Generalized Conditional Symmetries, Related Solutions and Conservation Laws of the Grad-Shafranov Equation with Arbitrary Flow

Abstract

The generalized conditional symmetry (GCS) method is applied to the case of a generalized Grad-Shafranov equation (GGSE) with incompressible flow of arbitrary direction. We investigate the conditions which yield the GGSE that admits a special class of second-order GCSs. Three GCS generators and the associated families of invariant solutions are pointed out. Several plots of the level sets or flux surfaces of the new solutions are displayed. These results extend the recent solutions with 5 parameters recently obtained on the basis of Lie-point symmetries. They could be useful in the study of plasma equilibrium, of transport phenomena, and of magnetohydrodynamic stability. Futher, by making use of the multiplier's method, three nontrivial conservation laws that are admitted by the concerned equation and which involve arbitrary functions, are highlighted.

Diffusion of ions in an electrostatic stochastic field and a space-dependent unperturbed magnetic field

“…Important results concerning the behavior of the magnetic field were obtained (see e.g. [34] and [35]) analyzing the Grad-Shafranov equation. From here, the possibility of constructing different magnetic fields appears.…”