Fluid-electromagnetic helicities and knotted solutions of the fluid-electromagnetic equations

TL;DR

This paper explores knotted solutions in fluid-electromagnetic systems, demonstrating that certain electromagnetic and fluid helicity configurations, including Hopfions, satisfy coupled equations and can be generated via conformal transformations.

Contribution

It introduces explicit knotted solutions in fluid-electromagnetic equations, linking fluid and electromagnetic helicities, and extends the analysis to nonlinear electromagnetism and lower dimensions.

Findings

Hopfion solutions carry fluid and electromagnetic helicities.

Knotted solutions satisfy coupled fluid-electromagnetic equations.

Conformal transformations generate knotted EM and fluid fields.

Abstract

In this paper we consider an Euler fluid coupled to external electromagnetism. We prove that the Hopfion fluid-electromagnetic knot, carrying fluid and electromagnetic (EM) helicities, solves the fluid dynamical equations as well as the Abanov Wiegmann (AW) equations for helicities, which are inspired by the axial-current anomaly of a Dirac fermion. We also find a nontrivial knot solution with truly interacting fluid and electromagnetic fields. The key ingredients of these phenomena are the EM and fluid helicities. An EM dual system, with a magnetically charged fluid, is proposed and the analogs of the AW equations are written down. We consider a fluid coupled to a nonlinear generalizations for electromagnetism. The Hopfions are shown to be solutions of the generalized equations. We write down the formalism of fluids in 2+1 dimensions, and we dimensionally reduce the 3+1 dimensional solutions. We determine the EM knotted solutions, from which we derive the fluid knots, by applying special conformal transformations with imaginary parameters on un-knotted null constant EM fields.