Normal form smoothing and spatial analyticity for the dispersion-generalized periodic KdV equation
Abstract
This paper investigates the persistence of spatial analyticity for solutions to the dispersion-generalized periodic Korteweg-de Vries equation ∂tu−∂x|Dx|αu+u∂xu=0, α≥2. For real-valued initial data u0∈ Gσ0,s(𝕋), with σ0>0 and s≥−α/4, we prove that the corresponding global solution satisfies u∈ C([−T,T];Gσ(T),s(𝕋)), σ(T)=min{σ0,c/T}, for every T>0. As a consequence, our result improves the lower bound for the radius of spatial analyticity established by Himonas, Kalisch, and Selberg (2017) for 2≤α<4, including the classical periodic KdV equation, and establishes a unified T−1 lower bound for the analyticity radius for every α≥2. The main novelties are twofold. First, we introduce a normal form reduction at the level of the Gevrey energy, which reveals an additional cancellation in the cubic Gevrey commutator and separates it into a non-accumulating boundary correction and a quartic remainder. Second, we establish a global-in-time periodic Lt,x4 estimate for the non-polynomial phase k|k|α, valid for every real α≥2. Together, these ingredients yield a refined almost conservation law and the T−1 lower bound for the radius of spatial analyticity.
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BibTeX
@misc{ahn2026normal,
title = {{Normal form smoothing and spatial analyticity for the dispersion-generalized periodic KdV equation}},
author = {Jaeseop Ahn and Chulkwang Kwak},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06313},
eprint = {2611.06313},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06313},
}
RIS
TY - UNPB
TI - Normal form smoothing and spatial analyticity for the dispersion-generalized periodic KdV equation
AU - Jaeseop Ahn
AU - Chulkwang Kwak
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06313
UR - https://arxiv.org/abs/2611.06313
AB - This paper investigates the persistence of spatial analyticity for solutions to the dispersion-generalized periodic Korteweg-de Vries equation ∂_tu−∂_x|D_x|^αu+u∂_xu=0, α≥2. For real-valued initial data u_0∈ G^σ_0,s(𝕋), with σ_0>0 and s≥−α/4, we prove that the corresponding global solution satisfies u∈ C([−T,T];G^σ(T),s(𝕋)), σ(T)=min{σ_0,c/T}, for every T>0. As a consequence, our result improves the lower bound for the radius of spatial analyticity established by Himonas, Kalisch, and Selberg (2017) for 2≤α<4, including the classical periodic KdV equation, and establishes a unified T^−1 lower bound for the analyticity radius for every α≥2. The main novelties are twofold. First, we introduce a normal form reduction at the level of the Gevrey energy, which reveals an additional cancellation in the cubic Gevrey commutator and separates it into a non-accumulating boundary correction and a quartic remainder. Second, we establish a global-in-time periodic L_t,x^4 estimate for the non-polynomial phase k|k|^α, valid for every real α≥2. Together, these ingredients yield a refined almost conservation law and the T^−1 lower bound for the radius of spatial analyticity.
ER -