Analyticity of solutions of the Korteweg-de Vries equation

Abstract

We consider the analytic smoothing effect for the KdV equation. That is to say, if the initial data given at $t = 0$ decays very rapidly, the solution to the Cauchy problem becomes analytic with respect to the space variable for $t > 0$. In this paper we show this effect by using the inverse scattering method which transforms the KdV equation to a linear dispersive equation whose analytic smoothing effect is shown through the properties of the Airy function.