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