Numerical investigations of singularity formation in non-linear wave equations in the adiabatic limit

Abstract

This dissertation deals with singularity formation in spherically symmetric solutions of the hyperbolic Yang Mills equations in (4+1) dimensions and in spherically symmetric solutions of C P^1 wave maps in (2+1) dimensions. These equations have known moduli spaces of time-independent (static) solutions. Evolution occurs close to the moduli space of static solutions. The evolution is modeled numerically using an iterative finite differencing scheme, and modeling is done close to the adiabatic limit, i.e., with small velocities. The stability of the numerical scheme is analyzed and growth is shown to be bounded, yielding a convergence estimate for the numerical scheme. The trajectory of the approach is characterized, as well as the shape of the profile at any given time during the evolution.