Almost periodicity in time of solutions of the KdV equation

Abstract

We study the Cauchy problem for the KdV equation ${\partial }_{t}u-6u{\partial }_{x}u+{\partial }_x^{3}u=0$ with almost periodic initial data $u(x,0)=V\left(x\right)$. 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.