  At present, there exists no physically plausible example of a quantum field
theory for which the existence of solutions has been proven mathematically. The
Clay Mathematics Institute has offered a prize for proving existence for a
class of Yang-Mills theories defined by Jaffe and Witten. This paper proposes a
multi-stage strategy for proving existence for a quantum version of the
"'tHooft Polyakov" (tP) field theory, and argues that this theory, while not
renormalizable or physically plausible as it stands, opens up a clear path to a
physically plausible well-defined theory. The key initial challenge is to first
prove stability for a classical version of this theory, in the spirit of Walter
Strauss. The stability results of Bogomolnyi for classical PDE systems are
widely cited as a foundation of string theory, but they leave key questions
unaddressed, and may even call for small modifications of the tP model itself.
This paper calls attention to the questions which require more attention, and
proposes some partial possible answers. It also specifies a new numerical
method to check for stability for the usual "hedgehog" types of solution, a
method which may have broader potential uses for any ODE problems where
accuracy in derivatives is as important as accuracy in function values.
