Petite Conjecture on temporal behavior of localized dispersive solutions

Prove the Petite Conjecture asserting that any sufficiently regular, spatially localized solution of a dispersive equation is (asymptotically) almost periodic in time.

Background

In discussing the possible long‑time behavior of localized states (including potential breathers), the author points to a conjectural structural property: temporal almost periodicity of localized, regular solutions.

Such a result would strongly constrain non‑radiative dynamics in nonlinear dispersive equations and clarify the landscape of possible coherent structures beyond solitons.