{"paper":{"title":"Long-time asymptotics for the integrable nonlocal focusing nonlinear Schr\\\"odinger equation for a family of step-like initial data","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["nlin.SI"],"primary_cat":"math.AP","authors_text":"Dmitry Shepelsky, Yan Rybalko","submitted_at":"2019-08-18T10:03:30Z","abstract_excerpt":"We study the Cauchy problem for the integrable nonlocal focusing nonlinear Schr\\\"odinger (NNLS) equation $\niq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\\bar{q}(-x,t)=0 $\nwith the step-like initial data close to the ``shifted step function'' $\\chi_R(x)=AH(x-R)$, where $H(x)$ is the Heaviside step function, and $A>0$ and $R>0$ are arbitrary constants. Our main aim is to study the large-$t$ behavior of the solution of this problem. We show that for $R\\in\\left(\\frac{(2n-1)\\pi}{2A},\\frac{(2n+1)\\pi}{2A}\\right)$, $n=1,2,\\dots$, the $(x,t)$ plane splits into\n  $4n+2$ sectors exhibiting different asymptotic beh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.06415","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/1908.06415/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"}