PhD advisor: Michael Sigal
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In this thesis we study the ZHK Chern-Simons equations which occur in the study of the fractional quantum hall effect of condensed matter physics. After stating basic properties of these equations, we first prove the existence of Abrikosov lattice solutions of them. Among these solutions, we find the physically interesting one whose lattice shape minimizes the average energy per lattice cell. In addition to the Abrikosov lattice solutions, we find solutions of the ZHK Chern-Simons equations on Riemann surfaces of higher genus $g$, by utilizing similar results for the Ginzburg-Landau equations.
Finally, we study the orbital stability of the Abrikosov lattice solutions under perturbations which preserve the lattice.