Yu. V. Brezhnev, “A $\tau$-function solution of the sixth Painlevé transcendent”, TMF, 161:3 (2009), 346–366; Theoret. and Math. Phys., 161:3 (2009), 1616

Loading [MathJax]/jax/output/SVG/config.js

Teoreticheskaya i Matematicheskaya Fizika

RUS ENG

JOURNALS PEOPLE ORGANISATIONS CONFERENCES SEMINARS VIDEO LIBRARY PACKAGE AMSBIB

JavaScript is disabled in your browser. Please switch it on to enable full functionality of the website

	General information
	Latest issue
	Archive
	Impact factor
	Guidelines for authors
	License agreement
	Submit a manuscript

	Search papers
	Search references

	RSS
	Latest issue
	Current issues
	Archive issues
	What is RSS

TMF:
Year:
Volume:
Issue:
Page:
	Find

Personal entry:
Login:
Password:
	Save password
	Enter
	Forgotten password?
	Register

Teoreticheskaya i Matematicheskaya Fizika, 2009, Volume 161, Number 3, Pages 346–366
DOI: https://doi.org/10.4213/tmf6446 (Mi tmf6446)

This article is cited in 12 scientific papers (total in 12 papers)

A $\tau$-function solution of the sixth Painlevé transcendent

Yu. V. Brezhnev

Tomsk State University, Tomsk, Russia

Full-text PDF (1043 kB) Citations (12)

References:

PDF

HTML

DOI: https://doi.org/10.4213/tmf6446

Abstract: We represent and analyze the general solution of the sixth Painlevé transcendent $\mathcal P_6$ in the Picard–Hitchin–Okamoto class in the Painlevé form as the logarithmic derivative of the ratio of $\tau$-functions. We express these functions explicitly in terms of the elliptic Legendre integrals and Jacobi theta functions, for which we write the general differentiation rules. We also establish a relation between the $\mathcal P_6$ equation and the uniformization of algebraic curves and present examples.

Keywords: Painlevé VI equation, elliptic function, theta function, uniformization, automorphic function.

Received: 04.03.2009
Revised: 08.04.2009

English version:
Theoretical and Mathematical Physics, 2009, Volume 161, Issue 3, Pages 1616–1633
DOI: https://doi.org/10.1007/s11232-009-0150-z

Bibliographic databases:

Language: Russian

Citation: Yu. V. Brezhnev, “A $\tau$-function solution of the sixth Painlevé transcendent”, TMF, 161:3 (2009), 346–366; Theoret. and Math. Phys., 161:3 (2009), 1616–1633

Citation in format AMSBIB

\Bibitem{Bre09}

\by Yu.~V.~Brezhnev

\paper A~$\tau$-function solution of the~sixth Painlev\'e transcendent

\jour TMF

\yr 2009

\vol 161

\issue 3

\pages 346--366

\mathnet{http://mi.mathnet.ru/tmf6446}

\crossref{https://doi.org/10.4213/tmf6446}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2642195}

\zmath{https://zbmath.org/?q=an:1185.37137}

\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2009TMP...161.1616B}

\transl

\jour Theoret. and Math. Phys.

\yr 2009

\vol 161

\issue 3

\pages 1616--1633

\crossref{https://doi.org/10.1007/s11232-009-0150-z}

\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000273561600005}

\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-76649090285}

Linking options:

https://www.mathnet.ru/eng/tmf6446

https://doi.org/10.4213/tmf6446

https://www.mathnet.ru/eng/tmf/v161/i3/p346

This publication is cited in the following 12 articles:

Gontsov R.R., Goryuchkina V I., “On the Convergence of Formal Exotic Series Solutions of An Ode”, Comput. Methods Funct. Theory, 20:2 (2020), 279–295
Chen Zh., Kuo T.-J., Lin Ch.-Sh., “Unitary monodromy implies the smoothness along the real axis for some Painlevé VI equation, I”, J. Geom. Phys., 116 (2017), 52–63
Brezhnev Yu.V., “The sixth Painlevé transcendent and uniformization of algebraic curves”, J. Differ. Equ., 260:3 (2016), 2507–2556
Davide Guzzetti, “A Review of the Sixth Painlevé Equation”, Constr Approx, 41:3 (2015), 495
Brezhnev Yu.V., “Non-Canonical Extension of Theta-Functions and Modular Integrability of Theta-Constants”, Proc. R. Soc. Edinb. Sect. A-Math., 143:4 (2013), 689–738
Guzzetti D., “Pole Distribution of Pvi Transcendents Close to a Critical Point”, Physica D, 241:23-24 (2012), 2188–2203
Brezhnev Yu.V., “Spectral/Quadrature Duality: Picard-Vessiot Theory and Finite-Gap Potentials”, Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics, Contemporary Mathematics, 563, eds. AcostaHumanez P., Finkel F., Kamran N., Olver P., Amer Mathematical Soc, 2012, 1–31
Davide Guzzetti, “Tabulation of Painlevé 6 transcendents”, Nonlinearity, 25:12 (2012), 3235
Blower G., “On linear systems and $\tau$ functions associated with Lamé's equation and Painlevé's equation VI”, J. Math. Anal. Appl., 376:1 (2011), 294–316
Goryuchkina I.V., “Reshenie pikara shestogo uravneniya penleve i asimptotiki, poluchennye s pomoschyu stepennoi geometrii”, Preprinty IPM im. M.V. Keldysha, 2011, no. 66, 1–8
I. V. Goryuchkina, “Reshenie Pikara shestogo uravneniya Penleve i asimptotiki, poluchennye s pomoschyu stepennoi geometrii”, Preprinty IPM im. M. V. Keldysha, 2011, 066, 8 pp.
Mangazeev V.V., Guttmann A.J., “Form factor expansions in the 2D Ising model and Painlevé VI”, Nuclear Phys. B, 838:3 (2010), 391–412

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Statistics & downloads:
Abstract page:	660
Full-text PDF :	255
References:	121
First page:	28

Registration to the website

Logotypes