Supercritical surface waves generated by negative or oscillatory forcing
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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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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 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 | 0.001 |
| 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 it