Periodic Korteweg-de Vries soliton potentials generate magnetic field strength with excellent quasisymmetry

Quasisymmetry (QS) is a hidden symmetry of the magnetic field strength, $B$, that confines charged particles effectively in a three-dimensional toroidal plasma equilibrium. Here, we show that QS has a deep connection to the underlying symmetry that makes solitons possible. We uncover a hidden lower dimensionality of $B$ on a magnetic flux surface, which could make stellarator optimization schemes significantly more efficient. Recent breakthroughs (M. Landreman and E. Paul, PRL 2022) have yielded configurations with excellent volumetric QS and surprisingly low magnetic shear. Our approach elucidates why the magnetic shear is low in these configurations. Given $B$, we deduce an upper bound on the maximum toroidal volume that can be quasisymmetric only from the properties of $B$ and verify it 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. The connection length diverges, indicating the possible presence of an X-point that could potentially be used as basis for a divertor. We present three independent approaches to demonstrate that quasisymmetric $B$ is described by well-known integrable systems such as the Korteweg-de Vries (KdV) and Gardner's equation. The first approach is weakly nonlinear multiscale perturbation theory. We show that the overdetermined problem of finding quasisymmetric vacuum fields admits solutions for which the rotational transform and the magnetic shear are highly constrained. Our second approach is non-perturbative and based on ensuring single-valuedness of $B$, which directly leads to its Painlevé property shared by the KdV equation. Our third machine-learning-based approach robustly recovers the KdV (and Gardner's equation) from a large dataset of numerically optimized quasisymmetric stellarators.