A reformulated Krein matrix for star-even polynomial operators with applications

TL;DR

This paper reformulates the Krein matrix to analyze spectral problems of star-even polynomial operators, extending its applicability to second-order Hamiltonian PDEs and studying bifurcations and eigenvalues in physical models.

Contribution

It introduces a generalized Krein matrix for operators with nontrivial kernels and extends its use to quadratic operators in Hamiltonian PDEs, enabling new spectral analysis methods.

Findings

Reformulated Krein matrix for operators with nontrivial kernel

Extended Krein matrix to quadratic star-even operators

Applied to spectral analysis of Hamiltonian PDE models

Abstract

In its original formulation the Krein matrix was used to locate the spectrum of first-order star-even polynomial operators where both operator coefficients are nonsingular. Such operators naturally arise when considering first-order-in-time Hamiltonian PDEs. Herein the matrix is reformulated to allow for operator coefficients with nontrivial kernel. Moreover, it is extended to allow for the study of the spectral problem associated with quadratic star-even operators, which arise when considering the spectral problem associated with second-order-in-time Hamiltonian PDEs. In conjunction with the Hamiltonian-Krein index (HKI) the Krein matrix is used to study two problems: conditions leading to Hamiltonian-Hopf bifurcations for small spatially periodic waves, and the location and Krein signature of small eigenvalues associated with, e.g., $n$-pulse problems. For the first case we consider in detail a first-order-in-time fifth-order KdV-like equation. In the latter case we use a combination of Lin's method, the HKI, and the Krein matrix to study the spectrum associated with $n$-pulses for a second-order-in-time Hamiltonian system which is used to model the dynamics of a suspension bridge.