Multidimensional partial differential equation systems: Generating new systems via conservation laws, potentials, gauges, subsystems
Bibliographic record
Abstract
For many systems of partial differential equations (PDEs), including nonlinear ones, one can construct nonlocally related PDE systems. In recent years, such nonlocally related systems have proven to be useful in applications. In particular, they have yielded systematically nonlocal symmetries, nonlocal conservation laws, noninvertible linearizations, and new exact solutions for many different PDE systems of interest. However, the overwhelming majority of new results and theoretical understanding pertain only to PDE systems with two independent variables. The situation for PDE systems with more than two independent variables turns out to be much more complicated due to gauge freedom relating potential variables. The current paper, together with the companion paper [A. F. Cheviakov and G. W. Bluman, J. Math. Phys. 51, 103522 (2010)], synthesizes and systematically extends known results for nonlocally related systems arising for multidimensional PDE systems, i.e., for PDE systems with three or more independent variables. The presented framework includes potential systems arising from lower-degree conservation laws of a given PDE system. Nonlocally related multidimensional PDE systems are discussed in terms of their construction, properties, and applications.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".