Generating magnetic field strength with exact quasisymmetry using soliton potentials

Quasisymmetry (QS) is a hidden symmetry of the magnetic field strength, B, that confines charged particles effectively in a nonsymmetric toroidal plasma equilibrium. Recent numerical breakthroughs have shown that excellent QS can be realized in a toroidal volume. Here, we show that the hidden symmetry of QS has a deep connection to the underlying symmetry that makes solitons possible. In particular, we demonstrate that a large class of quasisymmetric B is described by a periodic finite-gap soliton potential of the well-known Korteweg-de Vries (KdV) and Gardner equations, drastically reducing the number of independent B parameters on a magnetic flux surface to a few. We show this explicitly for a quasisymmetric vacuum field in slab geometry and extend the results to general geometry, leveraging the connection between Lewis-Ermakov and the parallel adiabatic invariants. We highlight the importance of magnetic shear in QS. Furthermore, we deduce an upper bound on the maximum toroidal volume that can be quasisymmetric. B approaches the form of the 1-soliton reflectionless potential in the neighborhood of the outermost surface. The hidden low dimensionality and the analytical insight into the role of magnetic shear could make stellarator optimization schemes significantly more efficient.