E-Book, Englisch, 286 Seiten
Proceedings of the International Conference, Saint Petersburg, Russia, June 17-23, 2011
E-Book, Englisch, 286 Seiten
Reihe: De Gruyter Proceedings in Mathematics
ISBN: 978-3-11-027566-7
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)
– Asymptotic forms and asymptotic expansions
– Connections of asymptotic forms of a solution near different points
– Convergency and asymptotic character of a formal solution
– New types of asymptotic forms and asymptotic expansions
– Riemann-Hilbert problems
– Isomonodromic deformations of linear systems
– Symmetries and transformations of solutions
– Algebraic solutions - Reductions of PDE to Painlevé equations and their generalizations - Ordinary Differential Equations systems equivalent to Painlevé equations and their generalizations - Applications of the equations and the solutions
Zielgruppe
Students, Graduates, Researchers, and Lecturers in Mathematics; Academic Libraries
Autoren/Hrsg.
Fachgebiete
Weitere Infos & Material
1;Preface;5
2;I Plane Power Geometry;15
2.1;1 Plane Power Geometry for One ODE and P1 – P6;17
2.1.1;1.1 Statement of the Problem;17
2.1.2;1.2 Computation of Truncated Equations;18
2.1.3;1.3 Computation of Expansions of Solutions to the Initial Equation (1.1) .;20
2.1.4;1.4 Extension of the Class of Solutions;21
2.1.5;1.5 Solution of Truncated Equations;21
2.1.6;1.6 Types of Expansions;24
2.1.7;1.7 Painlevé Equations Pl;25
2.2;2 New Simple Exact Solutions to Equation P6;27
2.2.1;2.1 Introduction;27
2.2.1.1;2.1.1 Power Geometry Essentials;27
2.2.1.2;2.1.2 Matching “Heads” and “Tails” of Expansions;28
2.2.2;2.2 Constructing the Template of an Exact Solution;29
2.2.3;2.3 Results;31
2.2.3.1;2.3.1 Known Exact Solutions to P6;31
2.2.3.2;2.3.2 Computed Solutions;31
2.2.3.3;2.3.3 Generalization of Computed Solutions;34
2.3;3 Convergence of a Formal Solution to an ODE;37
2.3.1;3.1 The General Case;37
2.3.2;3.2 The Case of Rational Power Exponents;38
2.3.3;3.3 The Case of Complex Power Exponents;39
2.3.4;3.4 On Solutions of the Sixth Painlevé Equation;39
2.4;4 Asymptotic Expansions and Forms of Solutions to P6;41
2.4.1;4.1 Asymptotic Expansions near Singular Points of the Equation;41
2.4.2;4.2 Asymptotic Expansions near a Regular Point of the Equation;44
2.4.3;4.3 Boutroux-Type Elliptic Asymptotic Forms;44
2.5;5 Asymptotic Expansions of Solutions to P5;47
2.5.1;5.1 Introduction;47
2.5.2;5.2 Asymptotic Expansions of Solutions near Infinity;49
2.5.3;5.3 Asymptotic Expansions of Solutions near Zero;49
2.5.4;5.4 Asymptotic Expansions of Solutions in the Neighborhood of the Nonsingular Point of an Equation;51
3;II Space Power Geometry;53
3.1;6 Space Power Geometry for one ODE and P1 – P4, P6;55
3.1.1;6.1 Space Power Geometry;55
3.1.2;6.2 Asymptotic Forms of Solutions to Painlevé Equations P1 – P4, P6;58
3.1.2.1;6.2.1 Equation P1;58
3.1.2.2;6.2.2 Equation P2;59
3.1.2.3;6.2.3 Equation P3 for cd . 0;60
3.1.2.4;6.2.4 Equation P3 for c = 0 and ad . 0;61
3.1.2.5;6.2.5 Equation P3 for c = d = 0 and ab . 0;62
3.1.2.6;6.2.6 Equation P4;63
3.1.2.7;6.2.7 Equation P6;64
3.2;7 Elliptic and Periodic Asymptotic Forms of Solutions to P5;67
3.2.1;7.1 The Fifth Painlevé Equation;67
3.2.2;7.2 The case d . 0;68
3.2.2.1;7.2.1 General Properties of the P5 Equation;68
3.2.2.2;7.2.2 The First Family of Elliptic Asymptotic Forms;69
3.2.2.3;7.2.3 The First Family of Periodic Asymptotic Forms;71
3.2.2.4;7.2.4 The Second Family of Periodic Asymptotic Forms;72
3.2.3;7.3 The Case d . 0, . . 0;73
3.2.3.1;7.3.1 General Properties;73
3.2.3.2;7.3.2 The Second Family of Elliptic Asymptotic Forms;74
3.2.3.3;7.3.3 The Third Family of Periodic Asymptotic Forms;76
3.2.3.4;7.3.4 The Fourth Family of Periodic Asymptotic Forms;77
3.2.4;7.4 The Results Obtained;78
3.3;8 Regular Asymptotic Expansions of Solutions to One ODE and P1–P5;81
3.3.1;8.1 Introduction;81
3.3.2;8.2 Finding Asymptotic Forms;82
3.3.3;8.3 Computation of Expansions (8.2);83
3.3.4;8.4 Equation P1;85
3.3.5;8.5 Equation P2;87
3.3.5.1;8.5.1 Elliptic Asymptotic Forms, Face G3(2);87
3.3.5.2;8.5.2 Periodic Asymptotic Forms, Face G4(2);88
3.3.6;8.6 Equation P3;89
3.3.6.1;8.6.1 Case cd . 0;89
3.3.6.2;8.6.2 Case c = 0, ad . 0;90
3.3.6.3;8.6.3 Case c = d = 0, ab . 0;91
3.3.7;8.7 Equation P4;91
3.3.7.1;8.7.1 Elliptic Asymptotic Forms, Face G3(2);92
3.3.7.2;8.7.2 Periodic Asymptotic Forms, Face G4(2);92
3.3.8;8.8 Equation P5;93
3.3.8.1;8.8.1 Case d . 0, Elliptic Asymptotic Forms, Face G1(2);93
3.3.8.2;8.8.2 Case d . 0, Periodic Asymptotic Forms, Face G2(2);95
3.3.8.3;8.8.3 Case d = 0, c . 0, Elliptic Asymptotic Forms, Face G1(2);95
3.3.8.4;8.8.4 Case d = 0, c . 0, Periodic Asymptotic Forms, Face G2(2);95
4;III Isomondromy Deformations;97
4.1;9 Isomonodromic Deformations on Riemann Surfaces;99
4.1.1;9.1 Introduction;99
4.1.2;9.2 The Space of Parameters T~;100
4.1.3;9.3 The Description of Bundles with Connections on a Riemann Surface;100
4.1.4;9.4 Isomonodromic Deformations;101
4.2;10 On Birational Darboux Coordinates of Isomonodromic Deformation Equations Phase Space;105
4.3;11 On the Malgrange Isomonodromic Deformations of Nonresonant Irregular Systems;109
4.3.1;11.1 Introduction;109
4.3.2;11.2 The Malgrange Isomonodromic Deformation of the Pair (E0, V¯0);110
4.3.3;11.3 Specificity of Meromorphic 2x2-Connections;112
4.4;12 Critical behavior of P6 Functions from the Isomonodromy Deformations Approach;115
4.4.1;12.1 Introduction;115
4.4.2;12.2 Behavior of y(x);116
4.4.3;12.3 Parameterization in Terms of Monodromy Data;118
4.5;13 Isomonodromy Deformation of the Heun Class Equation;121
4.5.1;13.1 Introduction;121
4.5.2;13.2 Gauge Transforms of Linear Differential Equations;122
4.5.3;13.3 Gauge Transforms of the Hypergeometric Class Equations;125
4.5.4;13.4 Gauge Transform of Heun Class Equations;126
4.5.4.1;13.4.1 Formulation of the Problem;126
4.5.4.2;13.4.2 Initial System of Equations and Equation Heunc2;127
4.5.4.3;13.4.3 Parameters of the Transformed Equation;128
4.5.5;13.5 Conclusion;130
4.6;14 Isomonodromy Deformations and Hypergeometric-Type Systems;131
4.6.1;14.1 Schlesinger Families of Fuchsian Systems;131
4.6.2;14.2 Schlesinger Systems;132
4.6.3;14.3 Upper-Triangular Schlesinger Systems;132
4.6.4;14.4 Jordan-Pochhammer Systems;134
4.6.5;14.5 The Basic Result;135
4.7;15 A Monodromy Problem Connected with P6;137
4.7.1;15.1 Preliminaries I;137
4.7.2;15.2 Preliminaries II;138
4.7.3;15.3 Main Result;139
4.7.4;15.4 Example;140
4.8;16 Monodromy Evolving Deformations and Confluent Halphen’s Systems;143
4.8.1;16.1 Introduction;143
4.8.2;16.2 Quadratic Systems and Nonassociative Algebras;145
4.8.3;16.3 Monodromy Evolving Deformations;147
4.8.4;16.4 Halphen’s Confluent Systems and Monodromy Evolving Deformations;149
4.9;17 On the Gauge Transformation of the Sixth Painlevé Equation;151
4.9.1;17.1 Linearizations of the Sixth Painlevé Equation;151
4.9.1.1;17.1.1 LODE LVI;152
4.9.1.2;17.1.2 LODE LVI;153
4.9.1.3;17.1.3 LODE LVI;154
4.9.1.4;17.1.4 Schlesinger System with Symmetric Gauge;156
4.9.1.5;17.1.5 Schlesinger System with Asymmetric Gauge;157
4.9.2;17.2 Schlesinger Transformation LVI . LVI;157
4.10;18 Expansions for Solutions of the Schlesinger Equation at a Singular Point;165
4.10.1;18.1 Introduction;165
4.10.2;18.2 Schlesinger Equation and Isomonodromic Deformations;168
4.10.3;18.3 Sketch of the Proof;169
5;IV Painlevé Property;173
5.1;19 Painleve Analysis of Lotka-Volterra Equations;175
5.2;20 Painlevé Test and Briot-Bouquet Systems;179
5.3;21 Solutions of the Chazy System;181
5.4;22 Third-Order Ordinary Differential Equations with the Painlevé Test;185
5.4.1;22.1 Introduction;185
5.4.2;22.2 Simplified Equation;186
5.4.3;22.3 Reduced Equations;188
5.4.3.1;22.3.1 Leading Order k =-1;188
5.4.3.2;22.3.2 Leading Order k = -2;195
5.4.3.3;22.3.3 Leading Order k = -3;196
5.4.3.4;22.3.4 Leading Order k = -4;197
5.5;23 Analytic Properties of Solutions of a Class of Third-Order Equations with an Irrational Right-Hand Side;199
6;V Other Aspects;205
6.1;24 The Sixth Painlevé Transcendent and Uniformizable Orbifolds;207
6.1.1;24.1 Algebraic Solutions of P6 and Uniformization Theory;207
6.1.2;24.2 On the General Solution to Equation (24.1);208
6.1.3;24.3 Calculus: Abelian Integrals and Affine (Analytic) Connections;209
6.2;25 On Uniformizable Representation for Abelian Integrals;213
6.2.1;25.1 Introduction;213
6.2.2;25.2 Schwarz Equation and Equations on Tori;214
6.2.3;25.3 Holomorphic Elliptic Integrals and Hypergeometric Functions;215
6.2.3.1;25.3.1 Lemniscate;215
6.2.3.2;25.3.2 Equi-Anharmonic Curve;217
6.2.4;25.4 Abelian Integrals for Genus g > 1;218
6.2.4.1;25.4.1 Higher Genera. Examples;219
6.3;26 Phase Shift for a Special Solution to the Korteweg-de Vries Equation in the Whitham Zone;223
6.3.1;26.1 Introduction;223
6.3.2;26.2 Evaluation of the Phase Shift;224
6.4;27 Fuchsian Reduction of Differential Equations;227
6.4.1;27.1 Fuchsian Reduction;229
6.4.1.1;27.1.1 Two Simple Examples;229
6.4.1.2;27.1.2 A More Complex Example;230
6.4.2;27.2 Two Applications: Astronomy and Relativity Theory;231
6.4.2.1;27.2.1 Astronomy. A Model of Gaseous Stars;231
6.4.2.2;27.2.2 Relativity. Gowdy Space-Time;232
6.4.3;27.3 Fuchsian Systems for Feynman Integrals;234
6.5;28 The Voros Coefficient and the Parametric Stokes Phenomenon for the Second Painlevé Equation;239
6.5.1;28.1 Introduction;239
6.5.2;28.2 Connection Formula for the Parametric Stokes Phenomenon;240
6.5.3;28.3 Derivation of the Connection Formulas Through the Analysis of the Voros Coefficient of (P2);242
6.6;29 Integral Symmetry and the Deformed Hypergeometric Equation;245
6.7;30 Integral Symmetries for Confluent Heun Equations and Symmetries of Painleve Equation P5;251
6.8;31 From the Tau Function of Painlevé P6 Equation to Moduli Spaces;255
6.9;32 On particular Solutions of q-Painlevé Equations and q-Hypergeometric Equations;261
6.9.1;32.1 Introduction;261
6.9.2;32.2 q-Difference Equation of the Hypergeometric Type;261
6.9.3;32.3 Hypergeometric Solutions of the q-Painlevé Equations;264
6.10;33 Derivation of Painlevé Equations by Antiquantization;267
6.11;34 Integral Transformation of Heun’s Equation and Apparent Singularity;271
6.11.1;34.1 Heun’s Equation and Integral Transformation;271
6.11.2;34.2 Apparent Singularity and Integral Representation of Solutions;272
6.11.3;34.3 Elliptical Representation of Heun’s Equation and Integral Transformation;273
6.12;35 Painlevé Analysis of Solutions to Some Nonlinear Differential Equations and their Systems Associated with Models of the Random-Matrix Type;277
6.12.1;35.1 Introduction;277
6.12.2;35.2 Model of the Random-Matrix Type with Airy Kernal;278
6.12.3;35.3 System of Differential Equations Associated with the Dyson Process;278
6.12.4;35.4 Solutions of the Traveling-Wave Form of a Partial Differential Equation;279
6.13;36 Reductions on the Lattice and Painlevé Equations P2, P5, P6;281
6.13.1;36.1 Introduction;281
6.13.2;36.2 Symmetries of the ABS Equations;282
6.13.3;36.3 Reduction on the Lattice and Discrete Painlevé Equations;283
6.13.4;36.4 Continuous Symmetry Reductions;283
7;Comments;285
