Supercritical surface waves generated by negative or oscillatory forcing
Bibliographic record
Abstract
The paper studies forced surface waves on an incompressible,inviscid fluid in a two-dimensional channel with a small negative oroscillatory bump on a rigid flat bottom. Such wave motions aredetermined by a non-dimensional wave speed $F$, called Froudenumber, and $F=1$ is a critical value of $F$. If $F= 1+ \lambda\epsilon $ with a small parameter $\epsilon> 0$, then a forced Korteweg-de Vries (FKdV)equation can be derived to model the wave motion on the freesurface. In this paper, the case $\lambda > 0$ (or $F> 1$, calledsupercritical case) is considered. The steady and unsteady solutionsof the FKdV equation with a negative bump function independent oftime are first studied both theoretically and numerically. It isshown that there are five steady solutions and only one of them,which exists for all $\lambda > 0$, is stable. Then, solutions ofthe FKdV equation with an oscillatory bump function posed on $R$ ora finite interval are considered. The corresponding linear problemsare solved explicitly and the solutions are rigorously shown to beeventually periodic as time goes to infinity, while a similar resultholds for the nonlinear problem posed on a finite interval withsmall initial data and forcing functions. The nonlinear solutionswith zero initial data for any forcing functions in the real line$R$ or large forcing functions in a finite interval are obtainednumerically. It is shown numerically that the solutions will becomeeventually periodic in time for a small forcing function. Thebehavior of the solutions becomes quite irregular as time goes toinfinity, if the forcing function is large.
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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 | 0.000 |
| 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 | 0.001 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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".