The Deift Conjecture: A Program to Construct a Counterexample

TL;DR

This paper proposes a program to construct a counterexample to the Deift conjecture, aiming to find an almost periodic function whose evolution under KdV is not almost periodic, using a dichotomy related to reflectionless potentials.

Contribution

It introduces a novel approach based on a dichotomy from the Kotani problem to construct counterexamples to the Deift conjecture.

Findings

A framework for constructing counterexamples to the Deift conjecture.

Identification of an analytic condition distinguishing reflectionless potentials.

Potential to challenge existing assumptions about almost periodic solutions in KdV.

Abstract

We describe a program to construct a counterexample to the Deift conjecture, that is, an almost periodic function whose evolution under the KdV equation is not almost periodic in time. The approach is based on a dichotomy found by Volberg and Yuditskii in their solution of the Kotani problem, which states that there exists an analytic condition that distinguishes between almost periodic and non-almost periodic reflectionless potentials with resolvent set given by a Widom domain.