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 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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| 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".