Effective stability and sharp logarithmic time scales for Hamiltonian PDEs on the circle

Abstract

We prove effective stability and matching logarithmic upper bounds for small solutions of Hamiltonian PDEs on the one-dimensional torus. An analytic coefficient-majorant calculus and an explicit normal-form iteration yield stability times with logarithms of order $L^{1+1/p}/(\log L)^{1/p}$ for Gevrey weights and $L^{1+(q-1)/(pq+1)}$ for logarithmic ultra-differentiable weights, where $L=\log(1/\varepsilon)$ and $p\geq1$ measures the order dependence of the small divisors. Even frequencies are organized into the exact blocks $\{n,-n\}$; elementary frequency gaps and constant weights on these blocks close the energy estimate. We verify the hypotheses for convolution NLS, fractional NLS with every fixed noninteger exponent $η>1/2$, and a beam equation with a metric parameter and a variational gradient nonlinearity. Two fixed analytic nonlocal Hamiltonian PDEs admit finite-support solutions whose first prescribed norm amplification occurs on the same logarithmic scales along sequences $\varepsilon_n\to0$. The common lower theorem applies to every sufficiently small datum of each admissible equation. Its combination with these examples establishes sharpness of both regularity-dependent scales under the same analytic and arithmetic hypotheses.