SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4705-4750, January 2020.
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.

中文翻译：

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星型多项式算子的重新构造的Kerin矩阵及其应用

SIAM数学分析杂志，第52卷，第5期，第4705-4750页，2020年1月。
在其原始公式中，Kerin矩阵用于定位两个算子系数都不是奇数的一阶星形偶数多项式算子的谱。在考虑一阶时间哈密顿量PDE时，自然会产生此类算子。这里，矩阵被重新构造以允许具有非平凡核的算子系数。此外，它被扩展以允许研究与二次星形偶数算符相关的光谱问题，这在考虑与二阶时间哈密顿量PDE相关的光谱问题时会出现。与汉密尔顿-克赖恩指数（HKI）结合使用Kerin矩阵来研究两个问题：导致小空间周期波的汉密尔顿-霍夫夫分叉的条件，以及与例如相关的小特征值的位置和Kerin签名 $ n $-脉冲问题。对于第一种情况，我们将详细考虑一阶实时五阶KdV类方程。在后一种情况下，我们使用Lin方法，HKI和Kerin矩阵的组合来研究二阶时间哈密顿系统的与$ n $脉冲相关的频谱，该系统用于模拟a的动力学。吊桥。