Solve the integro-differential edge-mode equation with time-dependent magnetic field

Solve the integro-differential boundary equation (Equation (e96)) for the edge-field F(t,R,θ) of a circular quantum Hall droplet subject to a time-dependent magnetic field, including the Dirichlet-to-Neumann kernel M(θ′,θ) and time-dependent coefficients, to obtain explicit edge-mode solutions or a characterization of their dynamics.

Background

From the constructed edge action, the authors derive a boundary equation for the edge deformation field F(t,R,θ) that includes chiral-boson-like local terms and nonlocal contributions via the Dirichlet-to-Neumann operator for the Laplacian on the disc. The coefficients depend on time through quantities determined by the Ermakov equations and the droplet radius R(t).

They state that solving this integro-differential equation is highly involved and remains undone. Obtaining solutions would elucidate edge dispersions and dynamics in quantum Hall systems driven by time-dependent magnetic fields.