{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/mechanisms-of-unstable-blowup-in-a-quadratic","title":"Mechanisms of unstable blowup in a quadratic nonlinear Schrödinger equation","arxiv_id":"2406.00762","date":"2024-06-02","proceeding":null,"authors":["Jonathan Jaquette"],"abstract":"In the work Cho et al. [Jpn. J. Ind. Appl. Math. 33 (2016): 145-166] the authors conjecture that the quadratic nonlinear Schr\\\"odinger equation (NLS) $i u_t = u_{xx} + u^2 $ for $ x \\in \\mathbb{T}$ is globally well-posed for real initial data. We identify initial data whose numerical solution blows up in contradiction of this conjecture. The solution exhibits self-similar blowup and potentially nontrivial self-similar dynamics, however the proper scaling ansatz remains elusive. Furthermore, the set of real initial data which blows up under the NLS dynamics appears to occur on a codimension-1 manifold, and we conjecture that it is precisely the stable manifold of the zero equilibrium for the nonlinear heat equation $u_t = u_{xx} + u^2 $. We apply the parameterization method to study the internal dynamics of this manifold, offering a heuristic argument in support of our conjecture.","url_abs":"https://arxiv.org/abs/2406.00762v2","url_pdf":"https://arxiv.org/pdf/2406.00762v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"mechanisms-of-unstable-blowup-in-a-quadratic","repo_url":"https://github.com/JCJaquette/Unstable-Blowup-NLS","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}