The Stability Spectrum for Elliptic Solutions to the Focusing NLS Equation

We present an analysis of the stability spectrum of all stationary elliptic-type solutions to the focusing Nonlinear Schr"{o}dinger equation (NLS). An analytical expression for the spectrum is given. From this expression, various quantitative and qualitative results about the spectrum are derived. Specifically, the solution parameter space is shown to be split into four regions of distinct qualitative behavior of the spectrum. Additional results on the stability of solutions with respect to perturbations of an integer multiple of the period are given, as well as a procedure for approximating the greatest real part of the spectrum.