Laguerre polynomials and transitional asymptotics of the modified Korteweg-de Vries equation for step-like initial data
Bibliographic record
Abstract
We consider the compressive wave for the modified Korteweg–de Vries equation with background constants \\(c > 0\\) for \\(x\\to-\\infty\\) and 0 for \\(x\\to+\\infty\\). We study the asymptotics of solutions in the transition zone \\(4c^2t-\\varepsilon t < x < 4c^2t-\\beta t^\\sigma\\) for \\(\\varepsilon > 0\\), \\(\\sigma \\in (0, 1)\\), \\(\\beta > 0\\). In this region we have a bulk of nonvanishing oscillations, the number of which grows as \\({\\varepsilon t \\over \\mbox{ln} t}\\). Also we show how to obtain Khruslov–Kotlyarov’s asymptotics in the domain \\(4c^2t-\\rho\\) ln \\(t<x<4c^2t\\) with the help of parametrices constructed out of Laguerre polynomials in the corresponding Riemann-Hilbert problem.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".