{"paper":{"title":"The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"nlin.SI","authors_text":"Da-jun Zhang, Dan-dan Xu, Songling Zhao","submitted_at":"2014-01-23T12:05:57Z","abstract_excerpt":"The paper is to reveal the direct links between the well known Sylvester equation in matrix theory and some integrable systems. Using the Sylvester equation $\\boldsymbol{K} \\boldsymbol{M}+\\boldsymbol{M} \\boldsymbol{K}=\\boldsymbol{r}\\, \\boldsymbol{s}^{T}$ we introduce a scalar function $S^{(i,j)}=\\boldsymbol{s}^{T}\\, \\boldsymbol{K}^j(\\boldsymbol{I}+\\boldsymbol{M})^{-1}\\boldsymbol{K}^i\\boldsymbol{r}$ which is defined as same as in discrete case. $S^{(i,j)}$ satisfy some recurrence relations which can be viewed as discrete equations and play indispensable roles in deriving continuous integrable e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.5949","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"}