# Knotted solutions for linear and nonlinear theories: electromagnetism and fluid dynamics

## TL;DR

This paper explores knotted solutions, like the Hopfion, in electromagnetism and fluid dynamics, revealing new solutions and connections between linear and nonlinear theories through mappings and reductions.

## Contribution

It introduces a mapping between electromagnetism and fluid dynamics to find new knotted solutions and extends these solutions to nonlinear and quantum-corrected theories.

## Key findings

- Found new knotted fluid solutions from electromagnetic knotted solutions.
- Demonstrated that knotted solutions persist in nonlinear generalizations of electromagnetism.
- Mapped electromagnetic knotted solutions to solutions of Euler's equations.

## Abstract

We examine knotted solutions, the most simple of which is the "Hopfion", from the point of view of relations between electromagnetism and ideal fluid dynamics. A map between fluid dynamics and electromagnetism works for initial conditions or for linear perturbations, allowing us to find new knotted fluid solutions. Knotted solutions are also found to to be solutions of nonlinear generalizations of electromagnetism, and of quantum-corrected actions for electromagnetism coupled to other modes. For null configurations, electromagnetism can be described as a null pressureless fluid, for which we can find solutions from the knotted solutions of electromagnetism. We also map them to solutions of Euler's equations, obtained from a type of nonrelativistic reduction of the relativistic fluid equations.