{"paper":{"title":"Soliton-like solutions of the Camassa--Holm equation with variable coefficients and a small dispersion","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"Asymptotic expansions construct soliton- and peakon-like solutions to arbitrary accuracy for the variable-coefficient Camassa-Holm equation with small dispersion.","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Valerii Samoilenko, Yuliia Samoilenko","submitted_at":"2026-04-19T09:44:07Z","abstract_excerpt":"The paper deals with the Camassa--Holm equation with variable coefficients (vcCH equation) that is a direct generalization of the well known Camassa--Holm equation. We focus on the mathematical description of particular solutions of the vcCH equation with a small dispersion that exhibit properties analogous to those of classical soliton and peakon solutions, and consider the construction of soliton- and peakon-like solutions in the form of asymptotic expansions, including both one-phase and two-phase cases.\n  The solution is expressed as the sum of a regular background common to all soliton- a"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Theorems on the asymptotic accuracy of the constructed asymptotic solutions have been proved; in the one-phase case the solvability of higher-order singular corrections is established in suitable functional spaces, enabling construction of asymptotic solutions to arbitrary accuracy in a small parameter.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The precise definition of the main singular term (the leading peakon or soliton profile) is chosen so that the resulting correction equations remain solvable at every order; this choice is stated to play a central role but is not derived from first principles within the paper.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The paper constructs asymptotic expansions for one-phase and two-phase soliton-like and peakon-like solutions of the variable-coefficient Camassa-Holm equation with small dispersion and proves their asymptotic accuracy.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Asymptotic expansions construct soliton- and peakon-like solutions to arbitrary accuracy for the variable-coefficient Camassa-Holm equation with small dispersion.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"dcddc5c60adb6d9db676c965e200f0815b79842c222c7daaeaf5a00fab7853d1"},"source":{"id":"2604.17348","kind":"arxiv","version":2},"verdict":{"id":"7673ef98-0102-430a-9f92-1d0b1a7ef96a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:23:02.433181Z","strongest_claim":"Theorems on the asymptotic accuracy of the constructed asymptotic solutions have been proved; in the one-phase case the solvability of higher-order singular corrections is established in suitable functional spaces, enabling construction of asymptotic solutions to arbitrary accuracy in a small parameter.","one_line_summary":"The paper constructs asymptotic expansions for one-phase and two-phase soliton-like and peakon-like solutions of the variable-coefficient Camassa-Holm equation with small dispersion and proves their asymptotic accuracy.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The precise definition of the main singular term (the leading peakon or soliton profile) is chosen so that the resulting correction equations remain solvable at every order; this choice is stated to play a central role but is not derived from first principles within the paper.","pith_extraction_headline":"Asymptotic expansions construct soliton- and peakon-like solutions to arbitrary accuracy for the variable-coefficient Camassa-Holm equation with small dispersion."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.17348/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":67,"sample":[{"doi":"10.1103/physrevlett.71.1661","year":1993,"title":"R. Camassa and D. Holm. An integrable shallow water equation with peaked soliton.Phys. Rev. Lett., 71(11):1661–1664, 1993. doi: 10.1103/PhysRevLett.71.1661","work_id":"359fe777-ebef-42e8-9d05-07a03129cfa7","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/j.physd.2022.133446","year":2022,"title":"H. Lundmark and J. Szmigielski. A view of the peakon world through the lens of approximation theory.Physica D: Nonlinear Phenomena, 440: 133446, 2022. doi: 10.1016/j.physd.2022.133446. 47","work_id":"c539fd84-3353-40ee-86e1-ef615f497321","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"H. Lundmark and B. Shuaib. Ghostpeakons and characteristic curves for theCamassa–Holm, Degasperis–ProcesiandNovikovequations.Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 15:","work_id":"78b6f0fb-ea77-4361-a344-df072a1204fe","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.3842/sigma.2019.017","year":2019,"title":"doi: 10.3842/SIGMA.2019.017","work_id":"588d82b3-7502-475f-a1ef-7f7435dd7e27","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/0167-2789(81)90004-x","year":1981,"title":"B. Fuchssteiner and A. S. Fokas. Symplectic structures, their Bäcklund transformations and hereditary symmetries.Physica D: Nonlinear Phe- nomena, 4(1):47–66, 1981/1982. doi: 10.1016/0167-2789(81)9000","work_id":"4472cbd9-e56e-49ca-a5d9-e175b97ee46a","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":67,"snapshot_sha256":"b0ae474739ec3c27e312837260112be67e0152d9d33983571dd7795c2560a3e8","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"74ecf43d4f94aca62324ee524bca997d6492e4d7ed616d5f604973f8986b3052"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}