{"paper":{"title":"On The Local Well-Posedness for Some Systems of Coupled KdV Equations","license":"","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Borys Alvarez-Samaniego, Xavier Carvajal","submitted_at":"2007-05-03T15:10:17Z","abstract_excerpt":"Using the theory developed by Kenig, Ponce, and Vega, we prove that the Hirota-Satsuma system is locally well-posed in Sobolev spaces $H^s(\\mathbb{R}) \\times H^{s}(\\mathbb{R})$ for $3/4<s\\le1$. We introduce some Bourgain-type spaces $X_{s,b}^a$ for $a\\not =0$, $s,b \\in \\mathbb{R}$ to obtain local well-posedness for the Gear-Grimshaw system in $H^s(\\mathbb{R})\\times H^s(\\mathbb{R})$ for $s>-3/4$, by establishing new mixed-bilinear estimates involving the two Bourgain-type spaces $X_{s,b}^{-\\alpha_-}$ and $X_{s,b}^{-\\alpha_+}$ adapted to $\\partial_t+\\alpha_-\\partial_x^3$ and $\\partial_t+\\alpha_+"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0705.0482","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"}