On the persistence of spatial analyticity for generalized KdV equation with higher order dispersion

Persistence of spatial analyticity is studied for solutions of the generalized Korteweg‐de Vries (KdV) equation with higher order dispersion ∂tu+(−1)j+1∂x2j+1u=∂xu2k+1,$$\begin{equation*} \partial _{t} u+(-1)^{j+1}\partial _{x}^{2j+1} u= \partial _x{\left(u^{2k+1} \right)}, \end{equation*}$$where j≥2$j\ge 2$, k≥1$k\ge 1$ are integers. For a class of analytic initial data with a fixed radius of analyticity σ0$\sigma _0$, we show that the uniform radius of spatial analyticity σ(t)$\sigma (t)$ of solutions at time t$t$ cannot decay faster than 1t$\frac{1}{\sqrt t}$ as t→∞$t\rightarrow \infty$. In particular, this improves a recent result due to Petronilho and Silva [Math. Nachr. 292 (2019), no. 9, 2032–2047] for the modified Kawahara equation (j=2$j=2$, k=1$k=1$), where they obtained a decay rate of order t−4+$ t^{-4 +}$. Our proof relies on an approximate conservation law in a modified Gevrey spaces, local smoothing, and maximal function estimates.