Hopf Solitons

Topological solitons are stable, finite-energy solutions of systems of nonlinear partial differential equations, with their stability being due in part to nontrivial topology. They occur in a variety of theories and describe a wide range of physical phenomena in areas including particle and nuclear physics, cosmology, and condensed matter physics, and have important and interesting potential applications.Most topological solitons are point-like, but Hopf solitons have a novel string-like structure, which means they can form complicated shapes including knots and links. Indeed for a prototype system it has been shown that knots and links are the preferred shapes for particular Hopf solitons to minimize their energy.This work will investigate the properties of Hopf solitons in a variety of theories modeling different physical situations, through the use of analytic and numerical methods. The shapes, interactions and dynamics of Hopf solitons will be investigated to address fundamental issues concerning universality, applicability, and the generic existence of knots and links.Knots play a vital role in numerous and diverse areas, from the study of DNA to quantum field theory --thus a completely new mathematical approach to their description, which automatically includes their interactions and dynamics, has dramatic potential in many fields.