Variational Principle and Stability of Nonmonotonic Vlasov-Poisson Equilibria

Abstract

The stability of nonmonotonic equilibria of the Vlasov-Poisson equation is assessed by using nonlinear constants of motion. The constants of motion make up the free energy of the system, which upon variation yields nonmonotonic equilibria. Such equilibria have not previously been obtainable from a variation principle, but here this is accomplished by the inclusion of a passively advected tracer field. Definiteness of the second variation of the free energy gives a sufficient condition for stability in agreement with Gardner's theorem [5]. Previously, we have argued that indefiniteness implies either spectral instability or negative energy modes, which are generically unstable when one adds dissipation or nonlinearity [6]. Such is the case for the nonmonotonic equilibria considered. © 1987, Walter de Gruyter. All rights reserved.