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Relativistic scattering of solitons in nonlinear field theory Gutierrez, Benjamin


This thesis presents results from numerical studies of the dynamics of three classical nonlinear field theories, each of which possesses stable, localized solutions called solitons. We focus on two of the theories, known as Skyrme models, which have had application in various areas of physics. The third, which treats a complex scalar field, is principally viewed as a model problem to develop solution techniques. In all cases, time dependent, nonlinear partial differential equations in several spatial dimensions are solved computationally. Simulation of high energy collisions of the solitons is of particular interest. The solitons of the complex scalar field theory are known as Q-balls and carry a charge Q. We investigate the scattering of these objects, reproducing previous findings for collisions where the Q-balls have charge of the same sign. We obtain new results for interactions where the Q-balls have opposite charge, and for scattering of Q-balls against potential obstructions. The chief contributions of this thesis come from simulations performed within the context of a Skyrme model in two spatial dimensions. This is a multi-scalar theory with solitons known as baby skyrmions. We concentrate on the rich phenomenology seen in high-energy scattering of pairs of these objects, each of which has a topological charge that can be either positive or negative. We extend the understanding of the role of different parameters of the model in governing the outcome of head-on and off-axis collisions. The study of instabilities seen in previous simulations of Skyrme models is of central interest. Our results confirm that the governing partial differential equations become of mixed hyperbolic-elliptic type for interactions at sufficiently high-energy, as originally suggested by Crutchfield and Bell. We present strong evidence for the loss of energy conservation and smoothness of the dynamical fields in these instances. This supports the conclusion that the initial value problem at hand becomes ill-posed, so that the observed instabilities result from the nature of the equations themselves, and are not numerical artifacts. Finally, we present preliminary results for the typical phenomenology seen in head-on scattering of solitons in the Skyrme model in three spatial dimensions.

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