Instability of the peaked travelling wave in a local model for shallow water waves
Bibliographic record
Abstract
Abstract The travelling wave with the peaked profile is usually considered as a limit in the family of travelling waves with the smooth profiles. We study the linear and nonlinear stability of the peaked travelling wave by using a local model for shallow water waves, which is an extended version of the Hunter–Saxton equation. The evolution problem is well-defined in the function space $H^1_{\rm per} \cap W^{1,\infty}$ , where we derive the linearised equations of motion and study the nonlinear evolution of co-periodic perturbations to the peaked periodic wave by using the method of characteristics. Within the linearised equations, we prove the spectral instability of the peaked travelling wave from the spectrum of the linearised operator in a Hilbert space, which completely covers the closed vertical strip with a specific half-width. Within the nonlinear equations, we prove the nonlinear instability of the peaked travelling wave by showing that the gradient of perturbations grows at the wave peak. By using numerical approximations of the smooth travelling waves and the spectrum of their associated linearised operator, we show that the spectral instability of the peaked travelling wave cannot be obtained as a limit in the family of the spectrally stable smooth travelling waves.
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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 | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".