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.

Keywords

  1. Krein matrix
  2. multipulses
  3. Hamiltonian PDEs

MSC codes

  1. 35P30
  2. 47A55
  3. 47A56
  4. 70H14

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Information & Authors

Information

Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 4705 - 4750
ISSN (online): 1095-7154

History

Submitted: 4 February 2019
Accepted: 29 June 2020
Published online: 30 September 2020

Keywords

  1. Krein matrix
  2. multipulses
  3. Hamiltonian PDEs

MSC codes

  1. 35P30
  2. 47A55
  3. 47A56
  4. 70H14

Authors

Affiliations

Funding Information

National Science Foundation https://doi.org/10.13039/100000001 : DMS-1108783, DMS-1148284, DMS-1714429

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