Traveling waves in periodic metric graphs via spatial dynamics
Bibliographic record
Abstract
The purpose of this work is to introduce a concept of traveling waves in the setting of periodic metric graphs. It is known that the nonlinear Schr{ö}dinger (NLS) equation on periodic metric graphs can be reduced asymptotically on long but finite time intervals to the homogeneous NLS equation, which admits traveling solitary wave solutions. In order to address persistence of such traveling waves beyond finite time intervals, we formulate the existence problem for traveling waves via spatial dynamics. There exist no spatially decaying (solitary) waves because of an infinite-dimensional center manifold in the spatial dynamics formulation. Existence of traveling modulating pulse solutions which are solitary waves with small oscillatory tails at very long distances from the pulse core is proven by using a local center-saddle manifold. We show that the variational formulation fails to capture existence of such modulating pulse solutions even in the singular limit of zero wave speeds where true (standing) solitary waves exist. Propagation of a traveling solitary wave and formation of a small oscillatory tail outside the pulse core is shown in numerical simulations of the NLS equation on the periodic graph.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".