Travelling wave solutions on a non-zero background for the generalized Korteweg–de Vries equation
Bibliographic record
Abstract
Abstract For the generalized p -power Korteweg–de Vries equation, all non-periodic travelling wave solutions with non-zero boundary conditions are explicitly classified for all integer powers p ⩾ 1. These solutions are shown to consist of: bright solitary waves and static humps on a non-zero background for odd p ; dark solitary waves on a non-zero background and kink (shock) waves for even p in the defocusing case; pairs of bright/dark solitary waves on a non-zero background, and also bright and dark heavy-tail waves (with power decay) on a non-zero background, for even p in the focussing case. An explicit physical parameterization is given for each type of solution in terms of the wave speed c , background size b , and wave height/depth h . The allowed kinematic region for existence of the solutions is derived, and other main kinematic features are discussed. Analytical formulas are presented in the higher power cases p = 3, 4, which are compared to the integrable cases p = 1, 2.
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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 | 0.000 | 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 | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".