Stability of periodic waves of 1D cubic nonlinear Schr{\\"o}dinger\n equations
Bibliographic record
Abstract
We study the stability of the cnoidal, dnoidal and snoidal elliptic functions\nas spatially-periodic standing wave solutions of the 1D cubic nonlinear\nSchr{\\"o}dinger equations. First, we give global variational characterizations\nof each of these periodic waves, which in particular provide alternate proofs\nof their orbital stability with respect to same-period perturbations,\nrestricted to certain subspaces. Second, we prove the spectral stability of the\ncnoidal waves against same-period perturbations (in a certain parameter range),\nand provide an alternate proof of this (known) fact for the snoidal waves,\nwhich does not rely on complete integrability. Third, we give a rigorous\nversion of a formal asymptotic calculation of Rowlands to establish the\ninstability of a class of real-valued periodic waves in 1D, which includes the\ncnoidal waves of the 1D cubic focusing nonlinear Schr{\\"o}dinger equation,\nagainst perturbations with period a large multiple of their fundamental period.\nFinally, we develop a numerical method to compute the minimizers of the energy\nwith fixed mass and momentum constraints. Numerical experiments support and\ncomplete our analytical results.\n
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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) | 0.000 | 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 | 0.001 |
| Open science | 0.001 | 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".