{"paper":{"title":"A study of variational single solitary waves governed by the conservative-extended KdV equation with applications to shallow water dispersive shocks","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Variational methods produce simple solitary wave solutions for the conservative extended KdV equation that agree with numerical simulations and apply to shallow water dispersive shocks.","cross_cats":["physics.flu-dyn"],"primary_cat":"nlin.PS","authors_text":"Hamid Said, Saleh Baqer","submitted_at":"2026-05-13T18:36:46Z","abstract_excerpt":"The extended KdV equation is a nonlinear dispersive wave model that is asymptotically or variationally derived from the full dispersive Euler shallow water waves equations when gravity-capillary and higher order nonlinear effects are taken into account, under weakly nonlinear and long-wave approximations. This reduction introduces four additional terms beyond the classical KdV equation: a nonlinear term (quadratic nonlinearity), two nonlinear-dispersive terms, and a fully dispersive term (fifth order dispersion). In this paper, we employ a variational approach based on averaged Lagrangians to "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Theoretical predictions show excellent agreement with numerical simulations for the solitary wave solutions applied to shallow water classical undular bores and non-classical resonant dispersive shocks.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The extended KdV equation with the four additional terms accurately captures the dynamics of the full dispersive Euler shallow water equations under the weakly nonlinear and long-wave approximations.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Variational solitary wave solutions for the conservative extended KdV equation agree with numerics and apply to shallow water dispersive shocks.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Variational methods produce simple solitary wave solutions for the conservative extended KdV equation that agree with numerical simulations and apply to shallow water dispersive shocks.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0cfc0f5192f3fe86764afdc7bb55fa2a98ba0fae012c65baa62a82943f2a3431"},"source":{"id":"2605.14024","kind":"arxiv","version":1},"verdict":{"id":"11fa2444-0498-4cbf-a1d7-b12580949d87","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T05:53:09.872980Z","strongest_claim":"Theoretical predictions show excellent agreement with numerical simulations for the solitary wave solutions applied to shallow water classical undular bores and non-classical resonant dispersive shocks.","one_line_summary":"Variational solitary wave solutions for the conservative extended KdV equation agree with numerics and apply to shallow water dispersive shocks.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The extended KdV equation with the four additional terms accurately captures the dynamics of the full dispersive Euler shallow water equations under the weakly nonlinear and long-wave approximations.","pith_extraction_headline":"Variational methods produce simple solitary wave solutions for the conservative extended KdV equation that agree with numerical simulations and apply to shallow water dispersive shocks."},"references":{"count":56,"sample":[{"doi":"","year":1974,"title":"G.B. Whitham, Linear and Nonlinear Waves, J. Wiley and Sons, New York (1974)","work_id":"713fd956-fb33-4c2b-9842-f84d6841721c","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2000,"title":"Kamchatnov, Nonlinear periodic waves and their modulations: an introdu ctory course, World Scientific, Singapore (2000)","work_id":"779def11-811c-4021-a4e8-1df2f76d476d","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"Shallow water waves–exten ded Korteweg-de Vries equa- tions,","work_id":"6f57edb7-2b56-448e-b7e1-cbee0c3b1e96","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2000,"title":"I.M. Gelfand and S.V. Fomin, Calculus of variations , Revised and translated by R. A. Silverman, Dover Publications, New York (2000)","work_id":"49c30910-eeb8-4fa9-8361-598158250a0f","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2000,"title":"Trefethen, Spectral methods in MATLAB","work_id":"fd0cf34b-2cba-4163-9354-80c3887c5d41","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":56,"snapshot_sha256":"de3ed942d856d2425c40f7637c16c35719ff8171b733d31fe8543514e7c5f57a","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"9f7d455075de8920c720cc8d65d21ec7fa6d9189102fd43f86b04240dc487561"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}