We study the Cauchy problem for the KdV equation with almost periodic initial data . We consider initial data , 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.
"Almost periodicity in time of solutions of the KdV equation." Duke Math. J. 167 (14) 2633 - 2678, 1 October 2018. https://doi.org/10.1215/00127094-2018-0015