Differential Equations and the Stokes Phenomenon.
Title:
Differential Equations and the Stokes Phenomenon.
Author:
Braaksma, B. L. J.
ISBN:
9789812776549
Personal Author:
Physical Description:
1 online resource (343 pages)
Contents:
Contents -- Preface -- Addresses of contributors -- Toward p-adic Stokes phenomena? Singularities of p-adic differential equations -- 1. p-adic differential equations. Why? -- 2. Singularities of p-adic differential equations -- 3. Slope filtrations -- 4. Relations between examples 3.2 and 3.4 -- 5. A 'p-adic real blow-up' and the question of Stokes phenomena -- Bibliography -- Moduli spaces for linear differential equations -- 1. Introduction -- 2. Definition of the moduli space -- 3. The moduli space is of finite type over C -- 4. Closed subspaces of moduli spaces -- 5. The Singer condition -- Bibliography -- Factorization of differential operators and application to differential Galois theory -- 1. Introduction -- 2. Solving differential equations in a valued field -- 3. Differential Hensel lemma -- 4. Factorization of differential operators -- 5. Application -- Bibliography -- Movable singularities of solutions of nonlinear differential and difference equations and the Painlevé property -- 1. Introduction -- 2. Nonlinear Stokes phenomena. Movable singularities -- 3. Difference equations: the isolated movable singularity property (IMSP) -- 4. Classification of some difference equations with IMSP Solvability -- Bibliography -- On a conjecture of Sophus Lie -- 1. Lie's Conjecture -- 2. A Realization Formula -- 3. Realizations with Polynomial Coefficients -- 4. Realizations with Coefficients in E -- 5. Transitive Lie Algebras in Few Variables -- 6. Beyond Three Variables -- Appendix -- Acknowledgements -- Bibliography -- A Tale of Three Structures: the Arithmetics of Multizetas, the Analysis of Singularities, the Lie algebra ARI. -- 1. Introduction -- 2. Singular systems and equational resurgence -- 3. Singularly perturbed systems and co-equational resurgence -- 4. Dimorphic monomials and monies.
5. The overarching structure: ARI/GARI -- 6. The arithmetics of multizetas -- 7. Conclusion and further vistas -- Bibliography -- A differential intermediate value theorem -- 1. Introduction -- 2. The field of real, grid-based transseries -- 3. The differential Newton polygon method -- 4. The intermediate value theorem -- 5. Perspectives -- Bibliography -- Integrable systems and number theory -- 1. Introduction -- 2. Scalar equations -- 3. Classification results -- 4. Systems of equations -- 5. Classification results -- 6. Conclusion -- Bibliography -- Towards the Galois groupoid of nonlinear O.D.E. -- 1. Singular holomorphic foliations on complex manifolds -- 2. The Riccati equation and a Theorem of Liouville -- 3. The transversal pseudo-group of a foliation -- 4. The generic rational differential equation -- 5. Geometric closure of a pseudo-group: first approach to Lie pseudo-groups -- 6. Saddle-node singularities -- 7. The Galois groupoid of nonlinear first order O.D.E. -- Bibliography -- Galois Theory of q-Difference Equations: the "Analytical" Approach. -- 1. Introduction -- 2. The Galois way with or without function theory -- 3. Galois theory with function theory, first attack -- 4. Galois theory with function theory, second attack -- Bibliography -- "Exact WKB integration" of the polynomial 1D Schrödinger (or Sturm-Liouville) problem -- 1. Standard asymptotic inputs -- 2. Exact inputs -- 3. Exact analysis of eigenvalue problem -- 4. Exact wave-function analysis -- Bibliography -- Une sommation discrète pour des équations aux q-différences linéaires et à coefficients analytiques: théorie générale et exemples -- 1. Transformations de Borel-Laplace q-analogues -- 2. Etude des séries hypergéométriques confluentes basiques -- Bibliography.
Abstract:
This volume is the record of a workshop on differential equations and the Stokes phenomenon, held in May 2001 at the University of Groningen. It contains expanded versions of most of the lectures given at the workshop. To a large extent, both the workshop and the book may be regarded as a sequel to a conference held in Groningen in 1995 which resulted in the book The Stokes Phenomenon and Hilbert's 16th Problem (B L J Braaksma, G K Immink and M van der Put, editors), also published by World Scientific (1996). Both books offer a snapshot concerning the state of the art in the areas of differential, difference and q -difference equations. Apart from the asymptotics of solutions, Painlevé properties and the algebraic theory, new topics addressed in the second book include arithmetic theory of linear equations, and Galois theory and Lie symmetries of nonlinear differential equations. Contents: Toward p -Adic Stokes Phenomena? Singularities of p -Adic Differential Equations (Y André); Moduli Spaces for Linear Differential Equations (M Berkenbosch); Factorization of Differential Operators and Application to Differential Galois Theory (M Bouffet); Movable Singularities of Solutions of Nonlinear Differential and Difference Equations and the Painlevé Property (O Costin & M Kruskal); On a Conjecture of Sophus Lie (J Draisma); A Tale of Three Structures: the Arithmetic of Multizetas, the Analysis of Singularities, the Lie Algebra ARI (J Ecalle); A Differential Intermediate Value Theorem (J van der Hoeven); Integrable Systems and Number Theory (P H van der Kamp et al.); Towards the Galois Groupoid of Nonlinear O D E (F Loray); Galois Theory of q -Difference Equations: The "Analytical" Approach (J Sauloy); "Exact WKB Integration" of the Polynomial 1D Schrödinger (or Sturm-Liouville) Problem (A Voros); Une Sommation Discrète Pour des Équations Aux q
-Différence Linéaires et à Coefficients Analytiques: Théorie Générale et Exemples (C-G Zhang). Readership: Graduate students, academics and researchers in analysis & differential equations, approximation theory and mathematical physics.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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