Pith Number

pith:O6UCFEME

pith:2026:O6UCFEMEWLNMDAJPFWSJPHT5OT

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Long-time stability for nonlinear Maryland models

Ruijie Cui, Zhiyan Zhao

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.

arxiv:2605.16624 v1 · 2026-05-15 · math-ph · math.DS · math.MP

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Record completeness

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Claims

C1strongest 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_*}.

C2weakest 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.

C3one 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.

References

36 extracted · 36 resolved · 1 Pith anchors

[1] Bambusi : Nekhoroshev theorem for small amplitude solutions in nonlinear Schr\"odinger equations, Math 1999

[2] height 2pt depth -1.6pt width 23pt: Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys., 234 (2003), pp. 253--285 2003

[3] D. Bambusi, J. Bernier, B. Gr\'ebert and R. Imekraz : Almost global existence for Hamiltonian PDEs on compact manifolds, arXiv:2502.17969

[4] 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 ( 2007

[5] D. Bambusi and B. Gr\'ebert : Forme normale pour NLS en dimension quelconque, C. R. Math. Acad. Sci. Paris, 337 (2003), pp. 409--414 2003

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:02:33.042830Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

77a8229184b2dac1812f2da4979e7d74cac32de68e040c05790c03109a5651ee

Aliases

arxiv: 2605.16624 · arxiv_version: 2605.16624v1 · doi: 10.48550/arxiv.2605.16624 · pith_short_12: O6UCFEMEWLNM · pith_short_16: O6UCFEMEWLNMDAJP · pith_short_8: O6UCFEME

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/O6UCFEMEWLNMDAJPFWSJPHT5OT \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 77a8229184b2dac1812f2da4979e7d74cac32de68e040c05790c03109a5651ee

Canonical record JSON

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    "abstract_canon_sha256": "ae1b57c5a16dc462f1370905eeadf32497eeec7401a4665658c0c9bdca814619",
    "cross_cats_sorted": [
      "math.DS",
      "math.MP"
    ],
    "license": "http://creativecommons.org/publicdomain/zero/1.0/",
    "primary_cat": "math-ph",
    "submitted_at": "2026-05-15T20:45:16Z",
    "title_canon_sha256": "b228f3dfde306a36e1bbd7f4c2df4a3f318f9da3dcd3199056a0e0d583f87986"
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  "source": {
    "id": "2605.16624",
    "kind": "arxiv",
    "version": 1
  }
}