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Teoreticheskaya i Matematicheskaya Fizika, 2012, Volume 172, Number 2, Pages 296–307
DOI: https://doi.org/10.4213/tmf6951
(Mi tmf6951)
 

This article is cited in 7 scientific papers (total in 7 papers)

Tronquée solutions of the Painlevé II equation

V. Yu. Novokshenov

Institute of Mathematics, RAS, Ufa, Russia
Full-text PDF (534 kB) Citations (7)
References:
Abstract: We study special solutions of the Painlevé II (PII) equation called tronquée solutions, i.e., those having no poles along one or more critical rays in the complex plane. They are parameterized by special monodromy data of the Lax pair equations. The manifold of the monodromy data for a general solution is a two-dimensional complex manifold with one- and zero-dimensional singularities, which arise because there is no global parameterization of the manifold. We show that these and only these singularities (together with zeros of the parameterization) are related to the tronquée solutions of the PII equation. As an illustration, we consider the known Hastings–McLeod and Ablowitz–Segur solutions and some other solutions to show that they belong to the class of tronquée solutions and correspond to one or another type of singularity of the monodromy data.
Keywords: Painlevé equation, tronquée solution, distribution of poles, Riemann–Hilbert problem, anharmonic oscillator, Bohr–Sommerfeld quantization, complex WKB method.
English version:
Theoretical and Mathematical Physics, 2012, Volume 172, Issue 2, Pages 1136–1146
DOI: https://doi.org/10.1007/s11232-012-0102-x
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: V. Yu. Novokshenov, “Tronquée solutions of the Painlevé II equation”, TMF, 172:2 (2012), 296–307; Theoret. and Math. Phys., 172:2 (2012), 1136–1146
Citation in format AMSBIB
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Linking options:
  • https://www.mathnet.ru/eng/tmf6951
  • https://doi.org/10.4213/tmf6951
  • https://www.mathnet.ru/eng/tmf/v172/i2/p296
  • This publication is cited in the following 7 articles:
    1. B. I. Suleimanov, “Isomonodromic quantization of the second Painlevé equation by means of conservative Hamiltonian systems with two degrees of freedom”, St. Petersburg Math. J., 33:6 (2022), 995–1009  mathnet  crossref
    2. Peter A. Clarkson, “Open Problems for Painlevé Equations”, SIGMA, 15 (2019), 006, 20 pp.  mathnet  crossref
    3. Peter D. Miller, “On the Increasing Tritronquée Solutions of the Painlevé-II Equation”, SIGMA, 14 (2018), 125, 38 pp.  mathnet  crossref
    4. Bothner T., Miller P.D., Sheng Yu., “Rational Solutions of the Painleve-III Equation”, Stud. Appl. Math., 141:4, SI (2018), 626–679  crossref  mathscinet  zmath  isi  scopus
    5. N. Steinmetz, “A unified approach to the Painlevé transcendents”, Ann. Acad. Sci. Fenn. Ser. A1-Math., 42:1 (2017), 17–49  crossref  mathscinet  zmath  isi
    6. I. Rumanov, “Painlevé representation of Tracy-Widom$_\beta$ distribution for $\beta=6$”, Commun. Math. Phys., 342:3 (2016), 843–868  crossref  mathscinet  zmath  adsnasa  isi
    7. B. Fornberg, J. A. C. Weideman, “A computational overview of the solution space of the imaginary Painlevé II equation”, Physica D, 309 (2015), 108–118  crossref  mathscinet  zmath  adsnasa  isi  elib
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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