{"paper":{"title":"The Cauchy problem for higher-order modified Camassa-Holm equations on the circle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Wei Yan, Xiaoping Zhai, Yimin Zhang, Yongsheng Li","submitted_at":"2016-05-09T04:09:17Z","abstract_excerpt":"In this paper, we investigate the Cauchy problem for the shallow water type equation \\begin{eqnarray*} u_{t}+\\partial_{x}^{2j+1}u + \\frac{1}{2}\\partial_{x}(u^{2})+ \\partial_{x}(1-\\partial_{x}^{2})^{-1}\\left[u^{2}+\\frac{1}{2}u_{x}^{2}\\right]=0 \\end{eqnarray*} with low regularity data in the periodic settings. Himonas and Misiolek (Communications in Partial Differential Equations, 23(1998), 123-139.) have proved that the problem is locally well-posed for small initial data in H^{s}(\\mathbf{T}) with s\\geq-\\frac{j}{2}+1,j\\in N^{+} with the aid of the standard Fourier restriction norm method. To th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.02412","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"}