New knotted solutions of Maxwell's equations

Abstract

In this note we have further developed the study of topologically non-trivial solutions of vacuum electrodynamics. We have discovered a novel method of generating such solutions by applying conformal transformations with complex parameters on known solutions expressed in terms of Bateman's variables. This has enabled us to get a wide class of solutions from the basic configuration like constant electromagnetic fields and plane-waves. We have introduced a covariant formulation of the Bateman's construction and discussed the conserved charges associated with the conformal group as well as a set of four types of conserved helicities. We have also given a formulation in terms of quaternions. This led to a simple map between the electromagnetic knotted and linked solutions into flat connections of $SU(2)$ gauge theory. We have computed the corresponding CS charge in a class of solutions and it takes integer values.

version accepted for publication in J. Phys. A. minor changes: references added, new figure added, typos corrected

References in corpus (3)

• Classification of Electromagnetic and Gravitational Hopfions by Algebraic Type
• Scaling Identities for Solitons beyond Derrick's Theorem
• Motion of charged particles in an electromagnetic knot

Cited by in corpus (7)

• Knotted solutions, from electromagnetism to fluid dynamics
• Anti-Self-Dual Spacetimes, Gravitational Instantons and Knotted Zeros of the Weyl Tensor
• Knotted solutions for linear and nonlinear theories: electromagnetism and fluid dynamics
• Electrodynamics of dual superconducting chiral medium
• On the existence of the field line solutions of the Einstein-Maxwell equations
• Nonlinear dynamics of a charged particle in a strong non-null knot wave background
• Finsler Geometries from Topological Electromagnetism