ANZIAM J.
44 (2002), 5159

Nonlinear electron solutions and their characteristics at infinity


Abstract

The MaxwellDirac 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). Wellbehaved solutions,
within reasonable Sobolev spaces, have been shown
to exist globally as recently as 1997 [12].
Exact solutions have not been foundexcept 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 wellbehaved
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 fourthorder ordinary
differential equation (ODE) [3].

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