Booth
  Centre for Bioinformation Science
  Australian National University
  Canberra ACT 0200

The Maxwell-Dirac equations model an electron in an electromagnetic field. The two equations are coupled via  the Dirac current which acts as a source in the Maxwell equation, resulting in a nonlinear system of partial differential equations (PDE's). Well-behaved solutions, within reasonable Sobolev spaces, have been shown to exist globally as recently as 1997 [12]. Exact solutions have not been found---except in some simple cases. We have shown analytically in [6, 18] that any spherical solution surrounds a Coulomb field and any cylindrical solution surrounds a central charged wire; and in [3] and [19] that in any stationary case, the surrounding electron field must  be equal and opposite to the central (external) field. Here we extend the numerical solutions in [6] to a family of orbits all of which are well-behaved numerical solutions satisfying the analytic results in [6] and [11]. These solutions die off exponentially with increasing distance from the central axis of symmetry. The results in [18] can be extended in the same way. A third case is included, with dependence on  z  only yielding a related fourth-order ordinary differential equation (ODE) [3].
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