Instability, Index Theorem, and Exponential Trichotomy for Linear Hamiltonian PDEs

Description

Consider a general linear Hamiltonian system ∂ t u = JLu in a Hilbert space X. We assume that L : X → X∗ induces a bounded and symmetric bi-linear form L · , on X, which has only finitely many negative dimensions n−(L). There is no restriction on the anti-self-dual operator J : X∗ ⊃ D(J) → X. We first obtain a structural decomposition of X into the direct sum of several closed subspaces so that L is blockwise diagonalized and JL is of upper triangular form, where the blocks are easier to handle. Based on this structure, we first prove the linear exponential trichotomy of e tJL . In particular, e tJL has at most algebraic growth in the finite co-dimensional center subspace. Next we prove an instability index theorem to relate n−(L) and the dimensions of generalized eigenspaces of eigenvalues of JL, some of which may be embedded in the continuous spectrum. This generalizes and refines previous results, where mostly J was assumed to have a bounded inverse. More explicit information for the indexes with pure imaginary eigenvalues are obtained as well. Moreover, when Hamiltonian perturbations are considered, we give a sharp condition for the structural instability regarding the generation of unstable spectrum from the imaginary axis. Finally, we discuss Hamiltonian PDEs including dispersive long wave models (BBM, KDV and good Boussinesq equations), 2D Euler equation for ideal fluids, and 2D nonlinear Schr¨odinger equations with nonzero conditions at infinity, where our general theory applies to yield stability or instability of some coherent states. 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 𝐿 does not have a positive lower bound on 𝑋₊ 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 𝐸_{𝑖𝜇} 8.1. Proof of Theorem 2.3: the index counting formula 8.2. Structures of subspaces 𝐸_{𝑖𝜇} of generalized eigenvectors 8.3. Subspace of generalized eigenvectors 𝐸₀ and index 𝑘₀^{≤0} 8.4. Non-degeneracy of ⟨𝐿⋅,⋅⟩ on 𝐸_{𝑖𝜇} 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 𝐿𝑢=𝜆𝑢′ 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