{"paper":{"title":"A Guide to Applications of $k$-Contact Geometry in Dissipative Field Equations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The k-contact Hamilton-De Donder-Weyl formalism supplies explicit Hamiltonian descriptions for many nonlinear dissipative PDEs.","cross_cats":["math.DG","math.MP"],"primary_cat":"math-ph","authors_text":"J. de Lucas, J. Lange, M. Krych","submitted_at":"2026-05-13T10:26:21Z","abstract_excerpt":"We study the practical scope of the $k$-contact Hamilton--De Donder--Weyl formalism as a geometric framework for dissipative field equations. In particular, our work focuses on canonical $k$-contact manifolds on $\\bigoplus^k {\\rm T}^*Q\\times\\mathbb{R}^k$ and $k$-contactifications of exact $k$-symplectic phase spaces. A special two-contactification of exact two-symplectic structures on cotangent bundles is defined and analysed. We also develop several tools for applications, including splitting results for the Hamilton--De Donder--Weyl equations on $k$-contactifications, regularity conditions f"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Our methods yield explicit Hamiltonian descriptions for several nonlinear nonconservative PDEs with polynomial dissipative terms, including damped Klein-Gordon, Allen-Cahn, generalized Burgers, porous medium equations with linear absorption, complex Ginzburg-Landau, damped nonlinear Schrödinger, Fisher-KPP, damped ϕ^4, damped sine-Gordon, and FitzHugh-Nagumo equations, and many others.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the k-contact Hamilton-De Donder-Weyl formalism on canonical k-contact manifolds and k-contactifications of exact k-symplectic phase spaces can be directly applied to produce well-defined Hamiltonian descriptions for the listed dissipative PDEs without additional ad-hoc adjustments.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"k-contact geometry supplies explicit Hamiltonian descriptions for multiple dissipative PDEs including damped Klein-Gordon, Allen-Cahn, Fisher-KPP, and complex Ginzburg-Landau equations.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The k-contact Hamilton-De Donder-Weyl formalism supplies explicit Hamiltonian descriptions for many nonlinear dissipative PDEs.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"04e8bb14ce7c11865c37a9ebbf871a74abad327ff721c5bd1598056cf29db7e4"},"source":{"id":"2605.13313","kind":"arxiv","version":1},"verdict":{"id":"4eb75e6f-13a0-42d8-bdbb-f96bacae7d8a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:27:15.147931Z","strongest_claim":"Our methods yield explicit Hamiltonian descriptions for several nonlinear nonconservative PDEs with polynomial dissipative terms, including damped Klein-Gordon, Allen-Cahn, generalized Burgers, porous medium equations with linear absorption, complex Ginzburg-Landau, damped nonlinear Schrödinger, Fisher-KPP, damped ϕ^4, damped sine-Gordon, and FitzHugh-Nagumo equations, and many others.","one_line_summary":"k-contact geometry supplies explicit Hamiltonian descriptions for multiple dissipative PDEs including damped Klein-Gordon, Allen-Cahn, Fisher-KPP, and complex Ginzburg-Landau equations.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the k-contact Hamilton-De Donder-Weyl formalism on canonical k-contact manifolds and k-contactifications of exact k-symplectic phase spaces can be directly applied to produce well-defined Hamiltonian descriptions for the listed dissipative PDEs without additional ad-hoc adjustments.","pith_extraction_headline":"The k-contact Hamilton-De Donder-Weyl formalism supplies explicit Hamiltonian descriptions for many nonlinear dissipative PDEs."},"references":{"count":70,"sample":[{"doi":"","year":2001,"title":"Pulsating solitons, chaotic soli- tons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg- Landau equation approach","work_id":"c3d6d527-34c1-4ca4-8a11-4df6c51a958b","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/0001-6160(79)90196-2","year":1979,"title":"Acta Metallurgica27(6), 1085–1095 (1979) https://doi.org/10.1016/0001-6160(79)90196-2","work_id":"797c4ed6-28f6-49c3-b439-035502517ed4","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1002/mma.9798","year":2024,"title":"Conservation laws and variational structure of damped nonlinear wave equations","work_id":"7c78c592-3a07-40d1-9848-767d55ad17b9","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1103/revmodphys.74","year":2002,"title":"The World of the Complex Ginzburg–Landau Equa- tion","work_id":"a9a6d6ad-5933-4b61-8478-a6df4cc6ed58","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/978-1-4757-1693-1","year":1989,"title":"V. I. Arnold.Mathematical Methods of Classical Mechanics. Vol. 60. Graduate Texts in Mathematics. 10.1007/978-1-4757-1693-1. New York: Springer, 1989.isbn: 0387968903","work_id":"3a2cc07b-d8b7-400c-8949-7db2642c424d","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":70,"snapshot_sha256":"e62aba49d95c1c0a948f3c1ecaca05cda84964381117b7058d53f50370a2a56c","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"}