Essentially different distributions of current in the inverse problem for the Grad-Shafranov equation

The paper answers a question debated by physicists for many years. It is proved that, for almost equal gradients of the magnetic flux u at its zero-level curve ∂ω, which is the piecewise smooth boundary of a simply-connected domain ω ⋐ ℝ², the inverse problem for the Grad-Shafranov equation of plasma equilibrium in a tokamak (in the cylindrical approximation) admits essentially different profiles of distributions f _u : ω ∋ (x, y) ↦ f(u(x, y)) = u _xx (x, y) + u _yy (x, y) ⩾ 0 in the class of third-order polynomials f(u) = Σ _m=0³ a _m u ^m .