Birkhoff normal form for the nonlinear Schrödinger equation
Bibliographic record
Abstract
This paper is intended to highlight the differences between the nonlinear Schrödinger equation (NLS) posed on a compact manifold (such as a torus \mathbb T^d ) in contrast to being posed on noncompact regions such as on all of \mathbb R^d . The point is to indicate a number of specific facts about the behavior of solutions in the former situation, in which they have the possibility for recurrence, and the latter, in which solutions have the tendency to disperse. This is the topic of the short article by McKean [8], in which the issue of resonance for partial differential evolution equations is discussed. The aspect of this question that we describe in the present paper is that there are different normal forms for these two cases, which rephrases the question as to which of the nonlinear terms are the resonant terms, and what is the appropriate Birkhoff normal form for the NLS. We show that, at least in a neighborhood of zero of an appropriate Hilbert space, the fourth order Birkhoff normal form transformation for the NLS equation is able to eliminate all of the nonresonant terms of the Hamiltonian, and as well, all of the resonant terms. The result is a prognosis, to the negative, for the formal theory of wave turbulence for Hamiltonian partial differential equations posed in Sobolev spaces over \mathbb R^d .
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".