Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs

Consider a general linear Hamiltonian system

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Bibliographic Details
Main Authors:	Lin, Zhiwu, Zeng, Chongchun
Format:	eBook Book
Language:	English
Published:	Providence, Rhode Island American Mathematical Society 2022
Edition:	1
Series:	Memoirs of the American Mathematical Society
Subjects:	Differential equations, Linear Dynamical systems and ergodic theory > Infinite-dimensional Hamiltonian systems > Stability problems. msc Hamiltonian systems Index theorems Operator theory > General theory of linear operators > Spectrum, resolvent. msc Partial differential equations > Qualitative properties of solutions > Stability. msc Partial differential equations > Spectral theory and eigenvalue problems > General topics in linear spectral theory. msc Hamiltonian perturbations index formula exponential trichotomy Linear Hamiltonian PDEs structural decomposition stability spectral mapping
ISBN:	9781470450441, 1470450445
ISSN:	0065-9266, 1947-6221
Online Access:	Get full text
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Table of Contents:

Introduction -- Main Results -- Basic properties of Linear Hamiltonian systems -- Finite dimensional Hamiltonian systems -- Invariant subspaces -- Structural decomposition -- Exponential trichotomy -- The index theorems and the structure of <inline-formula content-type="math/mathml"> E i μ E_{i\mu } </inline-formula> -- Perturbations -- Proof of Theorem where ( H2.b ) is weakened -- Hamiltonian PDE models -- Appendix
Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Main Results -- 2.1. Set-up -- 2.2. Structural decomposition -- 2.3. Exponential Trichotomy -- 2.4. Index Theorems and spectral properties -- 2.5. Structural stability/instability -- 2.6. A theorem where does not have a positive lower bound on ₊ -- 2.7. Some Applications to PDEs -- Chapter 3. Basic properties of Linear Hamiltonian systems -- Chapter 4. Finite dimensional Hamiltonian systems -- Chapter 5. Invariant subspaces -- 5.1. Maximal non-positive invariant subspaces (Pontryagin invariant subspaces) -- 5.2. Further discussions on invariant subspaces and invariant decompositions -- Chapter 6. Structural decomposition -- Chapter 7. Exponential trichotomy -- Chapter 8. The index theorems and the structure of _{ } -- 8.1. Proof of Theorem 2.3: the index counting formula -- 8.2. Structures of subspaces _{ } of generalized eigenvectors -- 8.3. Subspace of generalized eigenvectors ₀ and index ₀^{≤0} -- 8.4. Non-degeneracy of ⟨ ⋅,⋅⟩ on _{ } and isolated purely imaginary spectral points -- Chapter 9. Perturbations -- 9.1. Persistent exponential trichotomy and stability: Theorem 2.4 and Proposition 2.9 -- 9.2. Perturbations of purely imaginary spectrum and bifurcation to unstable eigenvalues -- Chapter 10. Proof of Theorem 2.7 where (H2.b) is weakened -- Chapter 11. Hamiltonian PDE models -- 11.1. Stability of Solitary waves of Long wave models -- 11.2. Stability of periodic traveling waves -- 11.3. Modulational Instability of periodic traveling waves -- 11.4. The spectral problem = ′ -- 11.5. Stability of steady flows of 2D Euler equation -- 11.6. Stability of traveling waves of 2-dim nonlinear Schrödinger equations with nonzero conditions at infinity -- Appendix A. Appendix -- Bibliography -- Back Cover