Nonlinear stability and instability of relativistic Vlasov‐Maxwell systems

We prove the nonlinear stability or instability of certain periodic equilibria of the 1½D relativistic Vlasov‐Maxwell system. In particular, for a purely magnetic equilibrium with vanishing electric field, we prove its nonlinear stability under a sharp criterion by extending the usual Casimir‐energy method in several new ways. For a general electromagnetic equilibrium we prove that nonlinear instability follows from linear instability. The nonlinear instability is macroscopic, involving only the L¹‐norms of the electromagnetic fields. © 2006 Wiley Periodicals, Inc.