On the Existence and Uniqueness of Global Solutions for the KdV Equation\n with Quasi-Periodic Initial Data
Bibliographic record
Abstract
We consider the KdV equation $$ \\partial_t u +\\partial^3_x u +u\\partial_x u=0\n$$ with quasi-periodic initial data whose Fourier coefficients decay\nexponentially and prove existence and uniqueness, in the class of functions\nwhich have an expansion with exponentially decaying Fourier coefficients, of a\nsolution on a small interval of time, the length of which depends on the given\ndata and the frequency vector involved. For a Diophantine frequency vector and\nfor small quasi-periodic data (i.e., when the Fourier coefficients obey $|c(m)|\n\\le \\varepsilon \\exp(-\\kappa_0 |m|)$ with $\\varepsilon > 0$ sufficiently small,\ndepending on $\\kappa_0 > 0$ and the frequency vector), we prove global\nexistence and uniqueness of the solution. The latter result relies on our\nrecent work \\cite{DG} on the inverse spectral problem for the quasi-periodic\nSchr\\"{o}dinger equation.\n
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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 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".