The stability and instabilities of stationary solutions to the nonlinear Schroedinger equation and the sine-Gordon equation

Abstract

I present an analysis of the stability spectrum of all stationary elliptic-type solutions to the focusing Nonlinear Schroedinger equation and 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. 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.