Almost periodicity in time of solutions of the KdV equation

Abstract

We study the Cauchy problem for the KdV equation ∂tu−6u∂xu+∂x3u=0 with almost periodic initial data u(x,0)=V(x). We consider initial data V, for which the associated Schrödinger operator is absolutely continuous and has a spectrum that is not too thin in a sense we specify, and we show the existence, uniqueness, and almost periodicity in time of solutions. This establishes a conjecture of Deift for this class of initial data. The result is shown to apply to all small analytic quasiperiodic initial data with Diophantine frequency vector.