Two-Dimensional and Three-Dimensional BGK Waves

Generalization of the Bernstein-Greene-Kruskal solution for a one-dimensional collisionless plasma to two and three-dimensional cases is demonstrated. For the case when the potential has the highest symmetry and has a single maximum at the center of the symmetry, explicit expressions for the trapped-particle distribution are obtained in terms of the potential and the other particle distribution functions for the untrapped particles. In particular, it is shown that when the distribution functions depend only on the particle energy, in the two-dimensional case the trapped-particle distribution can be analytic as a function of energy, but is never analytic in the three-dimensional case.