{"paper":{"title":"Long-time stability for nonlinear Maryland models","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"For almost every phase, small solutions to the nonlinear Maryland model remain of size order epsilon for times up to any negative power of epsilon.","cross_cats":["math.DS","math.MP"],"primary_cat":"math-ph","authors_text":"Ruijie Cui, Zhiyan Zhao","submitted_at":"2026-05-15T20:45:16Z","abstract_excerpt":"For the $d-$dimensional nonlinear Maryland model \\begin{equation}\\label{eq-abs} \\ri\\partial_t q_n=\\tan\\pi(n\\cdot\\varpi+x)q_n+\\epsilon(\\Delta q)_n+|q_n|^2q_n,\\quad n\\in{\\Z^d}, \\end{equation} with $d\\in\\N^*$, $\\epsilon\\in \\R$ and $\\varpi\\in\\R^d$ satisfying a suitable Diophantine condition, we establish polynomial long-time stability of polynomially weighted $\\ell^2$-norm $$\\|q(t)\\|_s:=\\left(\\sum_{n\\in{\\Z^d}}|q_n|^2 (1+|n|^2)^{s}\\right)^{\\frac{1}{2}},\\quad s>0. $$ More precisely, given any $M_*\\in\\N^*$, for phase parameters $x$ belonging to an almost full-measure subset of $\\R/\\Z$, if $|\\epsilon|"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Given any M_* ∈ N^*, for phase parameters x belonging to an almost full-measure subset of R/Z, if |ε| is sufficiently small, then solutions q(t) of the nonlinear Maryland model with sufficiently small initial weighted norm ε satisfy ||q(t)||_s = O(ε) for all |t| ≤ ε^{-1} ε^{-M_*}.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The frequency vector ϖ ∈ R^d satisfies a suitable Diophantine condition (as required for the Birkhoff normal form procedure to control resonances), together with the restriction to an almost full-measure set of phases x; if this condition fails, the normal form reduction and resulting stability bound may not hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Small solutions of the nonlinear Maryland model remain O(ε) in polynomially weighted ℓ² norm for times |t| ≤ ε^{-1} ε^{-M_*} under small ε and Diophantine conditions on ϖ for almost all x.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For almost every phase, small solutions to the nonlinear Maryland model remain of size order epsilon for times up to any negative power of epsilon.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"65c7b56d614829802768f4e03c3961f59dbf8cc518fc5d5e49559bfb88aa8ba5"},"source":{"id":"2605.16624","kind":"arxiv","version":1},"verdict":{"id":"0cdeaa76-bbd9-48f1-b98c-c080935db166","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T21:06:14.201810Z","strongest_claim":"Given any M_* ∈ N^*, for phase parameters x belonging to an almost full-measure subset of R/Z, if |ε| is sufficiently small, then solutions q(t) of the nonlinear Maryland model with sufficiently small initial weighted norm ε satisfy ||q(t)||_s = O(ε) for all |t| ≤ ε^{-1} ε^{-M_*}.","one_line_summary":"Small solutions of the nonlinear Maryland model remain O(ε) in polynomially weighted ℓ² norm for times |t| ≤ ε^{-1} ε^{-M_*} under small ε and Diophantine conditions on ϖ for almost all x.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The frequency vector ϖ ∈ R^d satisfies a suitable Diophantine condition (as required for the Birkhoff normal form procedure to control resonances), together with the restriction to an almost full-measure set of phases x; if this condition fails, the normal form reduction and resulting stability bound may not hold.","pith_extraction_headline":"For almost every phase, small solutions to the nonlinear Maryland model remain of size order epsilon for times up to any negative power of epsilon."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16624/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T21:31:19.413086Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T21:21:24.058709Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T19:21:56.761969Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.585878Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"f2cba5808823515910cf779e3dd67ca5a16585b037c9bc4dd4e9a3434f7360f5"},"references":{"count":36,"sample":[{"doi":"","year":1999,"title":"Bambusi : Nekhoroshev theorem for small amplitude solutions in nonlinear Schr\\\"odinger equations, Math","work_id":"415ed048-13ae-4bf5-a7af-756f56e3b846","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"height 2pt depth -1.6pt width 23pt: Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys., 234 (2003), pp. 253--285","work_id":"aa5f382d-e607-428c-9b1b-9280a4d1e7f0","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"D. Bambusi, J. Bernier, B. Gr\\'ebert and R. Imekraz : Almost global existence for Hamiltonian PDEs on compact manifolds, arXiv:2502.17969","work_id":"96d0b482-2ca6-4148-bb35-9038fc0726a8","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2007,"title":"D. Bambusi, J.-M. Delort, B. Gr\\'ebert and J. Szeftel : Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds, Comm. Pure Appl. Math., 60 (","work_id":"d357b291-0e3e-4cab-b2a0-b8af9c279f92","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"D. Bambusi and B. Gr\\'ebert : Forme normale pour NLS en dimension quelconque, C. R. Math. Acad. Sci. Paris, 337 (2003), pp. 409--414","work_id":"89a71bad-a326-4897-9b45-1a7d59510f08","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":36,"snapshot_sha256":"91eb3a8d3cdfbb7438508659745754f3d1c093a695b41d160b604b025a145228","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"086dfaec2c7bc29abdd535e28731202faf96fac81bc917fcc6a8d87fa0ca9ffa"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}