  This dissertation discusses solutions to the nonlinear Klein-Gordon equation
with symmetric and asymmetric double-well potentials, focusing on the collapse
and collision of bubbles and critical phenomena found therein. A new method is
presented that allows the solution of massive field equations on a (relatively)
small static grid. A coordinate transformation is used that transforms typical
flatspace coordnates to coordinates that move outward (near the outer boundary)
at nearly the speed of light. The outgoing radiation is compressed to nearly
the Nyquist limit of the grid where it is quenched by dissipation. The method
is implemented successfully in both spherically symmetric and axisymmetric
codes. New resonant oscillon solutions are discussed and the threshold of
expanding bubble formation is explored in both spherical and axial symmetry.
Approximate resonant oscillon solutions are obtained using trial function
methods and variational principles that are consistent with dynamical
evolutions.
