{"paper":{"title":"On well-posedness and wave operator for the gKdV equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ademir Pastor, Luiz Gustavo Farah","submitted_at":"2012-04-25T15:49:58Z","abstract_excerpt":"We consider the generalized Korteweg-de Vries (gKdV) equation $\\partial_t u+\\partial_x^3u+\\mu\\partial_x(u^{k+1})=0$, where $k>4$ is an integer number and $\\mu=\\pm1$. We give an alternative proof of the Kenig, Ponce, and Vega result in \\cite{kpv1}, which asserts local and global well-posedness in $\\dot{H}^{s_k}(\\R)$, with $s_k=(k-4)/2k$. A blow-up alternative in suitable Strichatz-type spaces is also established. The main tool is a new linear estimate. As a consequence, we also construct a wave operator in the critical space $\\dot{H}^{s_k}(\\R)$, extending the results of C\\^ote [2]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.5688","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"}