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

TL;DR

This paper develops a comprehensive framework for analyzing linear Hamiltonian PDEs, establishing exponential trichotomy, instability criteria, and spectral properties, applicable to various fluid and wave models.

Contribution

It introduces a structural decomposition of the phase space, proves exponential trichotomy, and refines instability index theorems for Hamiltonian PDEs, extending previous results to unbounded J operators.

Findings

Established exponential trichotomy for $e^{tJL}$ with algebraic growth in center subspace.

Derived a refined instability index theorem relating $n^-(L)$ to eigenvalue multiplicities.

Applied theory to stability analysis of dispersive wave models and fluid equations.

Abstract

Consider a general linear Hamiltonian system $\partial_{t}u=JLu$ in a Hilbert space $X$. We assume that$\ L: X \to X^{*}$ induces a bounded and symmetric bi-linear form $\left\langle L\cdot,\cdot\right\rangle $ on $X$, which has only finitely many negative dimensions $n^{-}(L)$. There is no restriction on the anti-self-dual operator $J: X^{*} \supset D(J) \to X$. 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^{-}\left( L\right) $ 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\"{o}dinger equations with nonzero condition at infinity, where our general theory applies to yield stability or instability of some coherent states.