The derivative nonlinear Schrödinger equation: Global well-posedness and soliton resolution
Bibliographic record
Abstract
We review recent results on global well-posedness and long-time behavior of smooth solutions to the derivative nonlinear Schrödinger (DNLS) equation. Using the integrable character of DNLS, we show how the inverse scattering tools and the method of Zhou [SIAM J. Math. Anal. 20 (1989), pp. 966–986] for treating spectral singularities lead to global well-posedness for general initial conditions in the weighted Sobolev space H 2 , 2 ( R ) H^{2,2}(\mathbb {R}) . For generic initial data that can support bright solitons but exclude spectral singularities, we prove the soliton resolution conjecture : the solution is asymptotic, at large times, to a sum of localized solitons and a dispersive component, Our results also show that soliton solutions of DNLS are asymptotically stable .
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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".