Chapter 20. Painlevé Test and Briot–Bouquet Systems

Chapters in this book

1. Frontmatter i
2. Preface v
3. Contents vii
4. Part I. Plane Power Geometry
5. Chapter 1. Plane Power Geometry for One ODE and P1–P6 3
6. Chapter 2. New Simple Exact Solutions to Equation P6 13
7. Chapter 3. Convergence of a Formal Solution to an ODE 23
8. Chapter 4. Asymptotic Expansions and Forms of Solutions to P6 27
9. Chapter 5. Asymptotic Expansions of Solutions to P5 33
10. Part II. Space Power Geometry
11. Chapter 6. Space Power Geometry for one ODE and P1–P4, P6 41
12. Chapter 7. Elliptic and Periodic Asymptotic Forms of Solutions to P5 53
13. Chapter 8. Regular Asymptotic Expansions of Solutions to One ODE and P1–P5 67
14. Part III. Isomondromy Deformations
15. Chapter 9. Isomonodromic Deformations on Riemann Surfaces 85
16. Chapter 10. On Birational Darboux Coordinates of Isomonodromic Deformation Equations Phase Space 91
17. Chapter 11. On the Malgrange Isomonodromic Deformations of Nonresonant Irregular Systems 95
18. Chapter 12. Critical behavior of P6 Functions from the Isomonodromy Deformations Approach 101
19. Chapter 13. Isomonodromy Deformation of the Heun Class Equation 107
20. Chapter 14. Isomonodromy Deformations and Hypergeometric-Type Systems 117
21. Chapter 15. A Monodromy Problem Connected with P6 123
22. Chapter 16. Monodromy Evolving Deformations and Confluent Halphen’s Systems 129
23. Chapter 17. On the Gauge Transformation of the Sixth Painlevé Equation 137
24. Chapter 18. Expansions for Solutions of the Schlesinger Equation at a Singular Point 151
25. Part IV. Painlevé Property
26. Chapter 19. Painleve Analysis of Lotka–Volterra Equations 161
27. Chapter 20. Painlevé Test and Briot–Bouquet Systems 165
28. Chapter 21. Solutions of the Chazy System 167
29. Chapter 22. Third-Order Ordinary Differential Equations with the Painlevé Test 171
30. Chapter 23. Analytic Properties of Solutions of a Class of Third-Order Equations with an Irrational Right-Hand Side 185
31. Part V. Other Aspects
32. Chapter 24. The Sixth Painlevé Transcendent and Uniformizable Orbifolds 193
33. Chapter 25. On Uniformizable Representation for Abelian Integrals 199
34. Chapter 26. Phase Shift for a Special Solution to the Korteweg–de Vries Equation in the Whitham Zone 209
35. Chapter 27. Fuchsian Reduction of Differential Equations 213
36. Chapter 28. The Voros Coefficient and the Parametric Stokes Phenomenon for the Second Painlevé Equation 225
37. Chapter 29. Integral Symmetry and the Deformed Hypergeometric Equation 231
38. Chapter 30. Integral Symmetries for Confluent Heun Equations and Symmetries of Painlevé Equation P5 237
39. Chapter 31. From the Tau Function of Painlevé P6 Equation to Moduli Spaces 241
40. Chapter 32. On particular Solutions of q-Painlevé Equations and q-Hypergeometric Equations 247
41. Chapter 33. Derivation of Painlevé Equations by Antiquantization 253
42. Chapter 34. Integral Transformation of Heun’s Equation and Apparent Singularity 257
43. Chapter 35. Painlevé Analysis of Solutions to Some Nonlinear Differential Equations and their Systems Associated with Models of the Random-Matrix Type 263
44. Chapter 36. Reductions on the Lattice and Painlevé Equations P2, P5, P6 267
45. Comments 271