Deift conjecture for almost periodic KdV dynamics

Determine whether almost periodic initial data for the Korteweg–de Vries equation generate solutions that are almost periodic in time or almost periodic in both space and time, subject to the relevant spectral hypotheses.

Background

The paper places its study of the defocusing modified Korteweg–de Vries equation within the broader Deift-conjecture program. The motivating issue is whether integrable dispersive equations preserve almost-periodic structure under their time evolution when the initial data are non-decaying and non-periodic.

The authors explain that the conjecture was originally formulated in the Korteweg–de Vries setting and has been established for certain classes of almost-periodic data, while also noting that it fails in full generality because solutions may lose spatial continuity at later times. The present paper proves an analogous result for defocusing mKdV under specified reflectionless and Craig-type spectral conditions, rather than resolving the unrestricted problem.

References

Integrability motivates more general questions: whether almost periodic initial data give solutions which are almost periodic in time, or spacetime almost periodic. This was conjectured by P. Lax in the setting of the KdV equation, supported by numerical evidence of M. Hyman that can be found in the appendix of . This problem was popularized by P. Deift and is now known as the Deift conjecture; it is particularly well-studied in the KdV setting.

— Almost periodic solutions of the defocusing mKdV equation  (2608.25283 - Li et al., 26 Aug 2026) in Section 1, Introduction and Main Results