Lie Group Analysis of Plasma-Fluid Equations

Lie group methods for nonlinear partial differential equations are implemented to study, analytically, a subset of the full solution space of a family of plasma-fluid models. The solutions obtained by this method are known as group invariant solutions. The basic set of equations considered comprise the three-field fluid model due to Hazeltine (HTFM), which was obtained to describe nonlinear large aspect ratio tokamak physics. This model contains as particular limits the physics of the Charney-Hasegawa -Mima equation (CHM) and reduced magnetohydrodynamics (RMHD), which are two other models known to describe some features of nonlinear behavior of tokamak plasmas. Lie's method requires a large number of systematic calculations to determine the Lie point symmetries of the system of differential equations. These symmetries form a Lie group and describe the geometrical invariance of the equations. The Lie symmetries have been calculated for the systems mentioned above by using a symbolic manipulation program. A detailed analysis of the physical meaning of these symmetries is given. Using the Lie algebraic properties of the generators of the symmetries, a reduction of the number of independent variables for the full nonlinear systems of equations is calculated, which in turn yields simplified equations that sometimes can be solved analytically. A discussion of some of the reductions and solutions generated by this technique is presented. The results show the feasibility of using Lie methods to obtain analytical results for complicated nonlinear systems of partial differential equations that describe physically interesting situations.