{"paper":{"title":"Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"Manwai Yuen","submitted_at":"2010-07-06T17:19:04Z","abstract_excerpt":"In this article, we study the self-similar solutions of the 2-component Camassa-Holm equations% \\begin{equation} \\left\\{ \\begin{array} [c]{c}% \\rho_{t}+u\\rho_{x}+\\rho u_{x}=0\n  m_{t}+2u_{x}m+um_{x}+\\sigma\\rho\\rho_{x}=0 \\end{array} \\right. \\end{equation} with \\begin{equation} m=u-\\alpha^{2}u_{xx}. \\end{equation} By the separation method, we can obtain a class of blowup or global solutions for $\\sigma=1$ or $-1$. In particular, for the integrable system with $\\sigma=1$, we have the global solutions:% \\begin{equation} \\left\\{ \\begin{array} [c]{c}% \\rho(t,x)=\\left\\{ \\begin{array} [c]{c}% \\frac{f\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.0962","kind":"arxiv","version":3},"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"}