Fate of polynomially decaying infinite-dimensional invariant tori

Determine the fate, in one-dimensional Hamiltonian partial differential equations, of infinite-dimensional invariant tori whose positive actions decay algebraically at the scale j^{-M}.

Background

The paper identifies the persistence or destruction of infinite-dimensional invariant tori with polynomially decaying actions as a longstanding problem in Hamiltonian PDEs. The cited formulation concerns tori with positive actions of order j^{-M}, corresponding to finite Sobolev regularity rather than the faster-than-polynomial decay commonly treated in existing KAM constructions.

The paper’s main theorem establishes a local nonpersistence result for a specific periodic quintic nonlinear Schrödinger equation and a specially constructed bounded Fourier multiplier. It does not settle the broader question of the fate of all such polynomially decaying infinite-dimensional tori across Hamiltonian PDEs.