Asymptotic lower bound for the radius of spatial analtyicity to solutions of KdV equation

It is shown that the uniform radius of spatial analyticity $\sigma (t)$ of solutions at time $t$ to the KdV equation cannot decay faster than $\mathrm{\mid }t{\mathrm{\mid }}^{-4\mathrm{/}3}$ as $\mathrm{\mid }t\mathrm{\mid }\to \mathrm{\infty }$ given initial data that is analytic with fixed radius ${\sigma }_0$. This improves a recent result of Selberg and Da Silva, where they proved a decay rate of $\mathrm{\mid }t{\mathrm{\mid }}^{-(4\mathrm{/}3+\epsilon )}$ for arbitrarily small positive $\epsilon$. The main ingredients in the proof are almost conservation law for the solution to the KdV equation in space of analytic functions and space-time dyadic bilinear $L^2$ estimates associated with the KdV equation.