Exact static soliton solutions of 3+1 dimensional integrable theory with nonzero Hopf numbers

TL;DR

This paper explicitly constructs an infinite family of static soliton solutions called Hopfions with non-zero Hopf numbers in a 3+1-dimensional integrable field theory, revealing their linked vortex structure and energy bounds.

Contribution

It provides explicit formulas for an infinite set of Hopfion solutions in a 3+1D integrable theory, linking topological charge to energy bounds.

Findings

Constructed explicit Hopfion solutions with non-zero Hopf numbers.

Demonstrated the linked vortex structure of the solutions.

Established the relation between Hopf charge and soliton energy.

Abstract

In this paper we construct explicitly an infinite number of Hopfions (static, soliton solutions with non-zero Hopf topological charges) within the recently proposed 3+1-dimensional, integrable and relativistically invariant field theory. Two integers label the family of Hopfions we have found. Their product is equal to the Hopf charge which provides a lower bound to the soliton's finite energy. The Hopfions are constructed explicitly in terms of the toroidal coordinates and shown to have a form of linked closed vortices.