Kinetic Flux Rope Solutions with both Electron and Ion Effects

Exact nonlinear solutions of the Vlasov-Poisson- Amp\`{e}re system of equations, known as two-dimensional Bernstein-Greene-Kruskal (BGK) modes, and having magnetic field structures in the form of small-scale kinetic flux ropes were found previously under the assumption of a uniform ion density [Ng, Phys. Plasmas {\bf 27}, 022301 (2020)]. In this work, we generalize the theory by including distribution functions for both electrons and ions. New calculations using both electrons and ions distributions with finite electron/ion temperature ratios show that solutions generally exist, including realistic temperature ratios commonly observed in space plasmas. We construct solutions with kinetic effects coming from electrons, or ions, or both, with either positive or negative electric potential, demonstrating a large range of possibilities for kinetic flux rope solutions.