Effective stability for Hamiltonian PDEs vanishing spectral gaps

2026/07/14 by Bingqi Yu, Yong Li
#math.AP #math.DS

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Abstract

This paper studies the effective stability of nearly integrable Hamiltonian PDEs with asymptotically vanishing spectral gaps (0 < α< 1). Under a unified high-low frequency decomposition, we construct a modified block clustering partition based on Bourgain's ideas. By leveraging the vanishing of spectral gaps to suppress high-frequency resonant contributions, the overall non-resonance property is maintained under high-regularity weights. This framework is applied to space fractional and fully dispersive Whitham-Schrödinger equations, uniformly yielding explicit stability estimates in Gevrey, logarithmic ultra-differentiable, and Sobolev spaces.

Citations

• An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS
• Nonlinear fractional Schrödinger equations in one dimension
• Invariant Cantor Manifolds of Quasi-Periodic Oscillations for a Nonlinear Schrodinger Equation

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