Applications of soliton theory to the study of 3D magnetic fields with nested flux surfaces*

The theory of solitons has been successfully applied in the past to understand various nonlinear plasma behavior. For example, the steady-state MHD equilibrium with the isodynamic constraint is described by the nonlinear Schrodinger equation (NLS) [1]. Soliton theory can shed light on how singular currents and island formation may be avoided in MHD. In this work, we show that the appearance of NLS-like soliton equations and their hierarchy of conserved quantities are ubiquitous in ideal MHD. In particular, we study three classes of MHD equilibrium: i) a circular cross-section near an arbitrary magnetic axis, ii) exact quasisymmetry near a planar flux-surface, iii) quasisymmetry close to isodynamic. In case i), we employ near-axis expansion to show that the rotational transform and its derivatives are related to the NLS invariants. In ii), we use near-surface expansion to obtain a class of quasisymmetric vacuum magnetic fields near a planar flux surface described by a reflectionless potential. Finally in iii), we study perturbations of exact isodynamic MHD equilibrium that are approximately quasisymmetric.

1. Schief, W. K. JPP. 69.6 (2003): 465-484