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

where $$j\ge 2$$, $$k\ge 1$$ are integers. For a class of analytic initial data with a fixed radius of analyticity $$\sigma _0$$, we show that the uniform radius of spatial analyticity $$\sigma (t)$$ of solutions at time $$t$$ cannot decay faster than $$\frac{1}{\sqrt t}$$ as $$t\rightarrow \infty$$. 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 ($$j=2$$, $$k=1$$), where they obtained a decay rate of order $$ t^{-4 +}$$. Our proof relies on an approximate conservation law in a modified Gevrey spaces, local smoothing, and maximal function estimates.