  Electrostatic structures have been observed in many regions of space plasmas,
including the solar wind, the magnetosphere, the auroral acceleration region,
and in association with shocks, turbulence, and magnetic reconnection. Due to
potentially large amplitude of electric fields within these structures, their
effects on particle heating, scattering, or acceleration can be important. One
possible theoretical description of some of these structures is the concept of
Bernstein-Greene-Kruskal (BGK) modes, which are exact nonlinear solutions of
the Vlasov-Poisson system of equations in collisionless kinetic theory. BGK
modes have been studied extensively for many decades, predominately in one
dimension (1D), although there have been observations showing that some of
these structures have clear 3D features. While there have been approximate
solutions of higher dimensional BGK modes, an exact 3D BGK mode solution in a
finite magnetic field has not been found yet. Recently we have constructed
exact solutions of 2D BGK modes in a magnetized plasma with finite magnetic
field strength in order to gain insights of the ultimate 3D theory [Ng,
Bhattacharjee, and Skiff, Phys. Plasmas 13, 055903 (2006)]. Based on the
analytic form of these solutions, as well as Particle-in-Cell (PIC)
simulations, we will present numerical studies of their stability for different
levels of background magnetic field strength.
