Solitons and Volume Preserving Flow

Abstract

Solitons arise as solutions to non-linear partial differential equations. These equations are only analytically solvable in very few special cases. Other solutions must be found numerically. A useful technique for obtaining static solutions is gradient flow. Gradient flow evolution is in a direction which never increases energy, leading to solutions which are local minima. Here, a modified version of gradient flow referred to as volume preserving flow is introduced. This flow is constructed to evolve solutions towards local minima, while leaving a number of global quantities unaltered. Volume preserving gradient flow will be introduced and demonstrated in some simple models.

Volume preserving flow will be used to investigate minimal surfaces in the context of double bubbles. Work will reproduce explicit results for double bubbles on the two-torus and construct a range of possible minimisers on the three-torus. Domain walls in a Wess-Zumino model with a triply degenerate vacuum will be used to represent the surfaces of the bubbles. Volume preserving flow will minimise the energy of the domain walls while maintaining the volumes of the space they contain. Global minima will represent minimal surfaces in the limit in which the domain wall thickness tends to zero.

Numerical simulations of solitons in models which have conformal symmetry are problematic. Discretisation breaks the zero modes associated with changes of scale to negative modes. These lead to the collapse of solutions. Volume preserving flow provides a framework in which minimisation occurs orthogonally to these zero modes, maintaining a scale for the minimisation.

Two such conformal models which permit Hopf solitons are the Nicole and AFZ models. They are comprised of the two components of the Skyrme-Faddeev model, taken to fractional powers to allow for solitons. Volume preserving flow will be used to find static solutions for a range of Hopf charges for each model. Comparisons will be made with the Skyrme-Faddeev model and general features of Hopf solitons will be discussed. A one parameter family of conformal Skyrme-Faddeev models will also be introduced. These models will be the set of linear combinations of the Nicole and AFZ models where the coefficients sum to one. Energy and topology transitions through this set of models will be investigated.