{"paper":{"title":"Rigorous Construction of Stop-and-Go Waves in the Optimal Velocity Model via a Difference-Differential Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The optimal velocity model admits rigorous heteroclinic traveling waves and large-period periodic stop-and-go solutions for sufficiently steep velocity functions.","cross_cats":[],"primary_cat":"math.DS","authors_text":"Kota Ikeda, Tomoyuki Miyaji","submitted_at":"2026-05-15T05:28:42Z","abstract_excerpt":"Nonlinear wave phenomena such as stop-and-go traffic patterns are widely observed in vehicular flow but remain challenging to describe within a rigorous mathematical framework. Motivated by this, we investigate nonlinear wave structures in the optimal velocity (OV) model, which is a fundamental microscopic traffic flow model describing the car-following dynamics on a circuit. Using a traveling-wave formulation for vehicle headways, we reduce the original ordinary differential system to a difference-differential equation. We focus on steep OV functions approaching a step function, which generat"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We rigorously prove the existence of heteroclinic traveling waves for sufficiently steep OV functions, the existence of homoclinic solutions with a necessary condition on the amplitude parameter, and the existence of large-period periodic solutions comprising alternating transition layers and quasi-uniform states under the global road-length constraint.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The optimal velocity function must be sufficiently steep and approach a step function so that the singular-limit analysis produces sharp, explicitly constructible transition layers; this premise is invoked in the traveling-wave formulation and the singular-limit construction.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Rigorous existence proofs for heteroclinic, homoclinic, and periodic traveling waves in the optimal velocity model via singular-limit analysis of a difference-differential equation.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The optimal velocity model admits rigorous heteroclinic traveling waves and large-period periodic stop-and-go solutions for sufficiently steep velocity functions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9c0c866f7ce2c33f835082bd2d48a216d252d0336cb600d4606bd7f348d63745"},"source":{"id":"2605.15629","kind":"arxiv","version":1},"verdict":{"id":"074e94f5-5cf1-4886-aebe-9e81865301a1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T19:50:22.961739Z","strongest_claim":"We rigorously prove the existence of heteroclinic traveling waves for sufficiently steep OV functions, the existence of homoclinic solutions with a necessary condition on the amplitude parameter, and the existence of large-period periodic solutions comprising alternating transition layers and quasi-uniform states under the global road-length constraint.","one_line_summary":"Rigorous existence proofs for heteroclinic, homoclinic, and periodic traveling waves in the optimal velocity model via singular-limit analysis of a difference-differential equation.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The optimal velocity function must be sufficiently steep and approach a step function so that the singular-limit analysis produces sharp, explicitly constructible transition layers; this premise is invoked in the traveling-wave formulation and the singular-limit construction.","pith_extraction_headline":"The optimal velocity model admits rigorous heteroclinic traveling waves and large-period periodic stop-and-go solutions for sufficiently steep velocity functions."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15629/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T20:01:29.614056Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T20:01:19.274757Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T19:33:35.679602Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T17:41:56.028863Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"06afd3299aad20172c740acd5b80c8fef48babaaa0454300755241390bfc86cb"},"references":{"count":42,"sample":[{"doi":"","year":1979,"title":"S. M. Allen and J. W. Cahn. A microscopic theory for antiphase bou ndary motion and its application to antiphase domain coarsening. Acta Metallurgica, 27(6):1085–1095, 1979","work_id":"af934e1c-b06e-4f21-ac12-a578bbc3a738","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1995,"title":"M. Bando, K. Hasebe, K. Nakanishi, A. Nakayama, A. Shibata, an d Y. Sugiyama. Phenomenological study of dynamical model of traffic flow. Journal de Physique I , 5(11):1389–1399, 1995","work_id":"2af16585-aa62-4957-8b0e-8963be94af3b","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1994,"title":"M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama . Structure stability of congestion in traffic dynamics. Japan Journal of Industrial and Applied Mathematics , 11:203–223, 1994","work_id":"3f6cc708-8aa7-428d-b458-602b2b59a56f","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1995,"title":"M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama . Dynamical model of traffic conges- tion and numerical simulation. Physical Review E , 51(2):1035, 1995","work_id":"30954348-da6e-4826-80ca-5e2f1cac888d","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2002,"title":"N. Bellomo, M. Delitala, and V. Coscia. On the mathematical theory of vehicular traffic flow I: Fluid dynamic and kinetic modelling. 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