GROUP-INVARIANT SOLUTIONS OF SEMILINEAR SCHRODINGER EQUATIONS IN MULTI-DIMENSIONS
Bibliographic record
Abstract
Symmetry group methods are applied to obtain all explicit group-invariant radial \nsolutions to a class of semilinear Schr¨odinger equations in dimensions n = 1. Both \nfocusing and defocusing cases of a power nonlinearity are considered, including the \nspecial case of the pseudo-conformal power p = 4/n relevant for critical dynamics. \nThe methods involve, first, reduction of the Schr¨odinger equations to group-invariant \nsemilinear complex 2nd order ordinary differential equations (ODEs) with respect \nto an optimal set of one-dimensional point symmetry groups, and second, use of \ninherited symmetries, hidden symmetries, and conditional symmetries to solve each \nODE by quadratures. Through Noether’s theorem, all conservation laws arising from \nthese point symmetry groups are listed. Some group-invariant solutions are found to \nexist for values of n other than just positive integers, and in such cases an alternative \ntwo-dimensional form of the Schr¨odinger equations involving an extra modulation \nterm with a parameter m = 2−n = 0 is discussed.
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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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".