Faculty of Engineering, University of Tokyo, Tokyo 113, Japan
(Received May 7, 1993; Accepted June 4, 1993)
Keywords: Plasma, Self-Organization, Eigenvalue Problem, Hamiltonian System, Cohomology, Magnetic Field
Abstract. Mathematical backgrounds for the theory of structures in a plasma are reviewed. The theory of differential geometry and its relevance to dynamical systems are invoked to study preferential structures of a plasma. A plasma wave such as the Langmuir wave produces a soliton, whose structure is determined by a semilinear eigenvalue problem. In a magnetized plasma the magnetic field profile in an equilibrium state also has an interesting structure. Two special cases are discussed. One is the case where the plasma has a symmetry. Then the field line equation is integrable, and the field structure is characterized by a second order semilinear eigenvalue problem, which resembles the eigenvalue problem of the soliton. The other is the force-free field, which is characterized by an eigenfunction of the curl operator. The cohomology class plays an essential role in the spectral theory of the curl operator.