{"paper":{"title":"Traveling waves & finite gap potentials for the Calogero-Sutherland Derivative nonlinear Schr\\\"odinger equation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Rana Badreddine","submitted_at":"2023-07-04T09:31:45Z","abstract_excerpt":"We consider the Calogero-Sutherland derivative nonlinear Schr\\\"odinger equation \\begin{equation}\\tag{CS}\n  i\\partial_tu+\\partial_x^2u\\,\\pm\\,\\frac{2}{i}\\,\\partial_x\\Pi(|u|^2)u=0\\,,\\qquad x\\in\\mathbb{T}\\,, \\end{equation} where $\\Pi$ is the Szeg\\H{o} projector $$\\Pi\\Big(\\sum_{n\\in \\mathbb{Z}}\\widehat{u}(n)\\mathrm{e}^{inx}\\Big)=\\sum_{n\\geq 0 }\\widehat{u}(n)\\mathrm{e}^{inx}\\,.$$ First, we characterize the traveling wave $u_0(x-ct)$ solutions to the defocusing equation (CS$^-$), and prove for the focusing equation (CS$^+$), that all the traveling waves must be either the constant functions or plane "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.01592","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2307.01592/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}