Deift's almost-periodic dynamics conjecture for the KdV equation

About 5 years old · traced to 2021

Let $u(t,x)$ solve the KdV equation

\partial_tu+\partial_x^3u+u\partial_xu=0.

Assume that the initial data $u(0,x)$ is almost periodic in $x$ . Deift's conjecture. The solution $u(t,x)$ is almost periodic in time $t$ and retains the same spatial almost periodicity as the initial data for all times. This conjecture concerns the long-time dynamics of KdV solutions with nondecaying almost-periodic initial data; the source presents it as a motivation for studying analogous existence, uniqueness, and asymptotic questions for nonlinear Schrödinger equations. Its resolution is not established in the supplied text.

References

Primary source

David Damanik, Yong Li and Fei Xu, “Existence, Uniqueness and Asymptotic Dynamics of Nonlinear Schrödinger Equations With Quasi-Periodic Initial Data: I. The Standard NLS”, arXiv:2405.19583 (2024).

Additional references

2 papers in this index state this conjecture (2021–2024). The statement above is taken from the most recent of them; the others are arXiv:2111.09345.

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2022 paper claims a counterexample showing the conjecture is false in general, although it remains true for several important restricted classes.

Deift proposed that KdV solutions from spatially almost-periodic data remain almost periodic in time and preserve spatial almost-periodicity; the source dates the conjecture to 2008.

Known results

The periodic case was proved by McKean and Trubowitz in the 1970s.
Almost-periodicity in time, existence, and uniqueness were proved for data whose associated Schrödinger operator has sufficiently thick absolutely continuous spectrum (2015).
The same work covers sufficiently small analytic quasiperiodic data with Diophantine frequencies (2015).
Related positive results hold for selected reflectionless and spectral classes.

September 2022 claimed counterexample

The paper Bounded solutions of KdV: uniqueness and the loss of almost periodicity claims a bounded solution with almost-periodic initial data whose spatial profile loses almost-periodicity at a later time. It also claims uniqueness in its class, so the failure is not attributed to choosing an abnormal solution. This would disprove the unrestricted conjecture, but the supplied sources provide no independent verification, referee assessment, or correction.

Current status (as of August 2026): The unrestricted conjecture is claimed false by a 2022 counterexample, unverified in the supplied record; restricted positive cases are established.

Sources

No solutions have been posted yet.