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Resonant dynamics within the nonlinear Klein-Gordon Equation: Much ado about Oscillons

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.