Formal and Analytic Solutions of Diff. Equations: FASdiff, Alcalá de Henares, Spain, September 2017, Selected, Revised Contributions

Description

These proceedings provide methods, techniques, different mathematical tools and recent results in the study of formal and analytic solutions to Diff. (differential, partial differential, difference, q-difference, q-difference-differential.... ) Equations. They consist of selected contributions from the conference "Formal and Analytic Solutions of Diff. Equations", held at Alcalá de Henares, Spain during September 4-8, 2017. Their topics include summability and asymptotic study of both ordinary and partial differential equations. The volume is divided into four parts. The first paper is a survey of the elements of nonlinear analysis. It describes the algorithms to obtain asymptotic expansion of solutions of nonlinear algebraic, ordinary differential, partial differential equations, and of systems of such equations. Five works on formal and analytic solutions of PDEs are followed by five papers on the study of solutions of ODEs. The proceedings conclude with five works on related topics, generalizations and applications. All contributions have been peer reviewed by anonymous referees chosen among the experts on the subject. The volume will be of interest to graduate students and researchers in theoretical and applied mathematics, physics and engineering seeking an overview of the recent trends in the theory of formal and analytic solutions of functional (differential, partial differential, difference, q-difference, q-difference-differential) equations in the complex domain. Preface Scientific Committee Organizing Committee Contents Contributors Part I A Survey on the Elements of Nonlinear Analysis Elements of Nonlinear Analysis 1 Introduction 2 Selection of the Leading Terms 2.1 Order of a Function Bruno1 2.2 Truncated Sums Bruno2,Bruno3 2.3 Variations Bruno4,Bruno3 3 Power Expansions of Solutions Bruno3,Bruno5 3.1 Statement of the Problem 3.2 Solving a Truncated Equation 3.3 Critical Numbers of the Truncated Solution 3.4 Computation of the Power Expansion of a Solution [§3]Bruno3 4 Complicated Expansions of Solutions [§5]Bruno3, Bruno7,Bruno8 4.1 Case of the Vertical Edge Γj(1) 4.2 Inclined Edge 5 Exponential Expansions of Solutions Bruno9,Bruno10 6 Generalizations 7 Applications References Part II Summability of Divergent Solutions of PDEs On the k-Summability of Formal Solutions for a Class of Higher Order Partial Differential Equations with Time-Dependent Coefficients 1 Result 2 Review of k-Summability 3 Construction of the Formal Solution of (CP) 3.1 Decomposition of PM 3.2 A Sequence of Cauchy Problems 4 Gevrey Order of the Formal Solution 5 Preliminaries for the Proof of Theorem1 6 Proof of Theorem1 7 Proof of Lemma3 7.1 A Canonical Form for Differential Equation of ν 7.2 Convolution Equations 7.3 Proof of Lemma3 7.4 Proof of Proposition3 References Singular Solutions to a System of Equations related to Ricci-Flat Kähler Metrics 1 Introduction 2 Preliminaries 3 Statement of the Problem 4 Existence of an tildemathcalO+ Solution 5 Uniqueness of the tildemathcalO+ Solution References Hyperasymptotic Solutions for Certain Partial Differential Equations 1 Introduction 2 Hyperasymptotic Expansions for the Heat Equation 2.1 0-Level Hyperasymptotic Expansion 2.2 N-Level Hyperasymptotic Expansion 2.3 Conclusion 3 Generalization to Linear PDEs with Constant Coefficients 3.1 Summability 3.2 Reduction of Linear PDEs with Constant Coefficients to Simple Pseudodifferential Equations 3.3 Summable Solutions of Simple Pseudodifferential Equations 3.4 Hyperasymptotic Expansion of Solution of Simple Pseudodifferential Equations References The Stokes Phenomenon for Certain PDEs in a Case When Initial Data Have a Finite Set of Singular Points 1 Introduction 2 Notation. Gevrey's Asymptotics and k-Summability 3 The Stokes Phenomenon and Hyperfunctions 3.1 The Stokes Phenomenon for k-Summable Formal Power Series 3.2 Laplace Type Hyperfunctions 3.3 The Description of Jumps Across the Stokes Lines in Terms of Hyperfunctions 4 Characterization of the Stokes Phenomenon in a Case When the Initial Data Have a Finite Set of Singular Points References Soliton Resolution for the Focusing Integrable Discrete Nonlinear Schrödinger Equation 1 Introduction 2 Inverse Scattering Transform 3 Main Results 4 Open Problem References Complicated and Exotic Expansions of Solutions to the Painlevé Equations 1 Introduction 2 Writing ODEs for Coefficients 2.1 Algebraic Case 2.2 Case of ODE 3 The Third Painlevé Equation P3 3.1 Truncated Equation and its Logarithmic Solutions 3.2 The Additional Complicated Family 3.3 The Main Complicated Family 3.4 Exotic Expansions for Equation P3 4 The Fifth Painlevé Equation P5 in Case I 4.1 Two Cases for Equation P5 4.2 Preliminary Transformations in Case I 4.3 Complicated Expansions 4.4 Exotic Expansions 5 The Fifth Painlevé Equation P5 in Case II 5.1 Preliminary Transformations 5.2 Complicated Expansions 5.3 Exotic Expansions 6 The Sixth Painlevé Equation P6 6.1 Preliminary Transformations 6.2 Complicated Expansions 6.3 Exotic Expansions 7 Conclusion References Part III Summability of Divergent Solutions of ODEs The Borel Transform of Canard Values and Its Singularities 1 Introduction 2 Statements 2.1 Canard Phenomenon 2.2 Formal Solution 2.3 Borel Summability 2.4 Exceptional Solutions 2.5 Composite Asymptotic Expansions 2.6 Asymptotic Expansion Versus Monodromy of Canard Values 2.7 First Singularity of the Borel Transform 2.8 Perspectives and Remarks 3 Proofs 3.1 Proof of Theorem 2.6 3.2 Proof of Proposition 2.10 3.3 Proof of Theorem 2.13 3.4 Proof of Theorem 2.15 References Quantization Conditions on Riemannian Surfaces and Spectral Series of Non-selfadjoint Operators 1 Introduction 2 Schrödinger Equation with a Complex Potential 3 Equation of Magnetic Induction 3.1 Torus 3.2 Sphere References Semilocal Monodromy of Rigid Local Systems 1 Introduction 2 Middle Convolution of a Sum of Residue Matrices 3 Semilocal Monodromy 4 Examples References On the Newton Polygon of a Moser-Irreducible Linear Differential System 1 Introduction 2 A Few Reminders 2.1 Newton Polygon and Polynomials 2.2 Moser-Irreducible Differential Systems 3 An Estimate of the Katz Invariant 3.1 Estimation of the Katz Invariant of a Moser-Irreducible System 3.2 Number of Coefficients Involved in the Computation of κ(A) 4 Computing the Katz Invariant and the Corresponding Newton Polynomial 4.1 Three Preliminary Lemmas 4.2 Main Results References Part IV Related Topics Uniqueness Property for ρ-Analytic Functions 1 Introduction and Statement of the Main Result 2 The Triangle Equality 3 Auxiliary Lemmas 4 Proof of Theorem 1 References On the Algebraic Study of Asymptotics 1 Introduction 2 Sheaves on a Subanalytic Site 3 Multi-Normal Deformation 4 Multi-Sectors 5 Multi-Specialization 6 Multi-Asymptotics 7 Multi-Specialization and Multi-Asymptotics References Deformations with a Resonant Irregular Singularity 1 Introduction 2 Main Results 2.1 Applications References Symmetric Semi-classical Orthogonal Polynomials of Class One on q-Quadratic Lattices 1 Motivation 2 Preliminary Results 2.1 The General Divided Difference Operator, q-Quadratic Lattices, and Orthogonal Polynomials 2.2 Semi-classical Orthogonal Polynomials on Non-uniform Lattices: The Class One and General Difference Equations 3 Main Results: The Symmetric Orthogonal Polynomials of Class One References Determinantal Form for Ladder Operators in a Problem Concerning a Convex Linear Combination of Discrete and Continuous Measures 1 Introduction 2 Connection Lemmas Between Qn and Lnα 3 Ladder Operators for the Laguerre–Krall SMOP 4 A Convex Linear Combination of Measures References