Critical behavior of P-6 Functions from the Isomonodromy Deformations Approach

Preface; I Plane Power Geometry; 1 Plane Power Geometry for One ODE and P1 - P6; 1.1 Statement of the Problem; 1.2 Computation of Truncated Equations; 1.3 Computation of Expansions of Solutions to the Initial Equation (1.1) .; 1.4 Extension of the Class of Solutions; 1.5 Solution of Truncated Equations; 1.6 Types of Expansions; 1.7 Painlevé Equations Pl; 2 New Simple Exact Solutions to Equation P6; 2.1 Introduction; 2.1.1 Power Geometry Essentials; 2.1.2 Matching "Heads" and "Tails" of Expansions; 2.2 Constructing the Template of an Exact Solution; 2.3 Results 2.3.1 Known Exact Solutions to P62.3.2 Computed Solutions; 2.3.3 Generalization of Computed Solutions; 3 Convergence of a Formal Solution to an ODE; 3.1 The General Case; 3.2 The Case of Rational Power Exponents; 3.3 The Case of Complex Power Exponents; 3.4 On Solutions of the Sixth Painlevé Equation; 4 Asymptotic Expansions and Forms of Solutions to P6; 4.1 Asymptotic Expansions near Singular Points of the Equation; 4.2 Asymptotic Expansions near a Regular Point of the Equation; 4.3 Boutroux-Type Elliptic Asymptotic Forms; 5 Asymptotic Expansions of Solutions to P5; 5.1 Introduction 5.2 Asymptotic Expansions of Solutions near Infinity5.3 Asymptotic Expansions of Solutions near Zero; 5.4 Asymptotic Expansions of Solutions in the Neighborhood of the Nonsingular Point of an Equation; II Space Power Geometry; 6 Space Power Geometry for one ODE and P1 - P4, P6; 6.1 Space Power Geometry; 6.2 Asymptotic Forms of Solutions to Painlevé Equations P1 - P4, P6; 6.2.1 Equation P1; 6.2.2 Equation P2; 6.2.3 Equation P3 for cd ≠ 0; 6.2.4 Equation P3 for c = 0 and ad ≠ 0; 6.2.5 Equation P3 for c = d = 0 and ab ≠ 0; 6.2.6 Equation P4; 6.2.7 Equation P6 7 Elliptic and Periodic Asymptotic Forms of Solutions to P57.1 The Fifth Painlevé Equation; 7.2 The case δ ≠ 0; 7.2.1 General Properties of the P5 Equation; 7.2.2 The First Family of Elliptic Asymptotic Forms; 7.2.3 The First Family of Periodic Asymptotic Forms; 7.2.4 The Second Family of Periodic Asymptotic Forms; 7.3 The Case δ ≠ 0, γ ≠ 0; 7.3.1 General Properties; 7.3.2 The Second Family of Elliptic Asymptotic Forms; 7.3.3 The Third Family of Periodic Asymptotic Forms; 7.3.4 The Fourth Family of Periodic Asymptotic Forms; 7.4 The Results Obtained 8 Regular Asymptotic Expansions of Solutions to One ODE and P1-P58.1 Introduction; 8.2 Finding Asymptotic Forms; 8.3 Computation of Expansions (8.2); 8.4 Equation P1; 8.5 Equation P2; 8.5.1 Elliptic Asymptotic Forms, Face Γ3(2); 8.5.2 Periodic Asymptotic Forms, Face Γ4(2); 8.6 Equation P3; 8.6.1 Case cd ≠ 0; 8.6.2 Case c = 0, ad ≠ 0; 8.6.3 Case c = d = 0, ab ≠ 0; 8.7 Equation P4; 8.7.1 Elliptic Asymptotic Forms, Face Γ3(2); 8.7.2 Periodic Asymptotic Forms, Face Γ4(2); 8.8 Equation P5; 8.8.1 Case d ≠ 0, Elliptic Asymptotic Forms, Face Γ1(2) 8.8.2 Case d ≠ 0, Periodic Asymptotic Forms, Face Γ2(2)