Group analysis of general Burgers–Korteweg–de Vries equations
Bibliographic record
Abstract
The complete group classification problem for the class of (1+1)-dimensional rth order general variable-coefficient Burgers–Korteweg–de Vries equations is solved for arbitrary values of r greater than or equal to two. We find the equivalence groupoids of this class and its various subclasses obtained by gauging equation coefficients with equivalence transformations. Showing that this class and certain gauged subclasses are normalized in the usual sense, we reduce the complete group classification problem for the entire class to that for the selected maximally gauged subclass, and it is the latter problem that is solved efficiently using the algebraic method of group classification. Similar studies are carried out for the two subclasses of equations with coefficients depending at most on the time or space variable, respectively. Applying an original technique, we classify Lie reductions of equations from the class under consideration with respect to its equivalence group. Studying alternative gauges for equation coefficients with equivalence transformations allows us not only to justify the choice of the most appropriate gauge for the group classification but also to construct for the first time classes of differential equations with nontrivial generalized equivalence group such that equivalence-transformation components corresponding to equation variables locally depend on nonconstant arbitrary elements of the class. For the subclass of equations with coefficients depending at most on the time variable, which is normalized in the extended generalized sense, we explicitly construct its extended generalized equivalence group in a rigorous way. The new notion of effective generalized equivalence group is introduced.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".