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Primarily motivated by the stability analysis of nonlinear waves insecond-order in time Hamiltonian systems, in this paper we develop aninstability index theory for quadratic operator pencils acting on a Hilbertspace. In an extension of the known theory for linear pencils, explicitconnections are made between the number of eigenvalues of a given quadraticoperator pencil with positive real parts to spectral information about theindividual operators comprising the coefficients of the spectral parameter inthe pencil. As an application, we apply the general theory developed here toyield spectral and nonlinear stability/instability results for abstractsecond-order in time wave equations. More specifically, we consider the problemof the existence and stability of spatially periodic waves for the "good"Boussinesq equation. In the analysis our instability index theory provides anexplicit, and somewhat surprising, connection between the stability of a givenperiodic traveling wave solution of the "good" Boussinesq equation and thestability of the same periodic profile, but with different wavespeed, in thenonlinear dynamics of a related generalized Korteweg-de Vries equation.

An Instability Index Theory for Quadratic Pencils and Applications