{"paper":{"title":"The periodic $\\mu$-$b$-equation and Euler equations on the circle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"Martin Kohlmann","submitted_at":"2010-10-09T10:39:15Z","abstract_excerpt":"In this paper, we study the $\\mu$-variant of the periodic $b$-equation and show that this equation can be realized as a metric Euler equation on the Lie group $\\Diff^{\\infty}(\\S)$ if and only if $b=2$ (for which it becomes the $\\mu$-Camassa-Holm equation). In this case, the inertia operator generating the metric on $\\Diff^{\\infty}(\\S)$ is given by $L=\\mu-\\partial_x^2$. In contrast, the $\\mu$-Degasperis-Procesi equation (obtained for $b=3$) is not a metric Euler equation on $\\Diff^{\\infty}(\\S)$ for any regular inertia operator $A\\in\\mathcal L_{\\text{is}}^{\\text{sym}}(C^{\\infty}(\\S))$. The paper"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.1832","kind":"arxiv","version":2},"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"}