Almost Periodicity in Time of Solutions of the KdV Equation

TL;DR

This paper proves the existence, uniqueness, and almost periodicity in time of solutions to the KdV equation with a broad class of almost periodic initial data, confirming a conjecture for small analytic quasiperiodic cases.

Contribution

It establishes the almost periodicity in time of KdV solutions for initial data with specific spectral properties, confirming Deift's conjecture for a wide class.

Findings

Solutions are almost periodic in time for specified initial data.

The results apply to small analytic quasiperiodic initial data with Diophantine frequencies.

Confirms Deift's conjecture for this class of initial data.

Abstract

We study the Cauchy problem for the KdV equation $\partial_t u - 6 u \partial_x u + \partial_x^3 u = 0$ with almost periodic initial data $u(x,0)=V(x)$. We consider initial data $V$, for which the associated Schr\"odinger 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.