The orbital stability of elliptic solutions of the Focusing Nonlinear Schr\"{o}dinger Equation

We examine the stability of the elliptic solutions of the focusing nonlinear Schr"odinger equation (NLS) with respect to subharmonic perturbations. Using the integrability of NLS, we discuss the spectral stability of the elliptic solutions, establishing that solutions of smaller amplitude are stable with respect to larger classes of perturbations. We show that spectrally stable solutions are orbitally stable by constructing a Lyapunov functional using higher-order conserved quantities of NLS.