The initial boundary problem for the Korteweg-de Vries equation on the negative quarter-plane
Bibliographic record
Abstract
The initial boundary–value problem for the Korteweg–de Vries (KdV) equation on the negative quarter–plane, x < 0 and t > 0, is considered. The formulation of this problem is different to the usual initial boundary–value problem on the positive quarter–plane, for which x > 0 and t > 0. Two boundary conditions are required at x = 0 for the negative quarter–plane problem, in contrast to the one boundary condition needed at x = 0 for the positive quarter–plane problem. Solutions of the KdV equation on the infinite line, such as the soliton, cnoidal wave, mean height variation and undular bore solution, are used to find approximate solutions to the negative quarter–plane problem. Five qualitatively different types of solution are found, depending on the relation between the initial and boundary values. Excellent comparisons are obtained between these solutions and full numerical solutions of the KdV equation.
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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 | 0.003 |
| 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 | 0.002 |
| Scholarly communication | 0.003 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 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".