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We use the theory of functions of noncommuting operators (noncommutative analysis) to solve an asymptotic problem for a partial differential equation and show how, starting from general constructions and operator formulas that seem to be rather abstract from the viewpoint of differential equations, one can end up with very specific, easy-to-evaluate expressions for the solution, useful, e.g., in the tsunami wave problem., Comment: 29 pages, 4 figures. Accepted for publication in The Vladimir Rabinovich Anniversary Volume of "Operator Theory: Advances and Applications" (Birkh\"auser)

Fluctuations of the order parameters of the Gardner model for any $\alpha, Comment: 17 pages, Latex

Analytical solutions of the Grad–Shafranov equilibrium equation in simply connected plasma configurations, comprised of toroidal magnetic surfaces and open surfaces connected to electrodes, are reviewed and generalised. The Grad–Shafranov equation is linearised introducing assumptions on plasma current and pressure, which preserve regularity of solutions on the symmetry axis, as required for a simply connected geometry. Particular solutions are found by separation of variables both in cylindrical coordinates and in spherical ones. Equilibria that model local or global features of PROTO-SPHERA plasmas are constructed by combining a few particular solutions.

A simple steady-state model for a 3-species mixture (ions, electrons, and neutrals) in a screw-pinch plasma configuration is developed. The model is applied to the central plasma column of the PROTO-SPHERA experiment. Degree of ionization, azimuthal current density, and azimuthal ion velocity are calculated. Full ionization is found at plasma temperatures above 1.5 eV, with neutrals confined in an outer shell where radial plasma flow develops and drives both azimuthal current and azimuthal flow.

The m = 1 resistive mode for a tokamak plasma with large aspect ratio is considered: the dynamic equations in a resistive layer are solved by means of an asymptotic expansion for values of the growth rate in a suitable range. The eigenvalues characterizing the perturbation are found by means of a series expansion and it is shown that the main contribution to the expression of the eigenvalues is given by the first and the second order of this expansion. This method is different from the one used in the paper [G. Ara et al., Ann. Phys. 112, 443 (1978)], and can be applied in more general situations. © EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2016.