Stable Directions for Degenerate Excited States of Nonlinear\n Schr\\"odinger Equations
Bibliographic record
Abstract
We consider nonlinear Schr\\"{o}dinger equations, $i\\partial_t \\psi = H_0 \\psi\n+ \\lambda |\\psi|^2\\psi$ in $\\mathbb{R}^3 \\times [0,\\infty)$, where $H_0 =\n-\\Delta + V$, $\\lambda=\\pm 1$, the potential $V$ is radial and spatially\ndecaying, and the linear Hamiltonian $H_0$ has only two eigenvalues $e_0 < e_1\n<0$, where $e_0$ is simple, and $e_1$ has multiplicity three. We show that\nthere exist two branches of small "nonlinear excited state" standing-wave\nsolutions, and in both the resonant ($e_0 < 2e_1$) and non-resonant ($e_0 >\n2e_1$) cases, we construct certain finite-codimension regions of the phase\nspace consisting of solutions converging to these excited states at time\ninfinity ("stable directions").\n
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| 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) | 0.002 | 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".