Non-Analyticity in Time of Solutions to the KdV Equation

Abstract

It is proved that formal power series solutions to the initial value problem ${\partial_tu = \partial_x^3u+\partial_x(u^2)}$, ${u(0,x)=\varphi(x)}$, with analytic data $\varphi$ belong to the Gevrey class ${G^2}$ in time. However, if ${\varphi(x)=1/(1+x^2)}$, 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.