Instability bands for periodic travelling waves in the modified Korteweg–de Vries equation
Bibliographic record
Abstract
Two families of periodic travelling waves exist in the focusing modified Korteweg–de Vries equation. Spectral stability of these waveforms with respect to co-periodic perturbations of the same period has been previously explored by using spectral analysis and variational formulation. By using tools of integrability, such as a relation between squared eigenfunctions of the Lax pair and eigenfunctions of the linearized stability problem, we revisit the spectral stability of these waveforms with respect to perturbations of arbitrary periods. In agreement with previous works, we find that one family is spectrally stable for all parameter configurations, whereas the other family is spectrally unstable for all parameter configurations. We show that the onset of the co-periodic instability for the latter family changes the instability bands from figure- 8 (crossing at the imaginary axis) into figure- ∞ (crossing at the real axis).
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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) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 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) | 0.003 | 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".