Identify additional global rugged invariants for GrNLS

Identify generic global rugged invariants, beyond mass M_-1 and momentum M_0, that are preserved by the Galerkin-regularized nonlinear Schrödinger (GrNLS) system and can be used to define critical sets of constrained functionals for realizing multi-frequency quasi-periodic solutions.

Background

The authors work with the Hamiltonian formulation of GrNLS, noting that only the Hamiltonian H, mass M_-1, and momentum M_0 are conserved under truncation. They construct certain quasi-periodic solutions using these rugged invariants and also introduce torus-specific invariants to build higher-dimensional tori.

They explicitly state that no other generic global rugged invariants are known, which limits the ability to define critical sets for more general multi-frequency solutions without resorting to torus-specific constructions. Discovering such invariants would broaden analytical tools for GrNLS and potentially clarify the structure of longulent attractors.