Persistence of maximal Sobolev tori

Determine whether maximal, rather than sparse or non-maximal, invariant tori with Sobolev-regularity action decay persist for Hamiltonian partial differential equations.

Background

The paper distinguishes maximal tori, which retain an independent angular variable at every Fourier mode, from sparse or non-maximal tori constructed using selected tangential modes. It notes that existing Sobolev constructions cited in the paper produce non-maximal tori, while a related mechanical result proves persistence of maximal tori only for a different system of infinitely many weakly interacting rotators.

The unresolved issue is therefore whether an analogous persistence theory can be established for maximal Sobolev tori in the Hamiltonian PDE setting. The paper’s nonpersistence theorem for a polynomial-action tube provides an obstruction in one quintic NLS setting but does not resolve this broader maximal-torus question.