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Painlevé Equations and Related Topics Proceedings of the International Conference, Saint Petersburg, Russia, June 17-23, 2011

:	Alexander D. Bruno, Alexander B. Batkhin
:	Painlevé Equations and Related Topics Proceedings of the International Conference, Saint Petersburg, Russia, June 17-23, 2011
:	Walter de Gruyter GmbH& Co.KG
:	9783110275667
:	De Gruyter Proceedings in Mathematics
:	1
:	CHF 177.40
:

:	Mathematik
:	English

:	286
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

< >This is a proceedings of the international conference 'Painlevé Equations and Related Topics' which was taking place at the Euler International Mathematical Institute, a branch of the Saint Petersburg Department of the SteklovInstitute of Mathematicsof theRussian Academy of Sciences, in Saint Petersburg on June 17 to 23, 2011. The survey articles discuss the following topics: general ordinary differentialequations, Painlevé equations and their generalizations, Painlevé property, discrete Painlevé equations, properties of solutions of all mentioned above equations, reductions ofpartial differential equationsto Painlevé equations and their generalizations,ordinary differentialequation systems equivalent to Painlevé equations and their generalizations, and applications of the equations and the solutions.

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< >Alexander D. Brunoand Alexander B. Batkhin, Russian Academy of Sciences, Moscow, Russia.

	Preface	5
	I Plane Power Geometry	15
	1 Plane Power Geometry for One ODE and P1 – P6	17
	1.1 Statement of the Problem	17
	1.2 Computation of Truncated Equations	18
	1.3 Computation of Expansions of Solutions to the Initial Equation (1.1) .	20
	1.4 Extension of the Class of Solutions	21
	1.5 Solution of Truncated Equations	21
	1.6 Types of Expansions	24
	1.7 Painlevé Equations Pl	25
	2 New Simple Exact Solutions to Equation P6	27
	2.1 Introduction	27
	2.1.1 Power Geometry Essentials	27
	2.1.2 Matching “Heads” and “Tails” of Expansions	28
	2.2 Constructing the Template of an Exact Solution	29
	2.3 Results	31
	2.3.1 Known Exact Solutions to P6	31
	2.3.2 Computed Solutions	31
	2.3.3 Generalization of Computed Solutions	34
	3 Convergence of a Formal Solution to an ODE	37
	3.1 The General Case	37
	3.2 The Case of Rational Power Exponents	38
	3.3 The Case of Complex Power Exponents	39
	3.4 On Solutions of the Sixth Painlevé Equation	39
	4 Asymptotic Expansions and Forms of Solutions to P6	41
	4.1 Asymptotic Expansions near Singular Points of the Equation	41
	4.2 Asymptotic Expansions near a Regular Point of the Equation	44
	4.3 Boutroux-Type Elliptic Asymptotic Forms	44
	5 Asymptotic Expansions of Solutions to P5	47
	5.1 Introduction	47
	5.2 Asymptotic Expansions of Solutions near Infinity	49
	5.3 Asymptotic Expansions of Solutions near Zero	49
	5.4 Asymptotic Expansions of Solutions in the Neighborhood of the Nonsingular Point of an Equation	51
	II Space Power Geometry	53
	6 Space Power Geometry for one ODE and P1 – P4, P6	55
	6.1 Space Power Geometry	55
	6.2 Asymptotic Forms of Solutions to Painlevé Equations P1 – P4, P6	58
	6.2.1 Equation P1	58
	6.2.2 Equation P2	59
	6.2.3 Equation P3 for cd . 0	60
	6.2.4 Equation P3 for c = 0 and ad . 0	61
	6.2.5 Equation P3 for c = d = 0 and ab . 0	62
	6.2.6 Equation P4	63
	6.2.7 Equation P6	64
	7 Elliptic and Periodic Asymptotic Forms of Solutions to P5	67
	7.1 The Fifth Painlevé Equation	67
	7.2 The case d . 0	68
	7.2.1 General Properties of the P5 Equation	68
	7.2.2 The First Family of Elliptic Asymptotic Forms	69
	7.2.3 The First Family of Periodic Asymptotic Forms	71
	7.2.4 The Second Family of Periodic Asymptotic Forms	72
	7.3 The Case d . 0, . . 0	73
	7.3.1 General Properties	73
	7.3.2 The Second Family of Elliptic Asymptotic Forms	74
	7.3.3 The Third Family of Periodic Asymptotic Forms	76
	7.3.4 The Fourth Family of Periodic Asymptotic Forms	77
	7.4 The Results Obtained	78
	8 Regular Asymptotic Expansions of Solutions to One ODE and P1–P5	81
	8.1 Introduction	81
	8.2 Finding Asymptotic Forms	82
	8.3 Computation of Expansions (8.2)	83
	8.4 Equation P1	85
	8.5 Equation P2	87
	8.5.1 Elliptic Asymptotic Forms, Face G3(2)	87
	8.5.2 Periodic Asymptotic Forms, Face G4(2)	88
	8.6 Equation P3	89
	8.6.1 Case cd . 0	89
	8.6.2 Case c = 0, ad . 0	90
	8.6.3 Case c = d = 0, ab . 0	91
	8.7 Equation P4	91
	8.7.1 Elliptic Asymptotic Forms, Face G3(2)	92
	8.7.2 Periodic Asymptotic Forms, Face G4(2)	92
	8.8 Equation P5	93
	8.8.1 Case d . 0, Elliptic Asymptotic Forms, Face G1(2)	93
	8.8.2 Case d . 0, Periodic Asymptotic Forms, Face G2(2)	95
	8.8.3 Case d = 0, c . 0, Elliptic Asymptotic Forms, Face G1(2)	95
	8.8.4 Case d = 0, c . 0, Periodic Asymptotic Forms, Face G2(2)	95
	III Isomondromy Deformations	97
	9 Isomonodromic Deformations on Riemann Surfaces	99
	9.1 Introduction	99
	9.2 The Space of Parameters T~	100
	9.3 The Description of Bundles with Connections on a Riemann Surface	100
	9.4 Isomonodromic Deformations	101
	10 On Birational Darboux Coordinates of Isomonodromic Deformation Equations Phase Space	105
	11 On the Malgrange Isomonodromic Deformations of Nonresonant Irregular Systems	109
	11.1 Introduction	109
	11.2 The Malgrange Isomonodromic Deformation of the Pair (E0, V¯0)	110
	11.3 Specificity of Meromorphic 2x2-Connections	112
	12 Critical behavior of P6 Functions from the Isomonodromy Deformations Approach	115
	12.1 Introduction	115
	12.2 Behavior of y(x)	116
	12.3 Parameterization in Terms of Monodromy Data	118
	13 Isomonodromy Deformation of the Heun Class Equation	121
	13.1 Introduction	121
	13.2 Gauge Transforms of Linear Differential Equations	122
	13.3 Gauge Transforms of the Hypergeometric Class Equations	125
	13.4 Gauge Transform of Heun Class Equations	126
	13.4.1 Formulation of the Problem	126
	13.4.2 Initial System of Equations and Equation Heunc2	127
	13.4.3 Parameters of the Transformed Equation