Hamiltonian spectral flows, the Maslov index, and the stability of standing waves in the nonlinear Schr\"{o}dinger equation

TL;DR

This paper uses the Maslov index to analyze the spectrum and stability of standing waves in the nonlinear Schrödinger equation on a compact interval, providing new stability criteria and spectral bounds.

Contribution

It introduces a novel application of the Maslov index to spectral stability analysis of nonlinear Schrödinger standing waves on finite domains, including formulas for eigenvalue curve concavity.

Findings

Derived new stability results analogous to Jones--Grillakis and Vakhitov--Kolokolov criteria.

Established a lower bound on positive real eigenvalues incorporating non-regular crossings.

Connected the Maslov index analysis with existing constrained eigenvalue count methods.

Abstract

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 non-regular 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 non-regular crossing in terms of the corresponding Jordan chains. This, along with homotopy techniques, enables the computation of the Maslov index at such a crossing. We apply our theory to study the spectral (in)stability of standing waves in the nonlinear Schr\"odinger equation on a compact spatial interval. We derive new 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 upon passing from the real line to the compact interval is the loss of translational invariance, in which case the zero eigenvalue of the linearised operator is 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.