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REVIEW 3 major objections 4 minor 39 references

Stability of Leapfrogging Vortex Pairs: A Semi-analytic Approach

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The leapfrogging four-vortex orbit loses stability exactly at $h=1/8$, obtained here by a convergent Hill-determinant sequence.

desk verdict Solid semi-analytic confirmation of a known bifurcation; the explicit Floquet reformulation is the real contribution, but the golden-ratio mechanism is still unexplained and the convergence claim is numerical, not proven. read the letter →

arxiv 1908.08618 v2 pith:EOFK6M6P submitted 2019-08-22 nlin.CD math.DSphysics.flu-dyn

Brandon M. Behring , Roy H. Goodman This is my paper · ORCID
classification nlin.CDmath.DSphysics.flu-dyn MSC 76B4737C2770H12
keywords leapfroggingvortexpairsfour-vortexproblemFloquetstabilityHill'smethodharmonicbalancegoldenratiobifurcationvaluepointdynamics
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 tries to explain why the periodic leapfrogging motion of four point vortices becomes unstable at one special parameter value: in the geometric breadth ratio this is $1/\alpha=\phi^2$ with $\phi$ the golden ratio, and in the energy variable used here it is $h=1/8$. It derives an exact Floquet stability problem in polar-angle coordinates, with coefficients given in closed form, so that the stability question no longer requires linearizing around a numerically computed orbit. Applying Hill's harmonic-balance method to this explicit system produces a sequence of polynomial equations whose relevant roots approach $h_c=1/8$ at roughly the rate $4^{-N}$. A sympathetic reader would care because this is the first semi-analytic route to the mysterious golden-ratio threshold, and the same machinery may transfer to unequal-strength pairs, larger vortex systems, and vortex rings.

What carries the argument

The load-bearing object is the explicit Floquet matrix $\tilde A_h(\theta)$ obtained in two steps: a canonical polar-coordinate change $X=\sqrt{2J}\cos\theta$, $Y=\sqrt{2J}\sin\theta$ that replaces time by the polar angle as the independent variable, and elimination of $J$ using the energy level $h$, leaving a $\pi$-periodic matrix in closed form. A Lyapunov transformation $W=B(\theta)Z$ with $B(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$ makes the leading-order term constant, converting the system into a Hill-type equation amenable to harmonic balance. The paper then inserts a symmetry-adapted Fourier ansatz containing only odd harmonics, with the first component a cosine series and the second a sine series, truncates both the Fourier series and the $h$-expansion at order $N$, and computes the determinant of the resulting linear system. The roots of these truncated determinants form the sequence converging to the bifurcation value; the adequacy of the ansatz is checked by showing that a more general Fourier ansatz factors so that only this symmetry-adapted factor vanishes.

What would settle it

Take the full untruncated Floquet system at $h=1/8$ and compute the monodromy matrix with rigorous interval arithmetic; if the certified trace differs from $2$ beyond numerical tolerance, the $h=1/8$ threshold is not exact. Alternatively, extend the truncated determinants to much larger $N$ and check whether the root errors continue to track the $4^{-N}$ rate and converge to $1/8$; any saturation or drift would falsify the convergence claim.

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Extended reading notes

Core claim

The central claim is that the stability threshold of the leapfrogging family is exactly $h_c=1/8$, equivalently $\alpha=\phi^{-2}$, and that this value is not just a numerical coincidence but the root of an explicitly constructed Hill determinant. The authors rewrite the linearized four-vortex equations as $dZ/d\theta=\tilde A_h(\theta)Z$, using the polar angle $\theta$ as the independent variable and eliminating the radial action by fixing the energy level $h$; the coefficient matrix is given in closed form (their equation 18), so no numerical orbit is needed. At $h=1/8$ the system possesses a $\pi$-periodic solution, verified by arbitrary-precision Taylor integration to agreement below $10^{-120}$. Expanding $\tilde A_h$ in powers of $h$, applying a Lyapunov transformation, truncating the Fourier series at order $N$, and computing the determinant of the resulting matrix yields roots $h_c^{(N)}$ that a least-squares fit shows converge to $1/8$ with error decaying like $4^{-N}$.

Load-bearing premise

The computation assumes that the observed symmetry of the perturbation at the threshold—its cosine/sine pattern with only odd harmonics—captures all relevant modes, and that truncating both the Fourier series and the $h$-expansion at the same order gives roots that converge to the true threshold; the authors check a more general ansatz at finite order but do not prove convergence.

Editorial extensions

If this is right

  • If the result is correct, the stability boundary of the leapfrogging family is the rational value $h_c=1/8$ in the natural energy coordinate, and the golden-ratio form $1/\alpha=\phi^2$ follows from the exact change of variables between $h$ and the geometric ratio $\alpha$.
  • At the threshold the monodromy matrix has a Floquet multiplier $+1$, so the linearized system has a nontrivial $\pi$-periodic perturbation; this is the periodic solution that the harmonic-balance calculation detects.
  • The explicit Floquet formulation lets one compute the stability threshold to arbitrary precision without integrating the nonlinear equations, as demonstrated by the agreement below $10^{-120}$ at $h=1/8$.
  • The same semi-analytic procedure can be applied to the two-parameter family with unequal vortex-pair strengths, producing stability curves in $(h,\lambda)$ space, and to $2N$-vortex analogues and vortices on a sphere.

Reading between the lines

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

  • The clean rational value $h_c=1/8$ in the energy variable suggests that the deeper statement is the rational threshold itself; the golden ratio enters only through the two-to-one map from $h$ to the geometric breadth ratio $\alpha$, so the apparent mystery may be partly a coordinate artifact.
  • A rigorous convergence proof for the doubly truncated Hill determinants is the natural next step; the observed $4^{-N}$ rate is consistent with analyticity of the Floquet coefficients in a neighborhood of $h=0$, so such a proof could likely be supplied by estimating exponential decay of the Fourier coefficients.
  • The footnote observation that the general Fourier ansatz factors, with only the symmetry-adapted factor vanishing at the threshold, could be developed into a structural explanation: if the factorization persists at all orders, the stability boundary is governed by a single scalar Hill-type equation rather than the full $2\times2$ system.
  • A direct test of the method's reach would be to apply the same Floquet-plus-Hill pipeline to unequal-strength vortex pairs and ask whether the stability curves in $(h,\lambda)$ space also exhibit rational or algebraic thresholds in an appropriate energy coordinate.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the linear stability of the leapfrogging periodic orbit of four point vortices. After reducing the four-vortex system to a one-degree-of-freedom Hamiltonian, the authors reformulate the linearized Floquet problem in canonical polar coordinates, so that the coefficients are given explicitly as functions of the polar angle theta and the energy parameter h. A Lyapunov transformation makes the leading-order term constant, and the coefficient matrix is then expanded in powers of h. The method of harmonic balance (Hill's determinant) is applied with a symmetry-adapted Fourier ansatz. The roots of the truncated Hill determinants form a sequence h_c^(N) that appears to converge to h_c = 1/8 at a rate of about 4^(-N), confirming the known bifurcation value (equivalently 1/alpha = phi^2). Direct numerical checks -- machine-precision root finding for tr(M_h)-2 and a periodic solution tracked to 10^(-120) at h = 1/8 -- provide strong independent support for the result.

Significance. If the result holds, this is a valuable contribution: it offers the first semi-analytic derivation of the known bifurcation value h_c = 1/8 for the leapfrogging four-vortex problem, and the explicit canonical-polar-angle formulation of the Floquet problem is likely to be useful for related N-vortex stability questions. The paper is careful and reports excellent numerical evidence, including a root found to machine precision and a periodic orbit integrated to 10^(-120). The authors are honest in the conclusion that no mechanism or exact formula for the golden-ratio value is obtained. The main value of the paper lies in the efficient semi-analytic computation and the explicit Floquet formulation, rather than in a rigorous proof or a conceptual explanation of the special value.

major comments (3)
  1. [Sec. V.C, Table I] The claimed convergence of the roots h_c^(N) to h_c = 1/8 at a rate of roughly 4^(-N) is inferred from a least-squares fit to N = 1,...,20, but no proof is given that the doubly truncated Hill determinants converge to the exact bifurcation value. The classical Hill determinant convergence theorem quoted in Section V.A applies to a scalar Hill equation with absolutely convergent Fourier coefficients; the present application is to the 2x2 Floquet system (22), with simultaneous truncation of the Fourier series and the h-expansion in (23). A concrete test would be to compare the roots with direct high-precision computations of tr(M_h)-2 near h = 1/8 and to study the error in h_c^(N) for larger N. At a minimum, the abstract's 'rapidly-converging sequence' should be qualified as a numerical observation, consistent with the more cautious wording in the conclusion that the sequence 'appears to converge exponentially.'
  2. [Sec. V.C, footnote [37] and Eqs. (30a)-(30b)] The parity-adapted Fourier ansatz is the basis of the entire harmonic-balance calculation, but it is introduced as 'based on mere observation,' and the claimed factorization of a more general ansatz is not shown. Since the omission of Fourier modes could in principle change the solvability condition, please include the explicit factorization or a proof that the omitted block cannot vanish, or state clearly that the ansatz is a numerically motivated conjecture. As it stands, a reader cannot verify that no relevant harmonics have been excluded.
  3. [Sec. IV.A, Eqs. (13)-(16)] The transformation as printed, X = sqrt(2J) cos(theta), Y = sqrt(2J) sin(theta), is inconsistent with the Hamiltonian H(J,theta) and the expressions for J_± and dtheta/dt that follow; those formulas are instead consistent with the standard action-angle form X = sqrt(2J) sin(theta), Y = sqrt(2J) cos(theta). Please correct the transformation or reconcile the subsequent formulas. The inconsistency does not appear to affect the numerical results, but it prevents a reader from reproducing the derivation.
minor comments (4)
  1. [Sec. III.B, Eq. (12)] The notation 'A(t) = A(t + T/2)' in Eq. (12) is confusing because the symbol A is used both for the coefficient matrix of the transverse system and for the matrix of the in-plane system; please clarify whether the transpose is intended or explain that the trace condition is invariant under transposition.
  2. [Abstract and Conclusion] The abstract states that the method constructs a 'rapidly-converging sequence of asymptotic approximations,' while the conclusion says the sequence 'appears to converge exponentially.' Please harmonize the language so that the level of certainty is consistent throughout.
  3. [Introduction and Conclusion] The introduction states that the study aims to 'explain the origin of this remarkable value,' but the conclusion acknowledges that no mechanism or exact formula is obtained. Consider adjusting the framing to emphasize the semi-analytic method rather than a full explanation of the golden-ratio value.
  4. [Table I] The table would be easier to assess if it listed a few more values of N (for example, N = 5, 10, 15, 20) or if the absolute errors |h_c^(N) - 1/8| were plotted against N, rather than showing only the first four rows and an ellipsis.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: h_c = 1/8 is used only as a post-hoc benchmark; the truncated Hill determinants are built from an explicitly derived Floquet matrix.

full rationale

The paper's central derivation is self-contained in the sense required: the Floquet matrix A_h(θ) in equation (18) is obtained by an exact canonical-polar variable substitution from the leapfrogging equations, expanded in h, and the truncated Hill determinants M^(N)(h) are computed by harmonic balance on that explicit system. The known bifurcation value h_c=1/8 is not an input to any of these constructions; it is used after the fact as a benchmark for the roots in Table I and for the least-squares estimate that |h_c^(N)-1/8| decays like 4^{-N}. The only place where numerical knowledge of the target informs the setup is footnote [37], which states that the symmetry-adapted Fourier ansatz (30) was 'based on mere observation from numerical simulations'; the authors report that a more general ansatz factors and that only the selected factor vanishes, which, if correct, means no root content is inserted by the ansatz. This is a reproducibility or verification caveat rather than a circular reduction: no equation is defined in terms of h_c, and no fitted parameter is renamed as a prediction. The skeptic's concern about unproved convergence of the doubly truncated harmonic-balance determinants is a rigor gap, not a circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The derivation uses standard Hamiltonian transformations, Floquet theory, and Hill's determinant. The only ad hoc element is the symmetry-adapted Fourier ansatz, which is checked numerically but not proven. No parameters are fitted to data; the bifurcation value emerges from the computation.

assumptions (6)
  • standard math The canonical polar coordinate transformation (13) preserves the Hamiltonian structure and the Floquet eigenvalue problem.
    Used in Sec. IV.A to rewrite the linearized system in terms of the polar angle; standard canonical transformation theory (Meyer, Hall, and Offin, reference [26]).
  • domain assumption The Aref-Eckhardt reduction (1) to the two-degree-of-freedom Hamiltonian (2) correctly describes the leapfrogging family and its stability.
    Sec. II relies on point-vortex Hamiltonian mechanics and the invariance of the plane P = Q = 0 for the leapfrogging orbits.
  • domain assumption Stability of the leapfrogging orbit is governed entirely by the second linearized system (10b), because the first system (10a) has Floquet multipliers identically equal to 1.
    Sec. III.B states this and cites the numerical Floquet analysis of Tophoj and Aref; it is standard for perturbations along the periodic family.
  • standard math The Maclaurin series expansion A_h = sum h^k A_k(θ) converges at the bifurcation value h = 1/8.
    The coefficient functions contain the square root 1 + 4h^2 + 4h cos 2θ, whose nearest singularities in the complex h-plane are at |h| = 1/2; the paper does not state this explicitly but relies on the expansion in Sec. IV.C.
  • ad hoc to paper The symmetry-adapted Fourier ansatz (30) captures the bifurcating periodic orbit with no loss of generality.
    Chosen from numerical observation that the first component is even and the second is odd (Sec. V.C); a general ansatz check is reported in footnote 37 as factoring so that only this factor vanishes, but this is a finite-order computation, not a proof.
  • standard math The roots of truncated Hill determinants converge to the roots of the infinite determinant as N increases.
    Poincare's proof of convergence of Hill's determinant, cited in reference [36]; the paper relies on this for the sequence h_c^(N).

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Pith. "Pith review of Stability of Leapfrogging Vortex Pairs: A Semi-analytic Approach." pith.science (2026). https://pith.science/paper/EOFK6M6P

@misc{pith2026190808618,
  author       = {Pith},
  title        = {Pith review of: Stability of Leapfrogging Vortex Pairs: A Semi-analytic Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EOFK6M6P}},
  note         = {Machine review of arXiv:1908.08618}
}
abstract

We investigate the stability of a one-parameter family of periodic solutions of the four-vortex problem known as `leapfrogging' orbits. These solutions, which consist of two pairs of identical yet oppositely-signed vortices, were known to W.\ Gr\"obli (1877) and A.\ E.\ H.\ Love (1883), and can be parameterized by a dimensionless parameter $\alpha$ related to the geometry of the initial configuration. Simulations by Acheson (2000) and numerical Floquet analysis by Toph{\o}j and Aref (2012) both indicate, to many digits, that the bifurcation occurs when $1/\alpha=\phi^2$, where $\phi$ is the golden ratio. This study aims to explain the origin of this remarkable value. Using a trick from the gravitational two-body problem, we change variables to render the Floquet problem in an explicit form that is more amenable to analysis. We then implement G. W. Hill's method of harmonic balance to high order using computer algebra to construct a rapidly-converging sequence of asymptotic approximations to the bifurcation value, confirming the value found earlier.

Figures

Figures reproduced from arXiv: 1908.08618 by the authors.

Figure 1
Figure 1. FIG. 1: (a) Opposite-signed vortices move in parallel along straight lines. (b) Like-signed vortices move along a circular path. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Motion in physical ( [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Motion in physical ( [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Level sets of the one-degree-of-freedom Hamiltonian ( [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: (a) The trace of the monodromy matrix as a function of the energy [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

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Reference graph

Works this paper leans on

39 extracted references · 36 canonical work pages

  1. [1]

    von Helmholtz, J

    H. von Helmholtz, J. Reine Angew. Math 55, 25 (1858)

    work page

  2. [2]

    We used MATLAB’s built in rootfinder, fzero along with the ODE Solver ode45 with a relative tolerance of 10−13, an absolute tolerance of 10−15 to solve the equation f(hc) = 0

    Let Mh be the monodromy matrix of the system (18), and define the function f(h) = tr Mh− 2. We used MATLAB’s built in rootfinder, fzero along with the ODE Solver ode45 with a relative tolerance of 10−13, an absolute tolerance of 10−15 to solve the equation f(hc) = 0. Using an ini- tial value of h = 0.1, the solver returned the numerical solution hc = 0.125 ...

    work page 2015

  3. [3]

    Aref, Journal of Mathematical Physics 48, 065401 (2007), ISSN 0022-2488

    H. Aref, Journal of Mathematical Physics 48, 065401 (2007), ISSN 0022-2488

    work page 2007

  4. [4]

    P. K. Newton, The N-Vortex Problem: Analytical Tech- niques, Applied Mathematical Sciences (Springer New York, 2013), ISBN 9781468492903

    work page 2013

  5. [5]

    Kirchhoff, Vorlesungen ¨ uber mathematische Physik: Mechanik, vol

    G. Kirchhoff, Vorlesungen ¨ uber mathematische Physik: Mechanik, vol. 1 of Vorlesungen ¨ uber mathematische Physik (Teubner, Leipzig, 1876)

    work page

  6. [6]

    Aref, Theor

    H. Aref, Theor. Comput. Fluid Dyn. 24, 1 (2010), ISSN 1432-2250

    work page 2010

  7. [7]

    Eckhardt and H

    B. Eckhardt and H. Aref, Philos. Trans. R. Soc. London, Ser. A (1988)

    work page 1988

  8. [8]

    The Hamiltonian (5) yields evolution equations dX dt = +∂H ∂Y = + Y (1−Y 2)2, dY dt =−∂H ∂X =− X (1 +X2)2, (8) whose phase plane is shown in Fig

    The two parameters are related by h = (1−α)2 8α . The Hamiltonian (5) yields evolution equations dX dt = +∂H ∂Y = + Y (1−Y 2)2, dY dt =−∂H ∂X =− X (1 +X2)2, (8) whose phase plane is shown in Fig. 4. In [12], Gr¨ obli integrated the equations of motion (8) to find an implicit formula for Xh(t). In our notation, this is given by t(X) = 1 2h2√ 1− 4h2F ( sin−1...

    work page

Show all 39 references
  1. [9]

    H. Aref, N. Rott, and H. Thomann, Annu. Rev. Fluid Mech. 24, 1 (1992), ISSN 0066-4189

    1992

  2. [10]

    Aref, Journal of Applied Mathematics and Physics (1989)

    H. Aref, Journal of Applied Mathematics and Physics (1989)

    1989

  3. [11]

    M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198 (1995)

    1995

  4. [12]

    M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 2498 (1999)

    1999

  5. [13]

    Navarro, R

    R. Navarro, R. Carretero-Gonz´ alez, P. J. Torres, P. G. Kevrekidis, D. J. Frantzeskakis, M. W. Ray, E. Altunta¸ s, and D. S. Hall, Phys. Rev. Lett. 110, 225301 (2013)

    2013

  6. [14]

    Gr¨ obli, Ph.D

    W. Gr¨ obli, Ph.D. thesis, Georg-August-Universit¨ at G¨ ottingen (1877)

  7. [15]

    A. E. H. Love, Proc. London Math. Soc. 1, 185 (1893)

  8. [16]

    A. V. Borisov, A. A. Kilin, and I. S. Mamaev, Discrete Contin. Dyn. Syst. 54, 100 (2005), ISSN 1078-0947

    2005

  9. [17]

    D. J. Acheson, Eur. J. Phys. 21, 269 (2000), ISSN 0143- 0807

    2000

  10. [18]

    Tophøj and H

    L. Tophøj and H. Aref, Phys. Fluids 20, 093605 (2008), ISSN 1070-6631

    2008

  11. [19]

    Tophøj and H

    L. Tophøj and H. Aref, Phys. Fluids 25, 014107 (2013), ISSN 1070-6631

    2013

  12. [20]

    J. D. Meiss, Differential Dynamical Systems (SIAM, Philadelphia, 2007)

    2007

  13. [21]

    Markus and H

    L. Markus and H. Yamabe, Osaka J. Math. 12, 305 (1960)

    1960

  14. [22]

    Whitchurch, P

    B. Whitchurch, P. G. Kevrekidis, and V. Koukouloyan- nis, Phys. Rev. Fluids 3, 014401 (2018)

    2018

  15. [23]

    N. Rott, J. Appl. Math. Phys. (ZAMP) 40, 473 (1989)

    1989

  16. [24]

    J. V. Jose and E. J. Saletan, Classical Dynamics: A Con- temporary Approach(Cambridge University Press, 1998)

    1998

  17. [25]

    Kaushal and H

    R. Kaushal and H. Korsch, Phys. Lett. A 276, 47 (2000), ISSN 0375-9601

    2000

  18. [26]

    Floquet, in Annales scientifiques de l’ ´Ecole normale sup´ erieure(1883), vol

    G. Floquet, in Annales scientifiques de l’ ´Ecole normale sup´ erieure(1883), vol. 12, pp. 47–88

  19. [27]

    Yakubovich and V

    V. Yakubovich and V. Starzhinskii, Linear Differential Equations with Periodic Coefficients Vol. 1 (John Wiley and Sons, 1975)

    1975

  20. [28]

    Meyer, G

    K. Meyer, G. Hall, and D. Offin, Introduction to Hamil- tonian Dynamical Systems and the N-body Problem (Springer, 2008)

    2008

  21. [29]

    J. A. Prez-Hernndez and L. Benet, PerezHz/TaylorIntegration.jl: TaylorIntegration v0.4.1 (2019), URL https://doi.org/10.5281/zenodo. 2562353

    2019 doi

  22. [30]

    E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, 1902)

    1902

  23. [31]

    G. W. Hill, Acta Math. 8, 1 (1886), ISSN 0001-5962

  24. [32]

    Poincar´ e, Les m´ ethodes nouvelles de la m´ ecanique c´ eleste

    H. Poincar´ e, Les m´ ethodes nouvelles de la m´ ecanique c´ eleste. Tome III(Gauthier-Villars, 1899)

  25. [33]

    Treschev, O

    D. Treschev, O. Zubelevich, D. Treschev, and O. Zubele- vich, Introduction to the Perturbation Theory of Hamilto- nian Systems (Springer-Verlag Berlin Heidelberg, 2009)

    2009

  26. [34]

    Bolotin and D

    S. Bolotin and D. Treschev, Russ. Math. Surv. 65, 191 (2010), ISSN 0036-0279

    2010

  27. [35]

    Wolfram Research, Inc., Mathematica, Version 12.0 (2019), champaign, IL

    2019

  28. [36]

    D. H. Wacks, A. W. Baggaley, and C. F. Barenghi, Phys. Fluids 26, 027102 (2014)

    2014

  29. [37]

    P. K. Newton and H. Shokraneh, Proc. R. Soc. London, Ser. A 464, 1525 (2008), ISSN 1364-5021

    2008

  30. [38]

    This is a regularization factor to guaranteehjj = 1 and is necessary for (27) to converge

    Note that equations (26) and (29) differ by a factor of 1 1+k2 . This is a regularization factor to guaranteehjj = 1 and is necessary for (27) to converge. 11

  31. [39]

    It would of course be possible to proceed with a more general Fourier ansatz

    This expansion contains only one-fourth of the possible non-zeros terms, and was based on mere observation from numerical simulations. It would of course be possible to proceed with a more general Fourier ansatz. We have done this, and found that the computed Hill determinant ...

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