Variational Principles for Nonstationary States of the Vlasov-Poisson System

Variational principles are used for studying nonstationary steady states of the Vlasov-Poisson system [1]. Our motivation is to understand the creation and long time stability properties of nonstationary steady states, motivated by recently discovered Kinetic Electrostatic Electron Nonlinear (KEEN) waves [2]. Unlike BGK modes, which are stationary, we seek steady states where the particles interact with a multiple-harmonic self-consistent nonstationary field via a sequence of trapping, detrapping, and retrapping transitions. While KEEN waves were discovered [2] by driving a Maxwellian plasma with a ponderomotive force caused by the beating of two laser beams [2,3], the variational formulations allow the investigation of other mechanisms that might lead to such states, besides a single mode drive of finite temporal duration or a Maxwellian initial state. Criteria for initial states and external fields that lead to nonstationary states can be obtained with variational procedures and reduced descriptions derived from it. [1] P.J.\ Morrison, Phys.\ Plasmas 12, 058102 (2005); H.\ Ye and P.J.\ Morrison, Phys.\ Fluids 4B, 771 (1992). [2] B.\ Afeyan et al., Proc. IFSA (Inertial Fusion Sciences and Applications 2003, Monterey, CA), 213, B.\ Hammel, et al., eds., Am.\ Nuc.\ Soc.\ (2004). [3] M.\ Mardirian et al., Savchenko et al., and Kline et al., this conference. Work Supported by the DOE SSAA Grant DE-FG03-03NA00059.