Non-Analyticity in Time of Solutions to the KdV Equation

Abstract

It is proved that formal power series solutions to the initial value problem ${\partial }_{t}u={\partial }_x^{3}u+{\partial }_x(u^2)$, $u(0,x)=\phi (x)$, with analytic data $\phi$ belong to the Gevrey class $G^2$ in time. However, if $\phi (x)=1/(1+x^2)$, the formal solution does not belong to the Gevrey class $G^s$ in time for $0\le s<2$, so it is not analytic in time. The proof is based on the estimation of a double sum of products of binomial coefficients.