Bounded solutions of KdV: uniqueness and the loss of almost periodicity

TL;DR

This paper proves the uniqueness of bounded solutions to the KdV equation without decay or regularity assumptions and demonstrates that almost periodic initial data can evolve into solutions that are no longer almost periodic.

Contribution

It establishes the uniqueness of bounded solutions to KdV and provides a counterexample showing almost periodicity is not preserved under KdV evolution.

Findings

Uniqueness of bounded KdV solutions without additional assumptions

Existence of almost periodic initial data evolving into non-almost periodic solutions

Counterexample for preservation of almost periodicity in KdV evolution

Abstract

We address two pressing questions in the theory of the Korteweg--de Vries (KdV) equation. First, we show the uniqueness of solutions to KdV that are merely bounded, without any further decay, regularity, periodicity, or almost periodicity assumptions. The second question, emphasized by Deift, regards whether almost periodic initial data leads to almost periodic solutions to KdV. Building on the new observation that this is false for the Airy equation, we construct an example of almost periodic initial data whose KdV evolution remains bounded, but fails to be almost periodic at a later time. Our uniqueness result ensures that the solution constructed is the unique development of this initial data.