{"paper":{"title":"The Calogero--Moser Derivative Nonlinear Schr\\\"odinger Equation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.MP","nlin.SI"],"primary_cat":"math.AP","authors_text":"Enno Lenzmann, Patrick G\\'erard","submitted_at":"2022-08-08T12:54:47Z","abstract_excerpt":"We study the Calogero--Moser derivative NLS equation $$ i \\partial_t u +\\partial_{xx} u + (D+|D|)(|u|^2) u =0 $$ posed on the Hardy-Sobolev space $H^s_+(\\mathbb{R})$ with suitable $s>0$. By using a Lax pair structure for this $L^2$-critical equation, we prove global well-posedness for $s \\geq 1$ and initial data with sub-critical or critical $L^2$-mass $\\| u_0 \\|_{L^2}^2 \\leq 2 \\pi$. Moreover, we prove uniqueness of ground states and also classify all traveling solitary waves. Finally, we study in detail the class of multi-soliton solutions $u(t)$ and we prove that they exhibit energy cascades"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.04105","kind":"arxiv","version":2},"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/2208.04105/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"}