Painlevé Equations and Related Topics (eBook)

Proceedings of the International Conference, Saint Petersburg, Russia, June 17-23, 2011

Alexander D. Bruno, Alexander B. Batkhin (Herausgeber)

eBook Download: PDF

2012
286 Seiten
De Gruyter (Verlag)
978-3-11-027566-7 (ISBN)

Infos

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.

Alexander D. Bruno and Alexander B. Batkhin, Russian Academy of Sciences, Moscow, Russia.

Autor

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Alexander D. Bruno and Alexander B. Batkhin, Russian Academy of Sciences, Moscow, Russia.

Inhalt

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 128
13.5 Conclusion 130
14 Isomonodromy Deformations and Hypergeometric-Type Systems 131
14.1 Schlesinger Families of Fuchsian Systems 131
14.2 Schlesinger Systems 132
14.3 Upper-Triangular Schlesinger Systems 132
14.4 Jordan-Pochhammer Systems 134
14.5 The Basic Result 135
15 A Monodromy Problem Connected with P6 137
15.1 Preliminaries I 137
15.2 Preliminaries II 138
15.3 Main Result 139
15.4 Example 140
16 Monodromy Evolving Deformations and Confluent Halphen’s Systems 143
16.1 Introduction 143
16.2 Quadratic Systems and Nonassociative Algebras 145
16.3 Monodromy Evolving Deformations 147
16.4 Halphen’s Confluent Systems and Monodromy Evolving Deformations 149
17 On the Gauge Transformation of the Sixth Painlevé Equation 151
17.1 Linearizations of the Sixth Painlevé Equation 151
17.1.1 LODE LVI 152
17.1.2 LODE LVI 153
17.1.3 LODE LVI 154
17.1.4 Schlesinger System with Symmetric Gauge 156
17.1.5 Schlesinger System with Asymmetric Gauge 157
17.2 Schlesinger Transformation LVI . LVI 157
18 Expansions for Solutions of the Schlesinger Equation at a Singular Point 165
18.1 Introduction 165
18.2 Schlesinger Equation and Isomonodromic Deformations 168
18.3 Sketch of the Proof 169
IV Painlevé Property 173
19 Painleve Analysis of Lotka-Volterra Equations 175
20 Painlevé Test and Briot-Bouquet Systems 179
21 Solutions of the Chazy System 181
22 Third-Order Ordinary Differential Equations with the Painlevé Test 185
22.1 Introduction 185
22.2 Simplified Equation 186
22.3 Reduced Equations 188
22.3.1 Leading Order k =-1 188
22.3.2 Leading Order k = -2 195
22.3.3 Leading Order k = -3 196
22.3.4 Leading Order k = -4 197
23 Analytic Properties of Solutions of a Class of Third-Order Equations with an Irrational Right-Hand Side 199
V Other Aspects 205
24 The Sixth Painlevé Transcendent and Uniformizable Orbifolds 207
24.1 Algebraic Solutions of P6 and Uniformization Theory 207
24.2 On the General Solution to Equation (24.1) 208
24.3 Calculus: Abelian Integrals and Affine (Analytic) Connections 209
25 On Uniformizable Representation for Abelian Integrals 213
25.1 Introduction 213
25.2 Schwarz Equation and Equations on Tori 214
25.3 Holomorphic Elliptic Integrals and Hypergeometric Functions 215
25.3.1 Lemniscate 215
25.3.2 Equi-Anharmonic Curve 217
25.4 Abelian Integrals for Genus g > 1
25.4.1 Higher Genera. Examples 219
26 Phase Shift for a Special Solution to the Korteweg-de Vries Equation in the Whitham Zone 223
26.1 Introduction 223
26.2 Evaluation of the Phase Shift 224
27 Fuchsian Reduction of Differential Equations 227
27.1 Fuchsian Reduction 229
27.1.1 Two Simple Examples 229
27.1.2 A More Complex Example 230
27.2 Two Applications: Astronomy and Relativity Theory 231
27.2.1 Astronomy. A Model of Gaseous Stars 231
27.2.2 Relativity. Gowdy Space-Time 232
27.3 Fuchsian Systems for Feynman Integrals 234
28 The Voros Coefficient and the Parametric Stokes Phenomenon for the Second Painlevé Equation 239
28.1 Introduction 239
28.2 Connection Formula for the Parametric Stokes Phenomenon 240
28.3 Derivation of the Connection Formulas Through the Analysis of the Voros Coefficient of (P2) 242
29 Integral Symmetry and the Deformed Hypergeometric Equation 245
30 Integral Symmetries for Confluent Heun Equations and Symmetries of Painleve Equation P5 251
31 From the Tau Function of Painlevé P6 Equation to Moduli Spaces 255
32 On particular Solutions of q-Painlevé Equations and q-Hypergeometric Equations 261
32.1 Introduction 261
32.2 q-Difference Equation of the Hypergeometric Type 261
32.3 Hypergeometric Solutions of the q-Painlevé Equations 264
33 Derivation of Painlevé Equations by Antiquantization 267
34 Integral Transformation of Heun’s Equation and Apparent Singularity 271
34.1 Heun’s Equation and Integral Transformation 271
34.2 Apparent Singularity and Integral Representation of Solutions 272
34.3 Elliptical Representation of Heun’s Equation and Integral Transformation 273
35 Painlevé Analysis of Solutions to Some Nonlinear Differential Equations and their Systems Associated with Models of the Random-Matrix Type 277
35.1 Introduction 277
35.2 Model of the Random-Matrix Type with Airy Kernal 278
35.3 System of Differential Equations Associated with the Dyson Process 278
35.4 Solutions of the Traveling-Wave Form of a Partial Differential Equation 279
36 Reductions on the Lattice and Painlevé Equations P2, P5, P6 281
36.1 Introduction 281
36.2 Symmetries of the ABS Equations 282
36.3 Reduction on the Lattice and Discrete Painlevé Equations 283
36.4 Continuous Symmetry Reductions 283
Comments 285

Produkt-Details

• Erscheint lt. Verlag: 31.8.2012
• Reihe/Serie: De Gruyter Proceedings in Mathematics
• Reihe/Serie: De Gruyter Proceedings in Mathematics
• Reihe/Serie: ISSN
• Reihe/Serie: ISSN
• Co-Autor: Yasin Adjabi, Tatsyana K. Andreeva, Dimitry V. Artamonov, Mikhail V. Babich, Alexander D. Batkhin, Natalie V. Batkhina, Yuliya P. Bibilo, Yurii V. Brezhnev, Alexander D. Bruno, Pantelis A. Damianou, Rustem N. Garifullin, Valentina A. Golubeva, Renat R. Gontsov, Irina V. Goryuchkina, Valerii I. Gromak, Davide Guzzetti, Kohei Iwaki, Alexander Ya. Kazakov, Arezki Kessi, Dmitry Korotkin, Vladimir P. Leksin, Ivan P. Martynov, Dmitrii P. Novikov, Yousuke Ohyama, Anastasya V. Parusnikova, Vyacheslav A. Pronko, Yoshikatsu Sasaki, Sergey Yu. Slavyanov, Kouichi Takemura, Vladimir Tsegel'nik, Ilya V. Vyugin, Pavlos Xenitidis, Peter Zograf
• Zusatzinfo: 25 b/w ill., 6 b/w tbl.
• Verlagsort: Berlin/Boston
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• Schlagworte: discrete Painlevé equations • ODE system • ordinary differential equation • painlevé equations • Painlevé Property • Panilevé Equation • PDE reduction • solution properties
• ISBN-10: 3-11-027566-X / 311027566X
• ISBN-13: 978-3-11-027566-7 / 9783110275667