Conjecture on torus-specific invariants and pseudo-integrability in GrNLS

Prove the conjecture that one can choose a set of torus-specific invariants for the Galerkin-regularized nonlinear Schrödinger (GrNLS) system whose total number equals the degrees of freedom of the truncation (i.e., the number of active Fourier modes), thereby establishing a form of pseudo-integrability that precisely specifies a longulent state.

Background

To better approximate or stabilize longulent states, the authors introduce torus-specific invariants (e.g., extensions of M_-1, M_0 with higher-order functionals) that are invariant on the defined torus but not globally. They note a lack of comprehensive theory for choosing these invariants and raise a conjecture inspired by prior work.

The conjecture proposes matching the number of torus-specific invariants to the full degrees of freedom of the truncated system, aiming to achieve a pseudo-integrable setup in which longulent states are precisely determined. Verifying this would provide a systematic route to construct and classify longulent attractors in GrNLS.

References

The conjecture follows again [4]: the "right" choice of the torus-specific invariants should be such as to have the total number be the degrees of freedom of the truncated system that a new kind of "pseudo-integrability" be established, in the sense of specifying precisely a longulent state.

— Constructing longulence in the Galerkin-regularized nonlinear Schrödinger and complex Ginzburg-Landau systems (2412.21142 - Zhu, 30 Dec 2024) in Section III.A (GrNLS longulence)