Classification of the real Painlev\'{e} I transcendents by zeros and connection problem: an asymptotic study

Wen-Gao Long; Yan Huang; Yu-Tian Li

REVIEW 3 major objections 5 minor 27 references

Classification of the real Painlev\'{e} I transcendents by zeros and connection problem: an asymptotic study

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read From the location and slope of a zero, this paper computes the Stokes multipliers of a real Painlevé I solution to all orders, and conversely predicts every large positive zero from the solution's type at negative infinity.

desk verdict A credible extension of the WKB/Stokes-multiplier program to zero data of real Painlevé I, with genuinely new expansions for Stokes multipliers and positive zero locations, but the branch-selection step behind Corollary 3 is asserted rather than proved. read the letter →

arxiv 2506.03648 v1 pith:ZG4C3OBD submitted 2025-06-04 math.CA

Yan Huang , Yu-Tian Li , Wen-Gao Long This is my paper

classification math.CA MSC 34M4033E1734A1234E0533C10

keywords PainlevéItranscendentzerosconnectionformulaStokesmultiplieruniformasymptoticscomplexWKBphasediagram

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 claims that for real solutions of the first Painlevé equation, the behavior at a zero—where y(r)=0 and y'(r)=b with b large—determines the Stokes multipliers to all asymptotic orders, and that, conversely, the solution type fixed by large-negative-time asymptotics determines the full asymptotic expansion of the locations and slopes of its large positive zeros. If correct, this turns a long-open connection problem into an explicit asymptotic dictionary between monodromy data and zero data. The paper also constructs a phase diagram in the (r,b)-plane separating oscillatory, separatrix, and singular solutions, and supports the expansions with numerical simulations.

What carries the argument

The argument runs through the Lax pair of the first Painlevé equation, reduced at a zero t=r to the second-order equation d²φ₁/dλ² = (4λ³+2λr+b²−b/λ+3/(4λ²))φ₁. Scaling λ=$ξ^{{2/5}}$z gives a WKB equation with leading term ξ²·4(z³+Az+1), whose turning points are the roots of the cubic. Complex WKB with Airy-function uniform approximation near the relevant turning point produces the asymptotic coefficients α_s; the sign of Re α_0 changes at the critical constant C_0≈2.00486, where the two turning points used for the matching coalesce in the Stokes geometry. At A=C_0, imposing the fixed Stokes multiplier s_1 allows the coefficients B_s and the sequence ξ_n to be inverted, yielding the zero expansions (3.21).

What would settle it

Numerically integrate a specific real PI solution with known Stokes multipliers—for instance the tritronquée solution with s_0=s_1=s_{−1}=i, s_2=s_{−2}=0—compute its large positive zeros r̂_n^+ for n up to 50 by high-precision integration, and compare with the leading term of (3.21) (taking n_1=0 as the numerics suggest). If the relative error fails to decrease with n as predicted, or if any zero's local A-value (r/2 $ξ^{{-4/5}}$ with ξ inferred from b) is not C_0, the expansion (3.21) is false.

Watch

Extended reading notes

Core claim

For a real PI solution satisfying y(r)=0 and y'(r)=b with b→±∞, the Stokes multipliers s_k admit full asymptotic expansions in powers of $ξ^{{-1}}$ (Theorems 1–4), where ξ is tied to the zero data by r=$2Aξ^{{4/5}}$ and $b^{2}$=$4ξ^{{6/5}}$(1+∑_{s≥1} B_s/ξ^s). The sign of Re α_0 distinguishes regimes: A below the critical value C_0 gives oscillatory-type behavior, A above C_0 gives singular-type behavior, and the critical case A=C_0 keeps the Stokes multipliers fixed at their large-negative values. In that critical case the paper inverts the expansions: the coefficients B_s are solved in terms of log|s_1|, and the sequence of large positive pole-region zeros is obtained from arg s_1. Corollary 3 then states that for any specific PI solution, the zero locations r̂_n^± and slopes b̂_n^± have the full asymptotic expansions (3.21), with |s_1| and arg s_1 expressed through the solution type's Stokes parameters. Thus the global large-negative asymptotic parameters and the asymptotic zero data determine each other.

Load-bearing premise

The expansion for specific solutions' pole-region zeros assumes those zeros are controlled by the critical turning point A=C_0 with r=$2C_0ξ^{{4/5}}$ and b=±$2ξ^{{3/5}}$(1+∑B_s/ξ^s)^{1/2}; if a real zero branch belongs to a different turning-point regime, formula (3.21) would not apply, and the ordering of zeros between consecutive poles is taken as a definition rather than proved.

Editorial extensions

If this is right

For any specific real PI solution, the location and slope of every large positive pole-region zero can be approximated to arbitrarily high asymptotic order directly from the solution's Stokes multiplier s_1.
The (r,b)-plane is partitioned into finger-like regions of oscillatory (type A) solutions separated by curves of separatrix (type B) solutions, with singular (type C) solutions in the remaining regions.
For A<−3/2^{2/3}, all sufficiently large-|b| zeros give oscillatory solutions; for A>C_0 they give singular solutions; only in the intermediate window do all three types occur.
The full expansions of the Stokes multipliers in Theorems 1–4 make the classification quantitative, not merely qualitative, and the numerical tables show relative errors around 10^{-3} already for the first few zeros.
The connection problem for PI is resolved asymptotically: specifying the large-negative parameters (d,θ), h, or (ρ,σ) fixes the large-positive zero data through Corollary 3.

Reading between the lines

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

The same WKB apparatus could be applied to complex zeros of PI by selecting other critical turning-point values, but the paper only treats real pole-region zeros; this is a natural testable extension.
Because the expansions are written purely in terms of monodromy data, they suggest a practical numerical route to Stokes multipliers: integrate the PI ODE, read off (r,b) at a large zero, and evaluate the expansions—an inverse of the paper's own tables.
The phase diagram implies a counting statement the authors do not state explicitly: the number of pole-region zeros of each type inside a large interval should be governed by the spacing of the curves Σ_n^±.
The undetermined integer n_1 in (3.21) is set to zero by the numerical evidence; if this holds for all solutions, the expansions become fully explicit predictions with no free parameters.

Share X Bluesky LinkedIn Reddit HN

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

Short version: this paper does real work and deserves referee time, but the main new result—the connection formula between large-negative asymptotic data and positive pole-region zeros—hangs on an ansatz that is asserted rather than proved.

What is new: Theorems 1–3 give Stokes multiplier asymptotics in terms of zero data (r,b) in the three generic A regimes; Theorem 4 handles A=C0 with the B_s corrections, and Corollary 3 inverts this to produce full asymptotic expansions for rhat_n^± and bhat_n^± for individual solutions in terms of |s1| and arg s1. That inversion is genuinely new. The earlier papers from the same group treated poles or fixed-point initial data; zeros are a different object. The derivations are not circular: WKB matching and Airy asymptotics produce the expansions, and the numerics for the tritronquée and one type-(C) solution show relative errors dropping as n grows. Theorems 1 and 2 have worked matching proofs.

Where it is soft. First, Lemma 1 is imported almost verbatim from [20] with proof deferred; that is acceptable if the lemma really is a minor modification, but the paper should say explicitly what changes. Second, Theorems 3 and 4 are not proved: Remark 8 sketches them, and Theorem 4 is the one that carries Corollary 3. Third, and most important, Corollary 3 simply assumes that the zeros counted between consecutive poles fall in the A=C0 regime, with r=2C0 ξ^{4/5} and b= ±2ξ^{3/5}(1+Σ B_s/ξ^s)^{1/2}. The fixed finite nonzero s1 rules out a different subsequential limit A, but it does not rule out A_n−C0 of order 1/ξ altering the expansion, nor does it prove the ordering of zeros matches the indexing of the fixed-Stokes equation. That is the hinge of the claimed connection problem, and it is unproved. Fourth, the integer shifts n0, m0, n1 are pinned by numerical observation, which is honest but leaves a finite indeterminacy in the expansions. Tables validate two solutions only; there is no type-(B) check.

Who gets value: people working on Painlevé connection problems and zero asymptotics; this is a natural sequel to [20]–[22], and the classification phase diagram in the (r,b)-plane is a useful picture. My own verdict is skeptical-conditional, but the skepticism is about the unproved branch selection, not about the machinery. I would send it to review, and I would ask for a proof that the indexed zeros actually lie in the A=C0 regime (or a corrected statement), plus full proofs of Theorems 3 and 4. If that branch selection is wrong, much of Corollary 3 collapses.

Referee Report

3 major / 5 minor

Summary. The paper studies the first Painlevé equation y''=6y^2+t for real solutions with a zero at t=r and large derivative b=y'(r). Using complex WKB and Airy-function matching, the authors derive formal full asymptotic expansions of the Stokes multipliers as b→∞ in four regimes of the parameter A, where r=2Aξ^{4/5} and b^2=4ξ^{6/5}(1+Σ B_s/ξ^s). These expansions are used to draw a phase diagram in the (r,b)-plane separating Kapaev's oscillatory (A), separatrix (B), and singular (C) types, and, via a fixed-Stokes-multiplier inversion at A=C0, to state full asymptotic expansions (Corollary 3) for the locations of large positive pole-region zeros and their slopes in terms of Stokes multipliers. Numerical comparisons for a tritronquée solution and a type-(C) solution are reported.

Significance. If correct, the result would give a substantially complete asymptotic connection between local zero data and global negative-infinity parameters for real PI solutions, going beyond previous pole-based connection results. The expansions are derived from WKB matching rather than fitted to the target quantities, and the two numerical tables show relative errors decreasing with n, which is genuine supporting evidence. The main limitations are that Theorem 4's proof is only sketched and that the zero-branch selection underpinning Corollary 3 is not proved; these limit the current certainty of the central connection claim.

major comments (3)

[§3.3, Corollary 3] The proof of Corollary 3 verifies only the auxiliary formulas (3.22)–(3.27) for |s1| and arg s1. The corollary's actual claim—that the zeros rhat_n^± defined in (i)–(ii) satisfy the A=C0 scaling (3.14) with ξ=ξ_n^± from (3.19)—is not proved. In particular, no argument rules out a zero branch with A≠C0, or with A→C0 at rate 1/ξ with a coefficient that changes the expansion. Since Theorems 1–3 show that other A-regimes have degenerate Stokes limits, the fixed finite value of s1 only excludes A≠C0 as a subsequential limit; it does not select the correct branch of zeros or prove the claimed ordering between consecutive poles. This is the load-bearing step in the connection problem and needs either a proof or an explicit, clearly marked assumption.
[§4, Theorem 4 and Remark 8] Theorems 3 and 4 are not proved in Section 4; Remark 8 states that they follow by the same strategy as Theorem 2, with Re α0<0 or Re α0=0. Theorem 4 is the basis of Corollary 3 and hence of the paper's central connection formula, and the matching calculation for A=C0 with nonzero B_s involves additional cancellations, for example in the expansions of s0 and s2 in (3.5), that are not a routine copy of the proof of Theorem 2. A full proof, or a precise reduction to a published argument, should be supplied.
[§3.3, definition of rhat_n^±] The definitions (i) and (ii) presuppose structural facts about real PI solutions: that for type (A)/(B) solutions there are exactly two zeros between consecutive large positive poles, and that for type (C) solutions the indexing by the (2n−1)-th and 2n-th zeros is well defined. These facts are not proved or referenced. If they fail for some parameter range, the indexing and the asymptotics (3.21) are not meaningful. A proof or reference for the zero/pole alternation in the large-positive regime should be added.

minor comments (5)

[§3.1, Theorem 4] The displayed formula for b^2 appears to be a typo: it reads b^2=4ξ^{6/5}Σ B_s/ξ^s, which is incompatible with (2.3); the factor (1+Σ B_s/ξ^s) is presumably intended.
[§3.2, Corollary 1] Corollary 1 states μ1=−4 Imα0 Imα2/π^2, while the proof concludes μ2=−4 Imα0 Imα2/π^2; the index should be made consistent.
[§3.3, Eq. (3.16)] The condition 'Re α_s=0' is presented as if it were a separate assumption; since |s1| is fixed, it actually follows from the vanishing of the ξ^{−s} coefficients in (3.15), and this logic should be spelled out.
[§2, Lemma 1] The proof of Lemma 1 is deferred to [20, Section 3]; a short statement of which parts are new and which are transferred from [20] would help the reader assess the adaptation.
[Figure 2 caption] The caption refers to 'fingertips' and green curves before Λ0 is defined in (3.13), making the figure hard to read without extensive cross-referencing.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the Stokes-multiplier expansions are WKB/Airy-derived and Corollary 3 inverts them algebraically against a fixed Stokes multiplier, with zero data checked afterwards.

full rationale

The derivation chain is not circular. Theorems 1-4 expand the Stokes multipliers using the complex WKB lemma and Airy-function matching with the canonical solutions of the Lax pair; the coefficients alpha_s are computed from the Lax pair after inserting the parametrization r=2A xi^{4/5}, b^2=4 xi^{6/5}(1+ sum B_s/xi^s), and are independent of the zero locations being predicted. Corollary 3 is an algebraic inversion: for a fixed solution, |s1| fixes B_1 (Eq. 3.17), arg s1 determines the sequences xi_n^pm through (3.18)-(3.19), and rhat,bhat are then obtained by evaluating the parametrization (3.14). The numerical zero positions are not used as inputs to this inversion; Tables 1-2 compare them afterward, so the prediction is not tautological. The formulas (3.22)-(3.27) merely rewrite the Stokes constraint s_k = i(1+s_{k+2}s_{k+3}) in polar form. The genuinely weak point is the unproved branch assumption in Section 3.3 that the indexed pole-region zeros belong to the A=C0 regime and that the integer n1 in (3.19) matches the left-to-right ordering; a zero branch approaching A != C0, or an incorrect n1, would invalidate (3.21). That is a correctness/justification gap, not a circular reduction. Self-citations [20]-[22] supply Lemma 1 and the threshold C0; since these are published results with independent proofs and the present paper sketches the adaptation (Remark 2), they do not make the argument circular under the stated rules. The proof of Theorems 3-4 is deferred to the same strategy as Theorem 2 (Section 4), a completeness omission rather than a circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities. The real load is carried by an unproved or borrowed WKB lemma, monodromy constraints, and the ansatz that pole-region zeros live at A=C0; the free constants are the B_s expansion coefficients and integer index shifts.

free parameters (2)

B_s (s>=1) = B1 = (log|s1| - 2 Re α1,2)/(4 Re α1,1); higher B_s = -Re α_s,2/α1,1
Expansion coefficients in b^2 = 4ξ^{6/5}(1 + Σ B_s/ξ^s). In Theorems 1-3 set to 0; in Corollary 3 they are solved from the known Stokes multiplier |s1|, so they are free degrees of the asymptotic parametrization.
n0, m0, n1 = 0
Undetermined integer shifts in the zero-index expansion (3.6), (3.10), (3.21); Remarks 6-7 set them to zero because numerical simulations suggest this, without a proof.

assumptions (5)

domain assumption Lemma 1: uniform Airy-type asymptotic representation (2.19)-(2.20) with error O(ξ^{-2n}) holds for Eq. (2.14) with modified a_s(ζ); proof is deferred to [20, Section 3].
Invoked in the proof of every theorem; without it the Stokes multiplier expansions (3.2)-(3.5) have no foundation.
domain assumption Monodromy constraints (A.5)-(A.7): s_{k+5}=s_k, s_k=i(1+s_{k+2}s_{k+3}), symmetry s_k(t,y,y')=-s_{-k}(conjugate), from [18,16].
Used to relate all Stokes multipliers to s0,s1 and to translate numbers into solution types; accepted from prior literature.
domain assumption Zero/pole ordering for Corollary 3: for type A/B solutions two zeros lie between nth and (n+1)th poles, and for type C the (2n-1)th and 2nth zeros are used.
The paper states this as the definition of rhat_n^± without proof; the asymptotic labeling of zeros depends on it.
domain assumption C0-ansatz for pole-region zeros: large positive pole-region zeros obey r = 2 C0 ξ^{4/5}, b = ±2ξ^{3/5}(1+...)^{1/2}, i.e., critical turning point A=C0.
Corollary 3 substitutes expressions from Theorem 4 into this ansatz; no proof shows non-critical A values cannot occur for pole-region zeros.
standard math Kapaev's large-negative asymptotics (1.2)-(1.7) and type classification by sign of Im s0.
Borrowed from [16]; forms the benchmark against which the zero-data classification is compared.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Classification of the real Painlev\'{e} I transcendents by zeros and connection problem: an asymptotic study." pith.science (2026). https://pith.science/paper/ZG4C3OBD

@misc{pith2026250603648,
  author       = {Pith},
  title        = {Pith review of: Classification of the real Painlev\'e I transcendents by zeros and connection problem: an asymptotic study},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZG4C3OBD}},
  note         = {Machine review of arXiv:2506.03648}
}

abstract

In this paper, we study the asymptotic behavior and connection problem of Painlev\'e I (PI) equation through a detailed analysis of the Stokes multipliers associated with its solutions. Focusing on the regime where the derivative at the real zeros of the solution becomes large, we apply the complex WKB method to derive full asymptotic expansions of the Stokes multipliers. These expansions allow us to classify real solutions of PI according to their behavior at the zeros, distinguishing between oscillatory, separatrix, and singular types solutions on the negative real axis. Furthermore, we resolve the connection problem between the large negative asymptotics and the location of positive zeros by establishing full asymptotic expansions of the zero parameters. Our approach enables the construction of a precise phase diagram in the $(r,b)$-plane, where $r$ is the location of a zero and $b$ is the derivative at that point. Numerical simulations are provided to validate the theoretical results. This work extends prior studies on monodromy asymptotics and contributes a comprehensive framework for understanding the global structure of real PI solutions through their local zero data.

Figures

Figures reproduced from arXiv: 2506.03648 by the authors.

**Figure 1.** Stokes geometry defined by Re R z zi f0(s) 1 2 ds = 0 for different cases of the parameter A. Theorem 1. Let r = 2Aξ2/5 and b 2 = 4ξ 6/5 , where A < −3/2 2/3 and is bounded away from −3/2 2/3 for sufficiently large ξ. Then Re α0 > 0, and as ξ → +∞, the Stokes multiplier s0 corresponding to the PI solution y(t) satisfying y(r) = 0 and y ′ (r) = b admits the asymptotic expansion s0 ∼ i exp ( − X∞ s=0 2αs ξ s−1 ) , (3… view at source ↗

**Figure 2.** Asymptotic and numerical descriptions of curves Σ [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Phase diagram of real PI solutions in the ( [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Two types of zeros of oscillatory solutions. Cyan asterisks ( [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

[20]

Long, Y.-J

W.-G. Long, Y.-J. Jiang and Y.-T. Li, Full asymptotic expansion of monodromy data for the first Painlev´ e transcendent: applications to connection problems,Nonlinearity,38(2025) 035005

work page 2025
[22]

Long, Y.-T

W.-G. Long, Y.-T. Li, and Q.-h. Wang, Connection problem of the first Painlev´ e transcendent between poles and negative infinity,SIAM J. Math. Anal., 55 (2023), 6676–6706

work page 2023
[1]

M. J. Ablowitz, S. Chakravarty, and R. G., Halburd, Integrable systems and reductions of the self-dual Yang–Mills equations,J. Math. Phys., 44 (8) (2003), 3147–3173

work page 2003
[2]

Barouch, B

E. Barouch, B. M. McCoy, and T. T. Wu, Zero-field susceptibility of the two-dimensional Ising model nearT c.Phys. Rev. Lett.31(1973), pp. 1409–1411

work page 1973
[3]

A. P. Bassom, P. A. Clarkson, C. K. Law and J. B. McLeod, Application of uniform asymptotics to the second Painlev´ e transcendent,Arch. Rational Mech. Anal.,143(1998), 241–271

work page 1998
[4]

C. M. Bender and J. Komijani, Painlev´ e transcendents and PT-symmetric Hamiltonians,J. Phys. A: Math. Theor.,48(2015), 475202, 15 pp

work page 2015
[5]

C. M. Bender, J. Komijani and Q. Wang, Nonlinear eigenvalue problems for generalized Painlev´ e equations,J. Phys. A: Math. and Theor.,52(2019), 315202. 24

work page 2019
[6]

Bertola and A

M. Bertola and A. Tovbis, Universality for the focusing nonlinear Schr¨ odinger equation at the gradient catastrophe point: rational breathers and poles of the tritronqu´ ee solution to Painlev´ e I,Comm. Pure Appl. Math., 66 (2013), pp. 678–752

work page 2013

Show all 27 references

[7]

P. A. Clarkson, Painlev´ e equations-nonlinear special functions,J. Comput. Appl. Math.,153 (2003), 127–140

2003
[8]

P. A. Clarkson, Painlev´ e equations-nonlinear special functions, inOrthogonal Polynomials and Special Functions, Springer, Berlin and Heidelberg, (2006), 331–411

2006
[9]

T. M. Dunster, Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point,Anal. Appl.,12(2014), 385-–402

2014
[10]

F. H. L. Essler, H. Frahm, A. R. Its, and V. E. Korepin, Painlev´ e transcendent describes quantum correlation function of the antiferromagnet away from the free-fermion point.J. Phys. A29(17) (1996), pp. 5619–5626

1996
[11]

A. S. Fokas, A. R. Its, A. A. Kapaev and V. Y. Novokshenov,Painlev´ e Transcendents: the Riemann-Hilbert Approach, American Mathematical Soc., R.I., 2006

2006
[12]

Jimbo, T

M. Jimbo, T. Miwa, Y. Mˆ ori, and M. Sato, Density matrix of an impenetrable Bose gas and the fifth Painlev´ e transcendent.Phys. D1(1) (1980), pp. 80–158

1980
[13]

Joshi and M

N. Joshi and M. D. Kruskal, The Painlev´ e connection problem: an asymptotic approach. I, Stud. Appl. Math.,86(1992), 315–376

1992
[14]

Joshi and A

N. Joshi and A. V. Kitaev, The Dirichlet boundary value problem for real solutions of the first Painlev´ e equation on segments in non-positive semi-axis.J. Reine Angew. Math.583(2005), 29–86

2005
[15]

Kanzieper, Replica field theories, Painlev´ e transcendents, and exact correlation functions, Phys

E. Kanzieper, Replica field theories, Painlev´ e transcendents, and exact correlation functions, Phys. Rev. Lett.89(25) (2002), article no. 250201

2002
[16]

A. A. Kapaev, Asymptotic behavior of the solutions of the Painlev´ e equation of the first kind, Differ. Uravn.,24(1988), 1684–1695

1988
[17]

A. A. Kapaev, Quasi-linear Stokes phenomenon for the Painlev´ e first equation,J. Phys. A: Math. Gen.,37(2004), 11149–11167

2004
[18]

A. A. Kapaev and A. V. Kitaev, Connection formulae for the first Painlev´ e transcendent in the complex domain,Lett. Math. Phys.,27(1993), 243–252

1993
[19]

Kawai and Y

T. Kawai and Y. Takei,Algebraic Analysis of Singular Perturbation Theory, American Math- ematical Soc., R.I., 2005

2005
[21]

Long and Y.-T

W.-G. Long and Y.-T. Li Connection problem of the first Painlev´ e transcendents with large initial data,J. Phys. A: Math. Theor.,56(2023), 175201. 25

2023
[23]

Long and J

W.-G. Long and J. Xia, On the asymptotics of real solutions for the Painlev´ e I equation,Anal. Appl., 2025, DOI: 10.1142/S0219530524500441

2025 doi
[24]

B. M. McCoy, Spin systems, statistical mechanics and Painlev´ e functions, in: Painlev´ e Tran- scendents: Their Asymptotics and Physical Applications, D. Levi and P. Winternitz (Eds.), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, (1992) pp. 377–391

1992
[25]

F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/, Release 1.2.0 of 2024-03-15

2024
[26]

C. A. Tracy and H. Widom, Painlev´ e functions in statistical physics,Publ. RIMS Kyoto Univ. 47(2011), 361–374

2011
[27]

T. T. Wu, B. M. McCoy, C. A. Tracy, and E. Barouch, Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region.Phys. Rev. B13(1976), pp. 316–374. 26

1976

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [20]

Long, Y.-J

W.-G. Long, Y.-J. Jiang and Y.-T. Li, Full asymptotic expansion of monodromy data for the first Painlev´ e transcendent: applications to connection problems,Nonlinearity,38(2025) 035005

work page 2025

[2] [22]

Long, Y.-T

W.-G. Long, Y.-T. Li, and Q.-h. Wang, Connection problem of the first Painlev´ e transcendent between poles and negative infinity,SIAM J. Math. Anal., 55 (2023), 6676–6706

work page 2023

[3] [1]

M. J. Ablowitz, S. Chakravarty, and R. G., Halburd, Integrable systems and reductions of the self-dual Yang–Mills equations,J. Math. Phys., 44 (8) (2003), 3147–3173

work page 2003

[4] [2]

Barouch, B

E. Barouch, B. M. McCoy, and T. T. Wu, Zero-field susceptibility of the two-dimensional Ising model nearT c.Phys. Rev. Lett.31(1973), pp. 1409–1411

work page 1973

[5] [3]

A. P. Bassom, P. A. Clarkson, C. K. Law and J. B. McLeod, Application of uniform asymptotics to the second Painlev´ e transcendent,Arch. Rational Mech. Anal.,143(1998), 241–271

work page 1998

[6] [4]

C. M. Bender and J. Komijani, Painlev´ e transcendents and PT-symmetric Hamiltonians,J. Phys. A: Math. Theor.,48(2015), 475202, 15 pp

work page 2015

[7] [5]

C. M. Bender, J. Komijani and Q. Wang, Nonlinear eigenvalue problems for generalized Painlev´ e equations,J. Phys. A: Math. and Theor.,52(2019), 315202. 24

work page 2019

[8] [6]

Bertola and A

M. Bertola and A. Tovbis, Universality for the focusing nonlinear Schr¨ odinger equation at the gradient catastrophe point: rational breathers and poles of the tritronqu´ ee solution to Painlev´ e I,Comm. Pure Appl. Math., 66 (2013), pp. 678–752

work page 2013