Lie group analysis of a generalized Krichever-Novikov differential-difference equation
Bibliographic record
Abstract
The symmetry algebra of the differential-difference equation \documentclass[12pt]{minimal}\begin{document}$\dot{u}_n = N/D$\end{document}u̇n=N/D with D = un+1 − un−1 and N = P(un)un+1un−1 + Q(un)(un+1 + un−1) + R(un), where P, Q, and R are arbitrary analytic functions is shown to have the dimension 1 ⩽ dimL ⩽ 5. When P, Q, and R are specific second order polynomials in un (depending on 6 constants) this is the integrable discretization of the Krichever–Novikov equation. We find 3 cases when the arbitrary functions are not polynomials and the symmetry algebra satisfies dimL = 2. These cases are shown not to be integrable. The symmetry algebras are used to reduce the equations to purely difference ones. The symmetry group is also used to impose periodicity un+N = un and thus to reduce the differential-difference equation to a system of N coupled ordinary three points difference equations.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".