Stable standing waves of nonlinear fractional Schrödinger equations
Bibliographic record
Abstract
We study the existence and orbital stability of standing waves of nonlinear fractional Schrödinger equations with a general nonlinear term \begin{document}$ \begin{equation*} \mathrm{i} u_t-\left(-\Delta\right)^s u +f\left(u\right) = 0, \ \left(t, x\right)\in\mathbb{R}_+\times\mathbb{R}^N. \end{equation*} $\end{document} We investigate the minimizing problem with \begin{document}$ L^2 $\end{document} -constraint: \begin{document}$ \begin{equation*} E_{\alpha} = \inf\Big\{\frac{1}{2}\int_{\mathbb{R}^N}\!|(-\Delta)^{\frac{s}{2}}u|^2\mathrm{d}x-\int_{\mathbb{R}^N}\!F(|u|)\mathrm{d}x\ \Big|\ u\in H^{s}(\mathbb{R}^N), \|u\|^2_{L^2(\mathbb{R}^N)} = \alpha\Big\}. \end{equation*} $\end{document} The existence and non-existence of global minimizers with respect to \begin{document}$ E_{\alpha} $\end{document} are established for all possible values of \begin{document}$ \alpha. $\end{document} Under some general assumptions on the nonlinear term \begin{document}$ f(u) $\end{document} , there exists a constant \begin{document}$ \alpha_0\ge 0 $\end{document} such that a global minimizer exists for \begin{document}$ E_\alpha $\end{document} for all \begin{document}$ \alpha>\alpha_0 $\end{document} , and there is no global minimizer with respect to \begin{document}$ E_{\alpha} $\end{document} for all \begin{document}$ 0<\alpha<\alpha_0. $\end{document} By virtue of concentration-compactness argument and the strict subadditivity of \begin{document}$ E_\alpha $\end{document} , the strong convergence of minimizing sequence is obtained. Moreover, we present some criteria which determine \begin{document}$ \alpha_0 = 0 $\end{document} or \begin{document}$ \alpha_0>0 $\end{document} , and the existence of global minimizers for \begin{document}$ E_{\alpha_0}. $\end{document} Besides, we show the orbital stability of the global minimizers set. Finally, we prove that an energy minimizer is a least action solution by Pohozaev identity.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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".