Stable Directions for Degenerate Excited States of Nonlinear\n Schr\\"odinger Equations
Why this work is in the frame
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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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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.001 |
| 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 it