Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation
Bibliographic record
Abstract
We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ≪1,K≫1,s>1, we construct smooth initial data u 0 with $\|u_{0}\|_{{H}^{s}}<\delta$ , so that the corresponding time evolution u satisfies $\|u(T)\|_{{H}^{s}}>K$ at some time T. This growth occurs despite the Hamiltonian’s bound on $\|u(t)\|_{\dot{H}^{1}}$ and despite the conservation of the quantity $\|u(t)\|_{L^{2}}$ . The proof contains two arguments which may be of interest beyond the particular result described above. The first is a construction of the solution’s frequency support that simplifies the system of ODE’s describing each Fourier mode’s evolution. The second is a construction of solutions to these simpler systems of ODE’s which begin near one invariant manifold and ricochet from arbitrarily small neighborhoods of an arbitrarily large number of other invariant manifolds. The techniques used here are related to but are distinct from those traditionally used to prove Arnold Diffusion in perturbations of Hamiltonian systems.
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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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".