HAMILTONIAN SPECTRAL FLOWS, THE MASLOV INDEX, AND THE STABILITY OF STANDING WAVES IN THE NONLINEAR SCHRODINGER EQUATION

We use the Maslov index to study the spectrum of a class of linear Hamiltonian differential operators. We provide a lower bound on the number of positive real eigenvalues, which includes a contribution to the Maslov index from a nonregular crossing. A close study of the eigenvalue curves, which represent the evolution of the eigenvalues as the domain is shrunk or expanded, yields formulas for their concavity at the nonregular crossing in terms of the corresponding Jordan chains. This enables the computation of the Maslov index at such a crossing via a homotopy ar-gument. We apply our theory to study the spectral (in)stability of standing waves in the nonlinear Schrodinger equation on a compact interval. We derive stability results in the spirit of the Jones-Grillakis instability theorem and the Vakhitov-Kolokolov criterion, both originally formulated on the real line. A fundamental difference on passing from the real line to the compact interval is the loss of translational invariance, in which case the zero eigenvalue of the linearized operator is (typically) geometrically simple. Consequently, the stability results differ depending on the boundary conditions satisfied by the wave. We compare our lower bound to existing results involving constrained eigenvalue counts, finding a direct relationship between the correction factors found therein and the objects of our analysis, including the second-order Maslov crossing form.