INFINITE DIMENSIONAL INVARIANT TORI FOR NONLINEAR SCHRÖDINGER EQUATIONS

Joackim Bernier, Benoît Grébert, Tristan Robert

correctmedium confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves items (i)–(iv) (finite-dimensional KAM tori with explicit frequency modulation, closeness, and measure estimates) and, crucially, item (v): the existence of infinite-dimensional nonresonant Kronecker tori accumulating the Kuksin–Pöschel tori, via a regularizing normal form, an opening step, and an infinite loop of KAM+Birkhoff steps culminating in an infinite-dimensional limit. The candidate solution reproduces only the standard finite-dimensional KAM scheme (including the quartic BNF and twist), and its argument for (v) incorrectly appeals to the density of Diophantine vectors in finite dimensions; it neither constructs nor justifies any infinite-dimensional torus or the limit process required by (v). Hence the model fails on the key new contribution.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The work delivers a substantive advance by constructing infinite-dimensional invariant tori for 1D NLS without external parameters and showing accumulation near finite-dimensional KAM tori. The architecture—regularizing normal form, strengthened KAM removing higher jets, and an opening/loop iteration—appears sound and is carefully documented. Some expository improvements (e.g., a compact overview of constants and parameter hierarchies) would enhance readability.