Three-Dimensional Bernstein-Greene-Kruskal Modes in a Finite Magnetic Field

We will present analytic forms and numerical solutions for three-dimensional (3D) Bernstein-Greene-Kruskal (BGK) modes in a magnetized plasma with a finite background uniform magnetic field. These are exact nonlinear solutions of the steady-state Vlasov equation with an electric potential localized in all three spatial dimensions that satisfies the Poisson equation self-consistently. Dynamics of both ions and electrons are included in the formulation. This new development is following our previous solutions for 3D BGK modes in an unmagnetized plasma [Ng & Bhattacharjee, Phys. Rev. Lett., 95, 245004 (2005)] and 2D BGK modes in a magnetized plasma with finite magnetic field [Ng, Phys. Plasmas, 27, 022301 (2020)]. These solutions have cylindrical symmetry with distribution functions of either ion or electron depending on the particle energy and the axial component of the canonical angular momentum. However, the functional forms of distribution functions for these solutions are very different from previous 3D BGK mode solutions that are based on the strong magnetic field assumption.