The stability spectrum for elliptic solutions to the sine-Gordon equation

We present an analysis of the stability spectrum for all stationary periodic solutions to the sine-Gordon equation. 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 regions of distinct qualitative behavior of the spectrum, in one of which the solutions are stable. Additional results on the spectral stability of solutions with respect to perturbations of an integer multiple of the solution period are given.