Abstract
We show that the well-known Hastings–McLeod solution to the second Painlevé equation is pole-free in the region \(\arg x \in [-\frac{\pi }{3},\frac{\pi }{3}]\cup [\frac{2\pi }{3},\frac{ 4 \pi }{3}]\), which proves an important special case of a general conjecture concerning pole distributions of Painlevé transcedents proposed by Novokshenov. Our strategy is to construct explicit quasi-solutions approximating the Hastings–McLeod solution in different regions of the complex plane and estimate the errors rigorously. The main idea is very similar to the one used to prove Dubrovin’s conjecture for the first Painlevé equation, but there are various technical improvements.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Adali, A., Tanveer, T.: Rigorous analytical approximation of tritronquee solution to Painleve-1 and the first singularity, preprint. arXiv:1412.3782
Baik, J., Deift, P., Johansson, K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Am. Math. Soc. 12, 1119–1178 (1999)
Bertola, M.: On the location of poles for the Ablowitz–Segur family of solutions to the second Painlevé equation. Nonlinearity 25, 1179–1185 (2012)
Bertola, M., Tovbis, A.: Universality for the focusing nonlinear Schrödinger equation at the gradient catastrophe point: rational breathers and poles of the tritronquée solution to Painlevé I. Commun. Pure Appl. Math. 66, 678–752 (2013)
Bleher, P., Its, A.: Double scaling limit in the random matrix model: the Riemann–Hilbert approach. Commun. Pure Appl. Math. 56, 433–516 (2003)
Boutroux, P.: Recherches sur les transcendantes de M. Painlevé et l’étude asymptotique des équations différentielles du second ordre. Ann. Sci. école Norm. Sup. (3) 30, 255–375 (1913)
Boutroux, P.: Recherches sur les transcendantes de M. Painlevé et l’étude asymptotique des équations différentielles du second ordre (suite). Ann. Sci. école Norm. 31, 99–159 (1914)
Claeys, T., Kuijlaars, A.B.J.: Universality of the double scaling limit in random matrix models. Commun. Pure Appl. Math. 59, 1573–1603 (2006)
Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1995)
Costin, O.: On Borel summation and Stokes phenomenon for rank-1 nonlinear differential systems of differential equations. Duke Math. J. 93, 289–344 (1998)
Costin, O., Costin, R.D.: On the formation of singularities of solutions of nonlinear differential systems in antistokes directions. Invent. Math. 145, 425–485 (2001)
Costin, O., Huang, M., Schlag, W.: On the spectral properties of \(L_{\pm }\) in three dimensions. Nonlinearity 25, 125–164 (2012)
Costin, O., Huang, M., Tanveer, S.: Proof of the Dubrovin conjecture and analysis of the tritronqué solutions of \(P_{\rm I}\). Duke Math. J. 163, 665–704 (2014)
Delvaux, S., Kuijlaars, A.B.J.: A phase transition for non-intersecting Brownian motions, and the Painlevé II equation. Int. Math. Res. Not. 2009, 3639–3725 (2009)
Delvaux, S., Kuijlaars, A.B.J., Zhang, L.: Critical behavior of non-intersecting Brownian motions at a tacnode. Commun. Pure Appl. Math. 64, 1305–1383 (2011)
Dubrovin, B., Grava, T., Klein, C.: On universality of critical behavior in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the tritonqueé solution to the Painlevé-I equation. J. Nonlinear Sci. 19, 57–94 (2009)
Duits, M., Geudens, D.: A critical phenomenon in the two-matrix model in the quartic/quadratic case. Duke Math. J. 162, 1383–1462 (2013)
Fokas, A.S., Its, A.R., Kapaev, A.A., Novokshenov, V.Y.: Painlevé transcendents: The Riemann–Hilbert approach, Mathematical Surveys and Monographs 128. Am. Math. Soc., Providence, RI (2006)
Fornberg, B., Weideman, J.A.C.: A numerical methodology for the Painlevé equations. J. Comput. Phys. 230, 5957–5973 (2011)
Fornberg, B., Weideman, J.A.C.: A computational exploration of the second Painlevé equation. Found. Comput. Math. 14, 985–1016 (2014)
Forrester, P.J., Witte, N.S.: Painlevé II in random matrix theory and related fields. Constr. Approx. 41, 589–613 (2015)
Hastings, S.P., McLeod, J.B.: A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation. Arch. Ration. Mech. Anal. 73, 31–51 (1980)
Its, A.R., Kapaev, A.A.: Quasi-linear Stokes phenomenon for the second Painlevé transcendent. Nonlinearity 16, 363–386 (2003)
Kapaev, A.A.: Global asymptotics of the second Painlevé transcendent. Phys. Lett. A 167, 356–362 (1992)
Kapaev, A.A.: Quasi-linear Stokes phenomenon for the Painlevé first equation. J. Phys. A Math. Gen. 37, 11149–11167 (2004)
Masoero, D.: Poles of intégrale tritronquée and anharmonic oscillators: a WKB approach. J. Phys. A 43, 095201 (2000)
Masoero, D.: Poles of intégrale tritronquée and anharmonic oscillators: asymptotic localization from WKB analysis. Nonlinearity 23, 2501–2507 (2010)
Novokshenov, V.Y.: Padé approximations for Painlevé I and II transcendents, (Russian) Teoret. Math. Fiz. 159, 515–526 (2009); translation in Theor. Math. Phys. 159, 853–862 (2009)
Novokshenov, V.Y.: Distributions of poles to Painlevé transcendents via Padé approximations. Constr. Approx. 39, 85–99 (2014)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010). Print companion to [DLMF]
Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Phys. Lett. B 305, 115–118 (1993)
Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994)
Acknowledgments
We would like to express our sincere gratitude and appreciation to Roderick Wong and Dan Dai for organizing a series of conferences and seminars that facilitated our collaboration on this project, as well as other related discussions. Min Huang was supported by the Early Career Scheme 21300114 of Research Grants Council of Hong Kong. Shuai-Xia Xu was supported in part by the National Natural Science Foundation of China under Grant No. 11201493 and 11571376, GuangDong Natural Science Foundation under Grant No. S2012040007824 and 2014A030313176, Postdoctoral Science Foundation of China under Grant No. 2012M521638, and the Fundamental Research Funds for the Central Universities under Grant No. 13lgpy41. Lun Zhang was partially supported by The Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (No. SHH1411007), by National Natural Science Foundation of China (No.11501120) and by Grant SGST 12DZ 2272800, EZH1411513 from Fudan University.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Percy A. Deift.
Rights and permissions
About this article
Cite this article
Huang, M., Xu, SX. & Zhang, L. Location of Poles for the Hastings–McLeod Solution to the Second Painlevé Equation. Constr Approx 43, 463–494 (2016). https://doi.org/10.1007/s00365-015-9307-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00365-015-9307-1