Borel-Pad\'e exponential asymptotics for the discrete nonlinear Schr\"odinger model with next-to-nearest neighbour interactions

Christopher J. Lustri; In\^es Aniceto; Panayotis G. Kevrekidis

REVIEW 3 major objections 5 minor 2 cited by

Borel-Pad\'e exponential asymptotics for the discrete nonlinear Schr\"odinger model with next-to-nearest neighbour interactions

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper derives the exponentially small translational eigenvalue that decides stability for on-site and inter-site standing waves in a discrete nonlinear Schrödinger lattice with competing next-to-nearest-neighbour coupling, and proves…

desk verdict Solid exponential-asymptotics paper with a real new formula and a real numerical-load-bearing step that is asserted rather than proven. read the letter →

arxiv 2506.21120 v1 pith:HBWACZFA submitted 2025-06-26 nlin.PS math-phmath.DSmath.MPnlin.SI

Christopher J. Lustri , Inês Aniceto , Panayotis G. Kevrekidis This is my paper · ORCID

classification nlin.PSmath-phmath.DSmath.MPnlin.SI MSC 34E0534E2037K60

keywords discretenonlinearSchrödingerequationnext-to-nearest-neighbourinteractionsexponentialasymptoticsStokesconstantsBorel-PadéapproximationstandingwavestranslationaleigenvaluePeierls-Nabarrobarrier

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies stationary solitary waves in a one-dimensional discrete nonlinear Schrödinger lattice with both nearest-neighbour and next-to-nearest-neighbour coupling, focusing on competing interactions, μ < 0. It claims that for -1/4 < μ < 0 the only standing waves are site-centred and inter-site-centred pulses, and that the translational eigenvalue controlling their stability is exponentially small in the lattice spacing ε. The sign of λ² is fixed by the pinning site: imaginary for on-site states (stable) and real for inter-site states (unstable), with magnitude $|\lambda| \sim 2^{5/4} 5^{1/2} \pi \beta^2 |\Lambda(\mu)|^{1/2} \varepsilon^{-5/2} e^{-\beta\pi^2/(2\varepsilon)}$ up to an explicit first correction. The paper also claims that no localized standing waves exist for μ < -1/4, because the competing next-to-nearest-neighbour interactions produce an oscillatory tail that no choice of site offset can cancel. These results matter because the eigenvalue depends on a Stokes constant that ordinary matched asymptotic expansions cannot reach, and the paper computes it with conformal Borel-Padé methods.

What carries the argument

The central object is the Borel transform of the inner transseries solution near the complex singularities of the leading-order sech pulse, combined with a conformal map, $\zeta = A_1 - (\xi - A_1)^2$, that converts the logarithmic branch point at $\zeta = A_1 = 2\pi i$ into a simple pole. The residue of that pole, extracted from a diagonal Padé approximant built from the first N even coefficients of the algebraic series, gives the Stokes constant $S_1$ through the identity $S_1 = 2i\Gamma(1/2) a_0^{(0)-1} \mathrm{Res}$. Matching this inner Stokes constant to the outer late-order constant via $2\pi i \Lambda(\mu) = S_1$ fixes the eigenvalue prefactor. This machinery is necessary because the χ₁,± contribution is beyond all orders even within the inner region, so it cannot be obtained by matched asymptotic expansions; the closer χ₂,± singulants must be resolved to isolate the pole at A₁.

What would settle it

Compute the residue defining $S_1$ for a fixed μ in (-1/4,0) using several Padé orders, such as N=150, 300, and 400, and different conformal maps, then compare the resulting $|\lambda|$ against high-precision numerical eigenvalues of the full lattice problem across a decade of ε values; if the residue drifts with N or the scaled quantity $\varepsilon^{5/2} e^{\beta\pi^2/(2\varepsilon)} \lambda$ does not approach the predicted constant, the central claim fails. Equivalently, a numerical search that finds a localized standing wave for μ just below -1/4 would refute the non-existence claim.

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Extended reading notes

Core claim

The central claim is that for μ in (-1/4,0), the translational eigenvalue λ of both on-site and inter-site standing waves is exponentially small, satisfying $|\lambda| \sim 2^{5/4} 5^{1/2} \pi \beta^2 |\Lambda(\mu)|^{1/2} \varepsilon^{-5/2} e^{-\beta\pi^2/(2\varepsilon)} \left(1 - \frac{\varepsilon \pi^2 (1+16\mu)}{48\beta^3}\right)$ as ε → 0, where $\beta = \sqrt{1+4\mu}$ and $\Lambda(\mu)$ is the Stokes constant fixed by a Borel-Padé residue calculation. The sign of λ² is determined by $\cos(2\pi(n-n_0))$: for on-site pinning, n₀ integer, λ² < 0 and the mode is imaginary and stable; for inter-site pinning, n₀ half-integer, λ² > 0 and the mode is real and unstable. The paper further establishes that for μ < -1/4 no localized standing wave exists, since the χ₂,± exponential contributions oscillate with a frequency that cannot be made commensurate with the lattice for either pinning choice. These claims are validated by numerical eigenvalue computations covering several negative μ values, including μ = -1/16, where the first correction term in ε vanishes exactly.

Load-bearing premise

The whole eigenvalue prediction rests on the assumption that the truncated diagonal Padé approximant of the conformally mapped Borel transform has converged to the true branch point at ζ=2πi and that its residue equals the actual Stokes constant $S_1$; the paper checks agreement between two truncation orders but does not prove convergence.

Editorial extensions

If this is right

For every -1/4 < μ < 0, the site-centred ground state has an imaginary translational eigenvalue pair and is linearly stable, while the inter-site state has a real pair and is unstable, uniformly in the small-ε regime covered by the asymptotic formula.
The Peierls-Nabarro barrier, set by the energy difference between the two families of standing waves, is exponentially small in the lattice spacing, scaling as $e^{-\beta\pi^2/\varepsilon}$ with the explicit prefactor and ε-correction computed in the paper.
At μ = -1/16, the model coincides with the fourth-order finite-difference discretization of the continuous nonlinear Schrödinger equation, and the first correction term in the eigenvalue expansion vanishes exactly.
For μ < -1/4, no localized standing wave exists because the competing next-to-nearest-neighbour interactions generate an oscillatory tail whose frequency cannot be made commensurate with the lattice by any site offset, so the tail cannot be cancelled.
The conformal Borel-Padé residue method provides a general template for extracting subdominant Stokes constants in parametric asymptotic problems where the relevant exponential is hidden beyond all orders in the inner expansion.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the residue-to-Stokes-constant identification is accepted, the same conformal Borel-Padé procedure should compute $\Lambda(\mu)$ for the cooperative case μ > 0 to comparable accuracy; the paper includes one positive-μ example but leaves a systematic study to future work.
Should the non-existence claim hold, the threshold μ = -1/4 should be observable in zigzag waveguide arrays: for μ just above the threshold the lattice supports pinned pulses, while below it localized pulses should fail to form.
Because the eigenvalue formula relies on a numerical Padé residue, a future exact derivation of $\Lambda(\mu)$ through a closed-form transseries or exact WKB analysis would turn the numerical evidence into proof and would also delimit the ε range where the first correction remains accurate.
The same conformal Borel-Padé template likely extends to other discrete nonlinear models with competing interactions, where the stability eigenvalue is exponentially small but ordinary matching cannot reach it; each extension would require its own singulant set and conformal map.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a solid applied-asymptotics paper with one load-bearing numerical step that rests on an assertion rather than a proof. The new results are real: the first exponentially-small eigenvalue formula for the NNN DNLS model, the beta = sqrt(1+4mu) structure, the existence boundary at mu = -1/4, and the first-order correction that vanishes at mu = -1/16. The derivation follows the standard factorial-over-power machinery cleanly: the singulant equation (15), the U equation (21), the Stokes switching in Appendix A, and the eigenvalue balance in (63)-(65) are coherent and cross-check against each other. The stability conclusion (on-site stable, inter-site unstable) for -1/4 < mu < 0 is exactly what the numerics show, and the mu = -1/16 cancellation is a nice touch.

The soft spot is exactly where the stress-test note points: the Stokes constant S1, and hence Lambda(mu), comes from the conformal Borel-Pade residue (50). The paper infers convergence by comparing N=300 with N=400 at a few representative mu values, and it admits in Section 4.4 that accuracy degrades as |mu| increases because the A2,pm branch points sit closer to the origin than A1. That admission is honest but it underscores the gap: there is no error estimate, no proof of convergence for the Pade approximants to the branch point, and no shipped code or coefficient data to reproduce the residue calculation. Since the eigenvalue formula (70) scales as sqrt(|Lambda(mu)|), any error in that residue propagates directly into the central prediction. The mu=0 check against [29] is genuine supporting evidence, but it is a single point, and the interesting regime is -1/4 < mu < 0.

That said, I would not call this manufactured. The derivation is careful, the numerics in Figure 5 are convincing across several mu values, and the general Borel-Pade template for subdominant Stokes multipliers is a useful methodological contribution. The circularity burden is low: nothing is fitted to the eigenvalue data. The paper deserves a serious referee.

Recommendation: send to peer review, but make acceptance conditional on the authors releasing the code and the coefficient series, and on a more systematic convergence study for the residue—for example, a range of N with error scaling, or an independent way of estimating S1 for at least one mu in the interval. If that holds up, this is publishable in SIAM J. Appl. Math. or similar.

Referee Report

3 major / 5 minor

Summary. The paper derives an exponentially small eigenvalue for the translational mode of on-site and inter-site standing waves in the DNLS equation with next-to-nearest-neighbor interactions. Using factorial-over-power exponential asymptotics, the authors identify two singulant families, \chi_1 and \chi_2, and show that the Stokes multiplier associated with \chi_1 is beyond all orders even in the inner region. They compute it via a conformal Borel-Pad\'e residue calculation, match it to the late-order constant \Lambda(\mu), and obtain the leading-order eigenvalue (65) and first correction (70). They also argue that no localized standing waves exist for \mu<-1/4. The prediction is compared with direct numerical eigenvalue computations for several \mu, including \mu=-1/16 and \mu=-1/30.

Significance. If the result holds, the paper makes a valuable methodological and physical contribution. The eigenvalue formula (70) is parameter-free, with no fitted constants; the \mu=0 limit reproduces the known nearest-neighbour result, and the independent numerical eigenvalue computations confirm the exponential scaling and the prefactor for representative values. The Borel-Pad\'e strategy for computing Stokes multipliers that are hidden beyond all orders in the inner expansion is a useful template for other discrete models. The main caveat, acknowledged by the authors, is the numerical nature of the Stokes constant, which is load-bearing for the quantitative prediction.

major comments (3)

[Section 4.4, Eq. (50)] The central prefactor in the eigenvalue prediction (65)/(70) is entirely determined by the Stokes constant S1, which is computed numerically as the residue of a diagonal Pad\'e approximant of order N=150\u2013400. The only convergence evidence offered is a comparison of N=300 and N=400 at representative values of \mu (Figure 4), and the paper itself states in Section 4.4 that accuracy degrades as |\mu| increases. Because |\lambda| \propto |\Lambda(\mu)|^{1/2}, an uncontrolled error in the residue propagates directly into the main prediction. I would like to see a quantitative convergence study, for example a table of S1(\mu;N) for several N and for \mu spread over the interval (-1/4,0), with estimated error bars, and ideally the coefficients a_\ell^{(0)} or the code used to generate them made available. The \mu=0 consistency check is reassuring but does not by itself establish reliability in the interval relevant to Eq. (70).
[Section 3.4] The claim that no localized standing waves exist for \mu<-1/4 rests on the statement that the \chi_2 oscillation, proportional to \sin(\omega(n-n0)) with 0<\omega<\pi, cannot be eliminated for any site offset. This conclusion assumes that the Stokes multiplier for the \chi_2 contribution is nonzero and that its amplitude is independent of n0; neither is demonstrated in the paper for \mu<-1/4. Since the nonexistence result is a headline claim of the abstract, please provide either a direct computation of the \chi_2 Stokes multiplier in Regime 1 or a numerical demonstration of the absence of stationary localized solutions for at least one \mu<-1/4.
[Section 5, Eq. (70)] The formula is stated for \mu in (-1/4,0), but as \mu \to -1/4, \beta \to 0 and the correction term \epsilon \pi^2(1+16\mu)/(48\beta^3) diverges unless \epsilon is taken extremely small relative to \beta^3. The paper does not discuss this non-uniformity, and Figure 4 suggests S1 varies significantly near that endpoint. Please comment on the expected range of validity in \mu and on the fate of the asymptotic prediction as \mu approaches -1/4; at minimum, state that \mu must be bounded away from -1/4 for the expansion to apply.

minor comments (5)

[Eq. (67)] The O(\epsilon) term is written as \epsilon\pi(1+16\mu)/(24\beta^2\sqrt{2}\,\eta), but the expansion near the singularity gives this term with \eta^2 in the denominator; the subsequent pole-shift calculation in Section 5.2 is consistent with the \eta^{-2} form. Please correct the displayed equation.
[Eqs. (25), (26), (65), (70)] There are typographical issues in the denominator notation: the bracket [2\pi i(\tilde z-\tilde z_s)2j+4] should read [2\pi i(\tilde z-\tilde z_s)]^{2j+4}, and the prefactor 25/451/2 should be rendered as 2^{5/4}5^{1/2}.
[Figure 3 caption] The caption appears to label both panels as (a); the second mention should be (b).
[Eq. (3)] The display of the fourth-order central difference operator contains an apparent duplication of the term -1/12 F_n and an unusual coefficient -5/2 F_n; please verify the coefficients.
[Section 4.4] The phrase "S1 decays rapidly as \mu grows in the negative direction" is ambiguous; it should say "as \mu decreases towards -1/4" or similar.

Circularity Check

0 steps flagged · score 0.0 of 10

The derivation is self-contained: the eigenvalue prediction is not fitted to the eigenvalue data, and the cited methods are external rather than load-bearing self-citations.

full rationale

The central chain is: outer late-order ansatz (13)/(25) introduces an undetermined constant Lambda(mu); the inner transseries (33) is solved independently from the inner equation (27), and its Borel transform (44) with conformal map (48) gives the Stokes constant S_1 via the residue formula (50); the matching relation (51) then fixes 2*pi*i*Lambda(mu) = S_1. Nothing in this chain fits Lambda(mu) to the eigenvalues predicted in (65) or (70). The predicted eigenvalue is subsequently compared with independent numerical computations in Section 6, including a rescaling that exposes exponential error. The mu = 0 check against the matched-asymptotics result [29] is an external consistency test, not a circular normalization. The Borel-Padé methodology is attributed to [3] (which shares an author), but it is a published technique also supported by independent references [6,9,21], and the convergence of the Padé residue to S_1 is a numerical-convergence assumption, not a reduction of the prediction to its input. No self-definitional, fitted-input, or self-citation-chain circularity is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard exponential-asymptotics assumptions and on the numerical convergence of a Padé-based residue computation. No parameters are fitted to the eigenvalue data being predicted, and no new physical entities are introduced.

assumptions (4)

domain assumption The late-order terms of the outer asymptotic series take the factorial-over-power form (13) and are dominated by the singulants with smallest |χ'|.
Invoked in Section 3.1 to derive χ and U; standard in exponential asymptotics but unproven for the NNN lattice.
domain assumption The inner equation (27) possesses a complete transseries (32)/(33) whose Stokes constants can be extracted from the Borel transform of the algebraic series.
Used in Section 4.1; the paper does not prove that the truncated transseries captures all exponentials, only that the leading ones match the outer singulants.
domain assumption The diagonal Padé approximant of the conformally mapped Borel transform converges to the true function near ζ = A₁, so the residue in Eq. (50) gives S₁.
Sections 4.3-4.4; convergence is checked numerically by comparing N = 300 and N = 400, not proven, and this is the weakest link.
domain assumption The matching between the inner Stokes constant S₁ and the outer late-order constant Λ(µ) stated in Eq. (51) is valid.
Section 4.4, after equation (51); this transfer converts the Borel-Padé residue into the eigenvalue formula.

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Cite this review

Pith. "Pith review of Borel-Pad\'e exponential asymptotics for the discrete nonlinear Schr\"odinger model with next-to-nearest neighbour interactions." pith.science (2026). https://pith.science/paper/HBWACZFA

@misc{pith2026250621120,
  author       = {Pith},
  title        = {Pith review of: Borel-Pad\'e exponential asymptotics for the discrete nonlinear Schr\"odinger model with next-to-nearest neighbour interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HBWACZFA}},
  note         = {Machine review of arXiv:2506.21120}
}

abstract

In the present work we study discrete nonlinear Schr{\"o}dinger models combining nearest (NN) and next-nearest (NNN) neighbor interactions, motivated by experiments in waveguide arrays. While we consider the more experimentally accessible case of positive ratio $\mu$ of NNN to NN interactions, we focus on the intriguing case of competing such interactions $(\mu<0)$, where stationary states can exist only for $-1/4 < \mu < 0$. We analyze the key eigenvalues for the stability of the pulse-like stationary (ground) states, and find that such modes depend exponentially on the coupling parameter $\eps$, with suitable polynomial prefactors and corrections that we analyze in detail. Very good agreement of the resulting predictions is found with systematic numerical computations of the associated eigenvalues. This analysis uses Borel-Pad\'{e} exponential asymptotics to determine Stokes multipliers in the solution; these multipliers cannot be obtained using standard matched asymptotic expansion approaches as they are hidden beyond all asymptotic orders, even near singular points. By using Borel-Pad\'{e} methods near the singularity, we construct a general asymptotic template for studying parametric problems which require the calculation of subdominant Stokes multipliers.

Figures

Figures reproduced from arXiv: 2506.21120 by the authors.

**Figure 2.** Stokes structure for the exponential contributions associated with (a) [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** (a) Singularity structure of the Borel transform of the algebraic series solution for [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Estimates of S1 obtained for different values of µ with N = 300, including some positive values. The estimated accuracy is shown for representative cases, obtained by comparing N = 300 to N = 400 and determining how many decimal places remained constant. As |µ| increas…

**Figure 5.** Figure 5: Asymptotic vs Numerical computation of the relevant exponentially small imaginary eigenvalue [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

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Works this paper leans on

37 extracted references · 33 canonical work pages · cited by 2 Pith papers

[29]

C. J. Lustri, P. G. Kevrekidis, and S. J. Chapman , Exponential asymptotics for translational modes in the discrete nonlinear Schr¨ odinger model, Q. Appl. Math. (In Press), (2025)

work page 2025
[1]

M. J. Ablowitz and J. T. Cole, Nonlinear optical waveguide lattices: Asymptotic analysis, solitons, and topological insulators, Physica D, 440 (2022), p. 133440, https://doi.org/https://doi.org/10. 1016/j.physd.2022.133440

work page arXiv 2022
[2]

Aniceto, G

I. Aniceto, G. Bas ¸ar, and R. Schiappa, A primer on resurgent transseries and their asymptotics , Phys. Rep., 809 (2019), pp. 1–135

work page 2019
[3]

Aniceto, J

I. Aniceto, J. Jankowski, B. Meiring, and M. Spali ´nski, The large proper-time expansion of Yang-Mills plasma as a resurgent transseries , J. High Energy Phys., 2019 (2019), pp. 1–36

work page 2019
[4]

Aniceto, R

I. Aniceto, R. Schiappa, and M. Vonk , The resurgence of instantons in string theory , Commun. Number Theory Phys., 6 (2012), pp. 339–496

work page 2012
[5]

V. A. Brazhnyi and V. V. Konotop , Theory of nonlinear matter waves in optical lattices , Mod. Phys. Lett. B, 18 (2004), pp. 627–651, https://doi.org/10.1142/S0217984904007190

work page doi:10.1142/s0217984904007190 2004
[6]

Caliceti, M

E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura , From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions, Phys. Rep., 446 (2007), pp. 1–96. 17

work page 2007
[7]

S. J. Chapman, J. R. King, and K. L. Adams, Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations, Proc. Roy. Soc. Lond. A, 454 (1998), pp. 2733–2755

work page 1998

Show all 37 references

[8]

Chong and P

C. Chong and P. G. Kevrekidis , Coherent Structures in Granular Crystals: From Experiment and Modelling to Computation and Mathematical Analysis , Springer International Publishing, 2018

2018
[9]

Costin and G

O. Costin and G. V. Dunne , Conformal and uniformizing maps in Borel analysis , The European Physical Journal Special Topics, 230 (2021), pp. 2679–2690

2021
[10]

Cruickshank, F

R. Cruickshank, F. Lorenzi, A. La Rooij, E. Kerr, T. Hilker, S. Kuhr, L. Salasnich, and E. Haller , Single-site and multi-site solitons of bright matter-waves in optical lattices , 2025, https://arxiv.org/abs/2504.11046

2025
[11]

Di Pietro and M

L. Di Pietro and M. Serone , Looking through the QCD conformal window with perturbation theory , J. High Energy Phys., 2020 (2020), pp. 1–35

2020
[12]

R. B. Dingle , Asymptotic Expansions: Their Derivation and Interpretation , Academic Press, New York, NY, USA, 1973

1973
[13]

Dreisow, A

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. T ¨unnermann, Second- order coupling in femtosecond-laser-written waveguide arrays , Opt. Lett., 33 (2008), pp. 2689–2691, https://doi.org/10.1364/OL.33.002689

2008 doi
[14]

Dreisow, G

F. Dreisow, G. W ang, M. Heinrich, R. Keil, A. T ¨unnermann, S. Nolte, and A. Szameit , Observation of anharmonic bloch oscillations , Opt. Lett., 36 (2011), pp. 3963–3965, https://doi.org/ 10.1364/OL.36.003963

2011 doi
[15]

N. K. Efremidis and D. N. Christodoulides, Discrete solitons in nonlinear zigzag optical waveguide arrays with tailored diffraction properties , Phys. Rev. E, 65 (2002), p. 056607, https://doi.org/10. 1103/PhysRevE.65.056607

2002
[16]

J. C. Eilbeck and M. Johansson , The discrete nonlinear Schr¨ odinger equation - 20 years on , Lo- calization and Energy Transfer in Nonlinear Systems, (2003), pp. pp. 44–67

2003
[17]

Eynard, E

B. Eynard, E. Garcia-F ailde, P. Gregori, R. Schiappa, and D. Lewanski , Resurgent asymptotics of jackiw-teitelboim gravity and the nonperturbative topological recursion , 2025, https: //arxiv.org/abs/2305.16940

2025 arXiv
[18]

Graves-Morris and G

P. Graves-Morris and G. Baker , Pad´ e approximants, encyclopedia of mathematics, 1981

1981
[19]

M. P. Heller, A. Serantes, M. Spali ´nski, V. Svensson, and B. Withers , Relativistic hydro- dynamics: a singulant perspective , Phys. Rev. X, 12 (2022), p. 041010

2022
[20]

J. Hu, S. Li, Z. Chen, J. L ¨u, B. Liu, and Y. Li , Discrete solitons in zigzag waveguide arrays with different types of linear mixing between nearest-neighbor and next-nearest-neighbor couplings, Phys. Lett. A, 384 (2020), p. 126448, https://doi.org/https://doi.org/10.1016/j....

2020
[21]

U. D. Jentschura and G. Soff , Improved conformal mapping of the Borel plane , J. Phys. A: Math. Gen., 34 (2001), p. 1451

2001
[22]

Johansson and S

M. Johansson and S. Aubry , Growth and decay of discrete nonlinear Schr¨ odinger breathers in- teracting with internal modes or standing-wave phonons , Phys. Rev. E, 61 (2000), pp. 5864–5879, https://doi.org/10.1103/PhysRevE.61.5864

2000 doi
[23]

Kapitula and P

T. Kapitula and P. G. Kevrekidis , Stability of waves in discrete systems , Nonlinearity, 14 (2001), p. 533, https://doi.org/10.1088/0951-7715/14/3/306

2001 doi
[24]

P. G. Kevrekidis , The Discrete Nonlinear Schr¨ odinger Equation, Springer-Verlag, Heidelberg, 2009, https://link.springer.com/book/10.1007/978-3-540-89199-4 . 18

2009 doi
[25]

J. R. King and S. J. Chapman , Asymptotics beyond all orders and Stokes lines in nonlinear differential-difference equations, Eur. J. Appl. Math., 12 (2001), p. 433–463

2001
[26]

Y. S. Kivshar and D. K. Campbell , Peierls-Nabarro potential barrier for highly localized nonlinear modes, Phys. Rev. E, 48 (1993), pp. 3077–3081, https://doi.org/10.1103/PhysRevE.48.3077

1993 doi
[27]

Lederer, G

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Sil- berberg, Discrete solitons in optics , Phys. Rep., 463 (2008), pp. 1–126, https://doi.org/https: //doi.org/10.1016/j.physrep.2008.04.004

2008 doi
[28]

C. J. Lustri, I. Aniceto, D. J. V andenHeuvel, and S. W. McCue, Locating complex singularities of Burgers’ equation using exponential asymptotics and transseries, Proc. Roy. Soc. Lond. A, 479 (2023), p. 20230516

2023
[30]

Mari ˜no, R

M. Mari ˜no, R. Miravitllas, and T. Reis , New renormalons from analytic trans-series , J. High Energy Phys., 2022 (2022), pp. 1–61

2022
[31]

Morsch and M

O. Morsch and M. Oberthaler , Dynamics of Bose-Einstein condensates in optical lattices , Rev. Mod. Phys., 78 (2006), pp. 179–215, https://doi.org/10.1103/RevModPhys.78.179

2006 doi
[32]

A. B. Olde Daalhuis, S. J. Chapman, J. R. King, J. R. Ockendon, and R. H. Tew , Stokes phenomenon and matched asymptotic expansions , SIAM J. Appl. Math., 55 (1995), pp. 1469–1483

1995
[33]

Remoissenet, Waves Called Solitons , Springer-Verlag, Berlin, 1999

M. Remoissenet, Waves Called Solitons , Springer-Verlag, Berlin, 1999

1999
[34]

V. M. Rothos, S. Anastassiou, and K. G. Hadjifotinou, Stationary solitons in discrete nonlinear Schr¨ odinger with non-nearest neighbour interactions, Proc. Roy. Soc. Lond. A, 481 (2025), p. 20240539, https://doi.org/10.1098/rspa.2024.0539

2025
[35]

Serone, G

M. Serone, G. Spada, and G. Villadoro , λϕ4 theory—Part I. the symmetric phase beyond NNNNNNNNLO, J. High Energy Phys., 2018 (2018), pp. 1–35

2018
[36]

Stahl, The convergence of Pad´ e approximants to functions with branch points , J

H. Stahl, The convergence of Pad´ e approximants to functions with branch points , J. Approximation Theory, 91 (1997), pp. 139–204

1997
[37]

Szameit, R

A. Szameit, R. Keil, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, and A. T ¨unnermann, Observation of discrete solitons in lattices with second-order interaction, Opt. Lett., 34 (2009), pp. 2838– 2840, https://doi.org/10.1364/OL.34.002838. 19

2009 doi

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Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [29]

C. J. Lustri, P. G. Kevrekidis, and S. J. Chapman , Exponential asymptotics for translational modes in the discrete nonlinear Schr¨ odinger model, Q. Appl. Math. (In Press), (2025)

work page 2025

[2] [1]

M. J. Ablowitz and J. T. Cole, Nonlinear optical waveguide lattices: Asymptotic analysis, solitons, and topological insulators, Physica D, 440 (2022), p. 133440, https://doi.org/https://doi.org/10. 1016/j.physd.2022.133440

work page arXiv 2022

[3] [2]

Aniceto, G

I. Aniceto, G. Bas ¸ar, and R. Schiappa, A primer on resurgent transseries and their asymptotics , Phys. Rep., 809 (2019), pp. 1–135

work page 2019

[4] [3]

Aniceto, J

I. Aniceto, J. Jankowski, B. Meiring, and M. Spali ´nski, The large proper-time expansion of Yang-Mills plasma as a resurgent transseries , J. High Energy Phys., 2019 (2019), pp. 1–36

work page 2019

[5] [4]

Aniceto, R

I. Aniceto, R. Schiappa, and M. Vonk , The resurgence of instantons in string theory , Commun. Number Theory Phys., 6 (2012), pp. 339–496

work page 2012

[6] [5]

V. A. Brazhnyi and V. V. Konotop , Theory of nonlinear matter waves in optical lattices , Mod. Phys. Lett. B, 18 (2004), pp. 627–651, https://doi.org/10.1142/S0217984904007190

work page doi:10.1142/s0217984904007190 2004

[7] [6]

Caliceti, M

E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura , From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions, Phys. Rep., 446 (2007), pp. 1–96. 17

work page 2007

[8] [7]

S. J. Chapman, J. R. King, and K. L. Adams, Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations, Proc. Roy. Soc. Lond. A, 454 (1998), pp. 2733–2755

work page 1998