# Periodic Korteweg-de Vries soliton potentials generate quasisymmetric magnetic fields

## TL;DR

This paper reveals a deep connection between quasisymmetry in magnetic fields and soliton theory, offering new insights for stellarator design and optimization through analytical and machine learning methods.

## Contribution

It uncovers a hidden symmetry linking quasisymmetry to soliton equations like KdV, enabling more efficient stellarator optimization and understanding of magnetic field structures.

## Key findings

- B approaches 1-soliton reflectionless potential near the outer surface.
- Diverging connection length suggests possible X-point or cusp formation.
- Machine learning recovers KdV and Gardner's equations from stellarator data.

## Abstract

Quasisymmetry (QS) is a hidden symmetry of the magnetic field strength, B, that effectively confines charged particles in a three-dimensional toroidal plasma equilibrium. Here, we show that QS has a deep connection to the underlying symmetry that makes solitons possible. Our approach uncovers a hidden lower dimensionality of B on a magnetic flux surface, which could make stellarator optimization schemes significantly more efficient. Recent numerical breakthroughs (M. Landreman and E. Paul, Phys. Rev. Lett. 128, 035001 (2022)) have yielded configurations with excellent volumetric QS and surprisingly low magnetic shear. Given B, it may be possible to deduce an upper bound on the maximum quasisymmetric toroidal volume which depends only on the properties of B. This has been verified for the Landreman-Paul precise quasiaxisymmetric (QA) stellarator configuration. In the neighborhood of the outermost surface, we show that B approaches the form of the 1-soliton reflectionless potential (I. Gjaja and A. Bhattacharjee, Phys. Rev. Lett. 68, 2413 (1992)). The connection length diverges, indicating the possible presence of an X-point or cusp that could potentially be used as a basis for a divertor. We present a non-perturbative approach based on ensuring single-valuedness of B, which directly leads to its Painleve property and the KdV and Gardner's equations. Finally, we use an approach based on machine learning, trained on a large dataset of numerically optimized quasisymmetric stellarators. We robustly recover the KdV and Gardner's equations from the data.