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