{"paper":{"title":"Self-Similar Blowup Solutions to the 2-Component Degasperis-Procesi Shallow Water System","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":"2010-08-13T10:17:02Z","abstract_excerpt":"In this article, we study the self-similar solutions of the 2-component Degasperis-Procesi water system:% [c]{c}% \\rho_{t}+k_{2}u\\rho_{x}+(k_{1}+k_{2})\\rho u_{x}=0 u_{t}-u_{xxt}+4uu_{x}-3u_{x}u_{xx}-uu_{xxx}+k_{3}\\rho\\rho_{x}=0.  By the separation method, we can obtain a class of self-similar solutions,% [c]{c}% \\rho(t,x)=\\max(\\frac{f(\\eta)}{a(4t)^{(k_{1}+k_{2})/4}},\\text{}0),\\text{}u(t,x)=\\frac{\\overset{\\cdot}{a}(4t)}{a(4t)}x \\overset{\\cdot\\cdot}{a}(s)-\\frac{\\xi}{4a(s)^{\\kappa}}=0,\\text{}a(0)=a_{0}% \\neq0,\\text{}\\overset{\\cdot}{a}(0)=a_{1} f(\\eta)=\\frac{k_{3}}{\\xi}\\sqrt{-\\frac{\\xi}{k_{3}}\\eta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.2282","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"}