Long-time action stability for perturbed KdV near finite-gap tori
Abstract
We prove action and spectral stability near finite-gap tori of the periodic Korteweg--de Vries equation under autonomous analytic Hamiltonian perturbations of size ε. The estimates are uniform over prescribed compact families, without small-amplitude or nonresonance assumptions, and allow nearby initial data with infinitely many open gaps. Assuming a strong solution and a uniform higher Sobolev bound on [0,T0/ε], we permit one-derivative loss in the perturbing Hamiltonian's normal gradient. For sufficiently small ε and initial normal Birkhoff size r0, the tangential action variations and weighted sum of normal actions are bounded by Cδ(r02+ε2δ), while the spectral-gap variation in ℓ2 is bounded by Cδ(r0+εδ), for every 0<δ<1/4. A key ingredient is coercivity of the negative Hessian of the nonlinear KdV Hamiltonian on the full ℓ2 action space at every finite-gap action. We combine this with resonance analysis and corrected action observables evaluated along the solution. The hypotheses are verified for several local and nonlocal Hamiltonian perturbations.
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BibTeX
@misc{huang2026longtime,
title = {{Long-time action stability for perturbed KdV near finite-gap tori}},
author = {Guan Huang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34601},
eprint = {2610.34601},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34601},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Long-time action stability for perturbed KdV near finite-gap tori
AU - Guan Huang
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34601
UR - https://arxiv.org/abs/2610.34601
AB - We prove action and spectral stability near finite-gap tori of the periodic Korteweg--de Vries equation under autonomous analytic Hamiltonian perturbations of size ε. The estimates are uniform over prescribed compact families, without small-amplitude or nonresonance assumptions, and allow nearby initial data with infinitely many open gaps. Assuming a strong solution and a uniform higher Sobolev bound on [0,T_0/ε], we permit one-derivative loss in the perturbing Hamiltonian's normal gradient. For sufficiently small ε and initial normal Birkhoff size r_0, the tangential action variations and weighted sum of normal actions are bounded by C_δ(r_0^2+ε^2δ), while the spectral-gap variation in ℓ^2 is bounded by C_δ(r_0+ε^δ), for every 0<δ<1/4. A key ingredient is coercivity of the negative Hessian of the nonlinear KdV Hamiltonian on the full ℓ^2 action space at every finite-gap action. We combine this with resonance analysis and corrected action observables evaluated along the solution. The hypotheses are verified for several local and nonlocal Hamiltonian perturbations.
ER -