Pinning in the Extended Lugiato–Lefever Equation
Bibliographic record
Abstract
Abstract. We consider a variant of the Lugiato–Lefever equation (LLE), which is a nonlinear Schrödinger equation on a one-dimensional torus with forcing and damping, to which we add a first-order derivative term with a potential [Formula: see text]. The potential breaks the translation invariance of LLE. Depending on the existence of zeroes of the effective potential [Formula: see text], which is a suitably weighted and integrated version of [Formula: see text], we show that stationary solutions from [Formula: see text] can be continued locally into the range [Formula: see text]. Moreover, the extremal points of the [Formula: see text]-continued solutions are located near zeros of [Formula: see text]. We therefore call this phenomenon pinning of stationary solutions. If we assume additionally that the starting stationary solution at [Formula: see text] is spectrally stable with the simple zero eigenvalue due to translation invariance being the only eigenvalue on the imaginary axis, we can prove asymptotic stability or instability of its [Formula: see text]-continuation depending on the sign of [Formula: see text] at the zero of [Formula: see text] and the sign of [Formula: see text]. The variant of the LLE arises in the description of optical frequency combs in a Kerr nonlinear ring-shaped microresonator which is pumped by two different continuous monochromatic light sources of different frequencies and different powers. Our analytical findings are illustrated by numerical simulations.
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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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.012 | 0.001 |
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".