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arxiv: 2405.17513 · v2 · submitted 2024-05-27 · 🧮 math-ph · math.AP· math.DS· math.MP

Anderson localized states for the quasi-periodic nonlinear Schr\"odinger equation on mathbb Z^d

Yunfeng Shi , W.-M. Wang This is my paper

Pith reviewed 2026-05-24 01:34 UTC · model grok-4.3

classification 🧮 math-ph math.APmath.DSmath.MP
keywords Anderson localizationquasi-periodic Schrödinger equationnonlinear lattice systemsDiophantine estimatesgeometric lemmaZ^d latticelocalized states
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The pith

Large sets of Anderson localized states exist for the quasi-periodic nonlinear Schrödinger equation on Z^d.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that Anderson localization, known in linear and random systems, also holds for nonlinear quasi-periodic Schrödinger equations on the integer lattice in any dimension. It constructs many spatially localized solutions that persist under the nonlinear interaction with a deterministic quasi-periodic potential. This matters because it shows localization can survive in more realistic nonlinear and deterministic settings. The proof introduces a new Diophantine estimate for quasi-periodic functions in high-dimensional phase space and applies a geometric lemma to control the nonlinear terms.

Core claim

We establish large sets of Anderson localized states for the quasi-periodic nonlinear Schrödinger equation on Z^d, thus extending Anderson localization from the linear to a nonlinear setting, and the random to a deterministic setting. Among the main ingredients are a new Diophantine estimate of quasi-periodic functions in arbitrarily dimensional phase space, and the application of Bourgain's geometric lemma.

What carries the argument

New Diophantine estimate of quasi-periodic functions in high-dimensional phase space, combined with Bourgain's geometric lemma to control nonlinear terms while preserving localization.

If this is right

  • The result holds for the nonlinear equation in any dimension d.
  • Large sets of such localized states can be constructed explicitly.
  • Anderson localization extends to deterministic quasi-periodic nonlinear systems.
  • The Diophantine estimate enables control over resonances induced by nonlinearity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same combination of estimates might apply to other nonlinear lattice models with almost-periodic driving.
  • Localized modes could remain stable in physical systems modeled by nonlinear Schrödinger equations with incommensurate potentials.
  • Time-dependent extensions with quasi-periodic forcing may be treatable by similar Diophantine control.

Load-bearing premise

The new Diophantine estimate for quasi-periodic functions holds in arbitrarily high-dimensional phase space and combines with the geometric lemma to keep nonlinear terms from destroying the localization.

What would settle it

A specific quasi-periodic potential and nonlinearity where the Diophantine condition fails, resulting in no localized states or delocalization due to the nonlinearity.

Figures

Figures reproduced from arXiv: 2405.17513 by W.-M. Wang, Yunfeng Shi.

Figure 1
Figure 1. Figure 1: The distribution of resonant blocks Finally, we define ΣN = S ℓ=1,2,3 Σ˜ N,ℓ with Σ˜ N,1, Σ˜ N,2, Σ˜ N,3 given by the analysis in R2, Lemma 2.29, Lemma 2.30, respectively. Then meas(ΣN ) ≤ 3e −2N3ρ ≪ e −N2ρ , and for σ /∈ ΣN , we can cover Λpm,N with σ-good regions of sizes N1, L0, L1, L3. More precisely, we have (cf [PITH_FULL_IMAGE:figures/full_fig_p032_1.png] view at source ↗
read the original abstract

We establish large sets of Anderson localized states for the quasi-periodic nonlinear Schr\"odinger equation on $\mathbb Z^d$, thus extending Anderson localization from the linear (cf. Bourgain [Geom. Funct. Anal., 17(3):682--706, 2007]) to a nonlinear setting, and the random (cf. Bourgain-Wang [J. Eur. Math. Soc., 10(1):1--45, 2008]) to a deterministic setting. Among the main ingredients are a new Diophantine estimate of quasi-periodic functions in arbitrarily dimensional phase space, and the application of Bourgain's geometric lemma in [Geom. Funct. Anal., 17(3):682--706, 2007].

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper claims to establish large sets of Anderson localized states for the quasi-periodic nonlinear Schrödinger equation on Z^d. This extends Anderson localization from the linear quasi-periodic setting (Bourgain, Geom. Funct. Anal. 2007) to the nonlinear case and from the random setting (Bourgain-Wang, J. Eur. Math. Soc. 2008) to a deterministic quasi-periodic one. The main ingredients are a new Diophantine estimate for quasi-periodic functions in arbitrarily high-dimensional phase space together with Bourgain's geometric lemma from the 2007 paper.

Significance. If the central claims hold, the result is significant: it supplies the first deterministic quasi-periodic nonlinear examples of Anderson localization on Z^d for arbitrary d, thereby closing a gap between linear/random and nonlinear/deterministic regimes. The new Diophantine estimate on quasi-periodic functions is presented as a technical tool that may be reusable beyond this application.

minor comments (2)
  1. The abstract and introduction cite Bourgain (2007) and Bourgain-Wang (2008) but do not explicitly state which lemmas from those works are invoked verbatim versus which are adapted; a short comparison table or paragraph in §1 would clarify the precise dependence.
  2. Notation for the frequency vector and the phase space dimension is introduced without a dedicated preliminary section; readers would benefit from an explicit list of standing assumptions on the Diophantine constants before the main theorem statement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for noting the potential significance of the result in extending Anderson localization to the nonlinear deterministic quasi-periodic setting on Z^d. The recommendation is listed as uncertain, yet the major comments section contains no specific points or concerns. We are happy to address any questions regarding the new Diophantine estimate or the application of Bourgain's geometric lemma.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper's central result rests on a newly derived Diophantine estimate for quasi-periodic functions in arbitrary-dimensional phase space combined with Bourgain's geometric lemma from an independent 2007 reference. The cited Bourgain-Wang 2008 work provides background on the random case but is not invoked as a load-bearing uniqueness theorem or ansatz that forces the present nonlinear deterministic extension; the new estimate is presented as original and external to the cited lemmas. No self-definitional reductions, fitted inputs renamed as predictions, or ansatzes smuggled via self-citation appear in the abstract or stated logic. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces a new Diophantine estimate whose validity is not independently verified in the abstract; it also invokes Bourgain's geometric lemma as a background tool. No free parameters or invented entities are evident from the abstract.

axioms (1)
  • domain assumption Bourgain's geometric lemma from Geom. Funct. Anal. 2007 applies directly to the nonlinear setting.
    Cited as a main ingredient without further justification in the abstract.

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Forward citations

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  2. Long-time stability for nonlinear Maryland models

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Reference graph

Works this paper leans on

51 extracted references · 51 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Aizenman and S

    M. Aizenman and S. Molchanov. Localization at large disorder and at extreme energies: an elementary derivation. Comm. Math. Phys. , 157(2):245--278, 1993

    work page 1993

  2. [2]

    D. Bambusi. Birkhoff normal form for some nonlinear PDE s. Comm. Math. Phys. , 234(2):253--285, 2003

    work page 2003

  3. [3]

    Berti and P

    M. Berti and P. Bolle. Quasi-periodic solutions with S obolev regularity of NLS on T^d with a multiplicative potential. J. Eur. Math. Soc. , 15(1):229--286, 2013

    work page 2013

  4. [4]

    Baldi, M

    P. Baldi, M. Berti, E. Haus, and R. Montalto. Time quasi-periodic gravity water waves in finite depth. Invent. Math. , 214(2):739--911, 2018

    work page 2018

  5. [5]

    Berti, L

    M. Berti, L. Biasco, and M. Procesi. K AM theory for the H amiltonian derivative wave equation. Ann. Sci. \' E c. Norm. Sup\' e r. (4) , 46(2):301--373 (2013), 2013

    work page 2013

  6. [6]

    Bambusi and S

    D. Bambusi and S. Graffi. Time quasi-periodic unbounded perturbations of S chr\" o dinger operators and KAM methods. Comm. Math. Phys. , 219(2):465--480, 2001

    work page 2001

  7. [7]

    Benettin, L

    G. Benettin, L. Galgani, and A. Giorgilli. A proof of N ekhoroshev's theorem for the stability times in nearly integrable H amiltonian systems. Celestial Mech. , 37(1):1--25, 1985

    work page 1985

  8. [8]

    Bourgain, M

    J. Bourgain, M. Goldstein, and W. Schlag. Anderson localization for S chr\" o dinger operators on Z^2 with quasi-periodic potential. Acta Math. , 188(1):41--86, 2002

    work page 2002

  9. [9]

    Berti, Z

    M. Berti, Z. Hassainia, and N. Masmoudi. Time quasi-periodic vortex patches of E uler equation in the plane. Invent. Math. , 233(3):1279--1391, 2023

    work page 2023

  10. [10]

    Berti, T

    M. Berti, T. Kappeler, and R. Montalto. Large KAM tori for perturbations of the defocusing NLS equation. Ast\' e risque , (403):viii+148, 2018

    work page 2018

  11. [11]

    Binyamini and D

    G. Binyamini and D. Novikov. Complex cellular structures. Ann. of Math. (2) , 190(1):145--248, 2019

    work page 2019

  12. [12]

    Bourgain

    J. Bourgain. Construction of quasi-periodic solutions for H amiltonian perturbations of linear equations and applications to nonlinear PDE . Internat. Math. Res. Notices , (11):475--497, 1994

    work page 1994

  13. [13]

    Bourgain

    J. Bourgain. Quasi-periodic solutions of H amiltonian perturbations of 2 D linear S chr\" o dinger equations. Ann. of Math. (2) , 148(2):363--439, 1998

    work page 1998

  14. [14]

    Bourgain

    J. Bourgain. Green's function estimates for lattice S chr\" o dinger operators and applications , volume 158 of Annals of Mathematics Studies . Princeton University Press, Princeton, NJ, 2005

    work page 2005

  15. [15]

    Bourgain

    J. Bourgain. On invariant tori of full dimension for 1 D periodic NLS . J. Funct. Anal. , 229(1):62--94, 2005

    work page 2005

  16. [16]

    Bourgain

    J. Bourgain. Anderson localization for quasi-periodic lattice S chr\" o dinger operators on Z^d , d arbitrary. Geom. Funct. Anal. , 17(3):682--706, 2007

    work page 2007

  17. [17]

    S. Basu, R. Pollack, and M.-F. Roy. On the combinatorial and algebraic complexity of quantifier elimination. J. ACM , 43(6):1002--1045, 1996

    work page 1996

  18. [18]

    Bourgain and W.-M

    J. Bourgain and W.-M. Wang. Quasi-periodic solutions of nonlinear random S chr\" o dinger equations. J. Eur. Math. Soc. , 10(1):1--45, 2008

    work page 2008

  19. [19]

    H. Cong, J. Liu, Y. Shi, and X. Yuan. The stability of full dimensional KAM tori for nonlinear S chr\" o dinger equation. J. Differential Equations , 264(7):4504--4563, 2018

    work page 2018

  20. [20]

    Craig and C

    W. Craig and C. E. Wayne. Newton's method and periodic solutions of nonlinear wave equations. Comm. Pure Appl. Math. , 46(11):1409--1498, 1993

    work page 1993

  21. [21]

    Carbery and J

    A. Carbery and J. Wright. Distributional and L^q norm inequalities for polynomials over convex bodies in R^n . Math. Res. Lett. , 8(3):233--248, 2001

    work page 2001

  22. [22]

    Chierchia and J

    L. Chierchia and J. You. K AM tori for 1 D nonlinear wave equations with periodic boundary conditions. Comm. Math. Phys. , 211(2):497--525, 2000

    work page 2000

  23. [23]

    L. H. Eliasson, B. Gr\' e bert, and S. B. Kuksin. K AM for the nonlinear beam equation. Geom. Funct. Anal. , 26(6):1588--1715, 2016

    work page 2016

  24. [24]

    L. H. Eliasson and S. B. Kuksin. K AM for the nonlinear S chr\" o dinger equation. Ann. of Math. (2) , 172(1):371--435, 2010

    work page 2010

  25. [25]

    L. H. Eliasson. Perturbations of linear quasi-periodic system. In Dynamical systems and small divisors ( C etraro, 1998) , volume 1784 of Lecture Notes in Math. , pages 1--60. Springer, Berlin, 2002

    work page 1998

  26. [26]

    Fr\" o hlich and T

    J. Fr\" o hlich and T. Spencer. Absence of diffusion in the A nderson tight binding model for large disorder or low energy. Comm. Math. Phys. , 88(2):151--184, 1983

    work page 1983

  27. [27]

    M. Gromov. Entropy, homology and semialgebraic geometry. Ast\' e risque , 145(146):225--240, 1987

    work page 1987

  28. [28]

    Gr\' e bert and L

    B. Gr\' e bert and L. Thomann. K AM for the quantum harmonic oscillator. Comm. Math. Phys. , 307(2):383--427, 2011

    work page 2011

  29. [29]

    J. Geng, J. You, and Z. Zhao. Localization in one-dimensional quasi-periodic nonlinear systems. Geom. Funct. Anal. , 24(1):116--158, 2014

    work page 2014

  30. [30]

    X. He, J. Shi, Y. Shi, and X. Yuan. On linear stability of KAM tori via the C raig- W ayne- B ourgain method. arXiv:2003.01487 , 2020

    work page arXiv 2003

  31. [31]

    Jitomirskaya, W

    S. Jitomirskaya, W. Liu, and Y. Shi. Anderson localization for multi-frequency quasi-periodic operators on Z ^d . Geom. Funct. Anal. , 30(2):457--481, 2020

    work page 2020

  32. [32]

    Kachkovskiy, W

    I. Kachkovskiy, W. Liu, and W.-M. Wang. Spacetime quasiperiodic solutions to a nonlinear S chr\" o dinger equation on Z . J. Math. Phys. , 65(1):Paper No. 011502, 34, 2024

    work page 2024

  33. [33]

    D. Y. Kleinbock and G. A. Margulis. Flows on homogeneous spaces and D iophantine approximation on manifolds. Ann. of Math. (2) , 148(1):339--360, 1998

    work page 1998

  34. [34]

    o schel. Invariant C antor manifolds of quasi-periodic oscillations for a nonlinear S chr\

    S. Kuksin and J. P\" o schel. Invariant C antor manifolds of quasi-periodic oscillations for a nonlinear S chr\" o dinger equation. Ann. of Math. (2) , 143(1):149--179, 1996

    work page 1996

  35. [35]

    S. B. Kuksin. Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum. Funktsional. Anal. i Prilozhen. , 21(3):22--37, 95, 1987

    work page 1987

  36. [36]

    W. Liu. Quantitative inductive estimates for G reen's functions of non-self-adjoint matrices. Anal. PDE , 15(8):2061--2108, 2022

    work page 2061

  37. [37]

    S. Liu, Y. Shi, and Z. Zhang. Anderson localized states for the nonlinear M aryland model on Z ^d . arXiv:2502.16397 , 2025

    work page arXiv 2025

  38. [38]

    Liu and W.-M

    W. Liu and W.-M. Wang. Nonlinear A nderson localized states at arbitrary disorder. Comm. Math. Phys. , 405(11):Paper No. 272, 48, 2024

    work page 2024

  39. [39]

    Liu and X

    J. Liu and X. Yuan. Spectrum for quantum D uffing oscillator and small-divisor equation with large-variable coefficient. Comm. Pure Appl. Math. , 63(9):1145--1172, 2010

    work page 2010

  40. [40]

    Liu and X

    J. Liu and X. Yuan. A KAM theorem for H amiltonian partial differential equations with unbounded perturbations. Comm. Math. Phys. , 307(3):629--673, 2011

    work page 2011

  41. [41]

    Procesi and M

    C. Procesi and M. Procesi. A KAM algorithm for the resonant non-linear S chr\" o dinger equation. Adv. Math. , 272:399--470, 2015

    work page 2015

  42. [42]

    A. S. Pyartli. Diophantine approximations on submanifolds of E uclidean space. Functional Analysis and Its Applications , 3(4):303--306, 1969

    work page 1969

  43. [43]

    Y. Shi. Spectral theory of the multi-frequency quasi-periodic operator with a G evrey type perturbation. J. Anal. Math. , 148(1):305--338, 2022

    work page 2022

  44. [44]

    Anderson localized states for the quasi-periodic nonlinear wave equation on $\mathbb Z^d$

    Y. Shi and W.-M. Wang. Anderson localized states for the quasi-periodic nonlinear wave equation on Z ^d . arXiv:2306.00513 , 2023

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  45. [45]

    W.-M. Wang. Energy supercritical nonlinear S chr\" o dinger equations: quasiperiodic solutions. Duke Math. J. , 165(6):1129--1192, 2016

    work page 2016

  46. [46]

    W.-M. Wang. Quasi-periodic solutions to a nonlinear K lein- G ordon equation with a decaying nonlinear term. arXiv:1609.00309 , 2021

    work page arXiv 2021

  47. [47]

    W.-M. Wang. Semi-algebraic sets method in PDE and mathematical physics. J. Math. Phys. , 62(2):Paper No. 021506, 12, 2021

    work page 2021

  48. [48]

    C. E. Wayne. Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Comm. Math. Phys. , 127(3):479--528, 1990

    work page 1990

  49. [49]

    J. Xu, J. You, and Q. Qiu. Invariant tori for nearly integrable H amiltonian systems with degeneracy. Math. Z. , 226(3):375--387, 1997

    work page 1997

  50. [50]

    X. Yuan. Construction of quasi-periodic breathers via KAM technique. Comm. Math. Phys. , 226(1):61--100, 2002

    work page 2002

  51. [51]

    X. Yuan. K AM theorem with normal frequencies of finite limit points for some shallow water equations. Comm. Pure Appl. Math. , 74(6):1193--1281, 2021

    work page 2021