Painlevé equations

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21: 32.7 Bäcklund Transformations

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§32.7(ii) Second Painlevé Equation

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§32.7(iii) Third Painlevé Equation

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§32.7(iv) Fourth Painlevé Equation

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§32.7(v) Fifth Painlevé Equation

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22: 32.6 Hamiltonian Structure

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§32.6(ii) First Painlevé Equation

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§32.6(iii) Second Painlevé Equation

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§32.6(iv) Third Painlevé Equation

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§32.6(v) Other Painlevé Equations

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23: Bibliography F

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On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

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External Links:

ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.13(i), §32.7(i), §32.7(iii).

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From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.

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External Links:

ISSN 0022-247X, Document, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§32.7(ii).

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Discrete Painlevé equations and their appearance in quantum gravity. Comm. Math. Phys. 142 (2), pp. 313–344.

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ISSN 0010-3616, Link, MathReview (Nikolai A. Kostov), ZentralBlatt

Referenced by:

§32.15.

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The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.

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External Links:

ISSN 0024-1318, MathReview (D. A. Lutz)

Referenced by:

§32.10(vi).

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A numerical methodology for the Painlevé equations. J. Comput. Phys. 230 (15), pp. 5957–5973.

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External Links:

ISSN 0021-9991, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§32.17.

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24: Bibliography Y

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On rational solutions of the second Painlevé equation. Vesti Akad. Navuk. BSSR Ser. Fiz. Tkh. Nauk. 3, pp. 30–35 (Russian).

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Referenced by:

§32.8(ii).

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25: Bibliography I

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On the asymptotic analysis of the Painlevé equations via the isomonodromy method. Nonlinearity 7 (5), pp. 1291–1325.

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External Links:

ISSN 0951-7715, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt

Referenced by:

§32.11(iii).

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The Isomonodromic Deformation Method in the Theory of Painlevé Equations. Lecture Notes in Mathematics, Vol. 1191, Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-16483-9, MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§32.11(iv), §32.4(i), Profile Alexander A. Its.

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26: 32.11 Asymptotic Approximations for Real Variables

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§32.11(i) First Painlevé Equation

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§32.11(ii) Second Painlevé Equation

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§32.11(iii) Modified Second Painlevé Equation

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§32.11(iv) Third Painlevé Equation

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§32.11(v) Fourth Painlevé Equation

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27: Bibliography K

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Asymptotic behavior of the solutions of the Painlevé equation of the first kind. Differ. Uravn. 24 (10), pp. 1684–1695 (Russian).

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Notes:

In Russian. English translation: Differential Equations 24(1988), no. 10, pp. 1107–1115 (1989)

External Links:

ISSN 0374-0641, MathReview, ZentralBlatt

Referenced by:

§32.11(i).

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The second Painlevé equation in electric probe theory. Some numerical solutions. Zh. Vychisl. Mat. Mat. Fiz. 38 (6), pp. 992–1000 (Russian).

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Notes:

In Russian. English translation: Comput. Math. Math. Phys. 38(1998), no. 6, pp. 950–958

External Links:

ISSN 0044-4669, MathReview, ZentralBlatt

Referenced by:

§32.17.

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The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface. Technical Physics 49 (1), pp. 1–7.

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External Links:

Document

Referenced by:

§32.17.

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Rational solutions of the fifth Painlevé equation. Differential Integral Equations 7 (3-4), pp. 967–1000.

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External Links:

ISSN 0893-4983, MathReview, ZentralBlatt

Referenced by:

§32.8(v).

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:

ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

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28: Bibliography L

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On a set of solutions of the sixth Painlevé equation. Differ. Uravn. 3 (3), pp. 520–523 (Russian).

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Notes:

In Russian. English translation: Differential Equations 3(3), pp. 264–266

External Links:

MathReview (J. Rausen), ZentralBlatt

Referenced by:

§32.10(vi).

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Elementary solutions of certain Painlevé equations. Differ. Uravn. 1 (3), pp. 731–735 (Russian).

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Notes:

In Russian. English translation: Differential Equations 1(3), pp. 561–564

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

§32.10(iii), §32.10(iv), §32.9(i).

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Theory of the fourth Painlevé equation. Differ. Uravn. 3 (5), pp. 771–780 (Russian).

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Notes:

In Russian. English translation: Differential Equations 3(5), pp. 395–399

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

§32.10(iv), §32.7(iv), §32.8(iv).

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Solutions of the fifth Painlevé equation. Differ. Uravn. 4 (8), pp. 1413–1420 (Russian).

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Notes:

In Russian. English translation: Differential Equations 4(8), pp. 732–735

External Links:

MathReview (C. E. Langenhop), ZentralBlatt

Referenced by:

§32.10(v), §32.8(v).

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The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).

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Notes:

In Russian. English translation: Differential Equations 7(6), pp. 853–854

External Links:

MathReview (Z. Rubinstein), ZentralBlatt

Referenced by:

§32.7(ii).

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29: 32.14 Combinatorics

… ►See Forrester and Witte (2001, 2002) for other instances of Painlevé equations in random matrix theory.

30: Bibliography J

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On Boutroux’s tritronquée solutions of the first Painlevé equation. Stud. Appl. Math. 107 (3), pp. 253–291.

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External Links:

ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§32.12(i).

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The Dirichlet boundary value problem for real solutions of the first Painlevé equation on segments in non-positive semi-axis. J. Reine Angew. Math. 583, pp. 29–86.

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External Links:

ISSN 0075-4102, Document, MathReview (Mehmet Can), ZentralBlatt

Referenced by:

§32.11(i).

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