{"paper":{"title":"Orbital stability of black solitons for quasilinear Schr\\\"odinger equations with nonzero conditions at infinity","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"The black soliton is orbitally stable in the energy space for quasilinear Schrödinger equations when the Vakhitov-Kolokolov slope condition holds.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Erwan Le Quiniou","submitted_at":"2026-05-13T14:57:07Z","abstract_excerpt":"We investigate the orbital stability of black solitons for a broad class of quasilinear Schr\\\"odinger equations in one space dimension, with nonzero boundary conditions at infinity. Namely, our framework handles general defocusing semilinear nonlinearities and focusing or defocusing quasilinear nonlinearities. First, we establish sufficient conditions on the quasi-linear nonlinearities ensuring the existence of a local branch of finite-energy solitons parameterized by their speed. Within this branch, the black soliton, also called kink, corresponds to the stationary solution. Our main result i"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Our main result is the orbital stability of the black soliton in the energy space, provided that the Vakhitov-Kolokolov (VK) slope condition holds; namely, that the derivative of the momentum with respect to the speed is negative at zero.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The analysis of minimizing sequences for the variational problem (infimum not attained) together with the sufficient conditions on the quasilinear nonlinearities that ensure existence of the local soliton branch parameterized by speed.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Black solitons are orbitally stable in the energy space when the derivative of momentum with respect to speed is negative at zero speed.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The black soliton is orbitally stable in the energy space for quasilinear Schrödinger equations when the Vakhitov-Kolokolov slope condition holds.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a7f5902e4720612279b9a7e2f1b55bb900235012d9ce3af0f0e2c1382ce308ea"},"source":{"id":"2605.13629","kind":"arxiv","version":1},"verdict":{"id":"2634c0c0-e20e-4906-aa46-e94bac767129","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:01:17.950365Z","strongest_claim":"Our main result is the orbital stability of the black soliton in the energy space, provided that the Vakhitov-Kolokolov (VK) slope condition holds; namely, that the derivative of the momentum with respect to the speed is negative at zero.","one_line_summary":"Black solitons are orbitally stable in the energy space when the derivative of momentum with respect to speed is negative at zero speed.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The analysis of minimizing sequences for the variational problem (infimum not attained) together with the sufficient conditions on the quasilinear nonlinearities that ensure existence of the local soliton branch parameterized by speed.","pith_extraction_headline":"The black soliton is orbitally stable in the energy space for quasilinear Schrödinger equations when the Vakhitov-Kolokolov slope condition holds."},"references":{"count":60,"sample":[{"doi":"","year":2024,"title":"M. A. Alejo and A. J. Corcho. Orbital stability of the black soliton for the quintic Gross- Pitaevskii equation.Rev. Mat. Iberoam., 40(5):1731–1780, 2024","work_id":"94bcc07d-f886-4344-96ec-30166334021c","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"C. O. Alves, Y. Wang, and Y. Shen. Soliton solutions for a class of quasilinear Schrödinger equations with a parameter.J. Differential Equations, 259(1):318–343, 2015","work_id":"992c471c-c378-48a6-8ea5-5f0b8cc767e3","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"C. Audiard. Small energy traveling waves for the Euler-Korteweg system.Nonlinearity, 30(9):3362–3399, 2017","work_id":"fe40e712-90a7-4560-89cd-9d75bccbf08b","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"C. Audiard and B. Haspot. Global well-posedness of the Euler-Korteweg system for small irrotational data.Comm. Math. Phys., 351(1):201–247, 2017","work_id":"1f49974e-8c34-411d-95df-261c539d0170","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"L. Baldelli, B. Bieganowski, and J. Mederski. Traveling waves for nonlinear Schrödinger equations, 2024. Preprint arXiv:2406.03910. 44","work_id":"c6746181-7f15-4984-bf26-78956bd200f2","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":60,"snapshot_sha256":"544ed9d3fc1cf9e9f59438e22453085ffe22ce5b858431a2243b23536324285c","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"16c250f163f16551c644b41340156353d76632539c27702924f1871dccb4b8f8"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}