{"paper":{"title":"Newton's Method and Symmetry for Semilinear Elliptic PDE on the Cube","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"nlin.PS","authors_text":"James W. Swift, John M. Neuberger, Nandor Sieben","submitted_at":"2013-01-29T22:14:00Z","abstract_excerpt":"We seek discrete approximations to solutions $u:\\Omega \\to R$ of semilinear elliptic partial differential equations of the form $\\Delta u + f_s(u) = 0$, where $f_s$ is a one-parameter family of nonlinear functions and $\\Omega$ is a domain in $R^d$. The main achievement of this paper is the approximation of solutions to the PDE on the cube $\\Omega=(0,\\pi)^3 \\subseteq R^3$. There are 323 possible isotropy subgroups of functions on the cube, which fall into 99 conjugacy classes. The bifurcations with symmetry in this problem are quite interesting, including many with 3-dimensional critical eigens"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.7085","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}