Almost periodicity in time of solutions of the KdV equation

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