Normal form smoothing and spatial analyticity for the dispersion-generalized periodic KdV equation

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, cT\, 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.