Divergent Series, Summability and Resurgence III: Resurgent Methods and the First Painlevé Equation PDF
Preview Divergent Series, Summability and Resurgence III: Resurgent Methods and the First Painlevé Equation
Lecture Notes in Mathematics 2155 Eric Delabaere Divergent Series, Summability and Resurgence III Resurgent Methods and the First Painlevé Equation Lecture Notes in Mathematics 2155 Editors-in-Chief: J.-M.Morel,Cachan B.Teissier,Paris AdvisoryBoard: CamilloDeLellis,Zurich MariodiBernardo,Bristol AlessioFigalli,Austin DavarKhoshnevisan,SaltLakeCity IoannisKontoyiannis,Athens GaborLugosi,Barcelona MarkPodolskij,Aarhus SylviaSerfaty,ParisandNew York CatharinaStroppel,Bonn AnnaWienhard,Heidelberg Moreinformationaboutthisseriesathttp://www.springer.com/series/304 Eric Delabaere Divergent Series, Summability and Resurgence III Resurgent Methods and the First Painlevé Equation Eric Delabaere Département de Mathématiques Université d’Angers Angers, France ISSN 0075-8434 ISSN1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-319-28999-1 ISBN 978-3-319-29000-3 (eBook) DOI 10.1007/978-3-319-29000-3 Libra ry of Congress Control Number: 2016940058 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland TomychildrenDanae´ andAdrian,tomywife Fabienne. Avant-Propos Le sujet principal traite´ dans la se´rie de volumes Divergent Series, Summability and Resurgence est la the´orie des de´veloppements asymptotiques et des se´ries di- vergentes applique´e aux e´quations diffe´rentielles ordinaires (EDO) et a` certaines e´quationsauxdiffe´rencesdanslechampcomplexe. Lese´quationsdiffe´rentiellesdanslechampcomplexe,etdanslecadreholomor- phe,sontunsujettre`sancien.Lathe´orieae´te´ tre`sactivedansladeuxie`memoitie´ duXIX-e`mesie`cle.Encequiconcernelese´quationsline´aires,lesmathe´maticiens de cette e´poque les ont subdivise´es en deux classes. Pour la premie`re, celle des e´quationsa` pointssinguliersre´guliers(oudeFuchs),ge´ne´ralisantlese´quationshy- perge´ome´triques d’Euler et de Gauss, ils ont enregistre´ “des succe`s aussi de´cisifs quefaciles”commel’e´crivaitRene´Garnieren1919.Enrevanche,pourlaseconde, celledese´quationsditesa` pointssinguliersirre´guliers,commel’e´crivaitaussiGar- nier,“leurseffortsrestentimpuissantsa` e´difieraucunethe´oriege´ne´rale”.Laraison centrale de ce vif contraste est que toute se´rie entie`re apparaissant dans l’e´criture d’unesolutiond’unee´quationdiffe´rentielledeFuchsestautomatiquementconver- gentetandisquepourlese´quationsirre´gulie`rescesse´riessontge´ne´riquementdiver- gentes et que l’on ne savait qu’en faire. La situation a commence´ a` changer graˆce a` un travail magistral de Henri Poincare´ entrepris juste apre`s sa the`se, dans lequel il “donne un sens” aux solutions divergentes des EDO line´aires irre´gulie`res en in- troduisant un outil nouveau, et qui e´tait appele´ a` un grand avenir, la the´orie des de´veloppementsasymptotiques.Ilaensuiteutilise´cetoutilpourdonnerunsensaux se´ries divergentes de la me´canique ce´leste, et remporte´ de tels succe`s que presque toutlemondeaoublie´ l’originedel’histoire,c’est-a`-direlesEDO!Lestravauxde Poincare´ont(unpeu...)remisa`l’honneurl’e´tudedesse´riesdivergentes,abandonne´e parlesmathe´maticiensapre`sCauchy.L’Acade´miedesSciencesasoumiscesujetau concoursen1899cequifuta`l’origined’untravailimportantd’E´mileBorel.Celui- ciestlasourcedenombredestechniquesutilise´esdansDivergentSeries,Summabil- ityandResurgence.PourrevenirauxEDOirre´gulie`res,lesujetafaitl’objetdenom- breuxetimportantstravauxdeG.D.BirkhoffetR.Garnierdurantlepremierquart duXX-e`mesie`cle.Onretrouveraicidenombreuxprolongementsdesme´thodesde Birkhoff. Apre`s 1940, le sujet a e´trangement presque disparu, la the´orie e´tant, je vii viii Avant-Propos nesaistroppourquoi,conside´re´ecommeacheve´e,toutcommecelledese´quations de Fuchs. Ces dernie`res ont re´e´merge´ au de´but des anne´es 1970, avec les travaux deRaymondGe´rard,puisunlivredePierreDeligne.Lese´quationsirre´gulie`resont suiviavecdestravauxdel’e´coleallemandeetsurtoutdel’e´colefranc¸aise.Denom- breuses techniques comple`tement nouvelles ont e´te´ introduites (de´veloppements asymptotiques Gevrey, k-sommabilite´, multisommabilite´, fonctions re´surgentes...) permettant en particulier une vaste ge´ne´ralisation du phe´nome`ne de Stokes et sa miseenrelationaveclathe´oriedeGaloisdiffe´rentielleetleproble`medeRiemann- Hilbert ge´ne´ralise´. Tout ceci a depuis rec¸u de tre`s nombreuses applications dans des domaines tre`s varie´s, allant de l’inte´grabilite´ des syste`mes hamiltoniens aux proble`mes de points tournant pour les EDO singulie`rement perturbe´es ou a` divers proble`mes de modules. On en trouvera certaines dans Divergent Series, Summa- bilityandResurgence,commel’e´tudere´surgentedesgermesdediffe´omorphismes analytiques du plan complexe tangents a` l’identite´ ou celle de l’EDO non-line´aire Painleve´ I. Lesujetrestaitaujourd’huidifficiled’acce`s,lelecteurnedisposantpas,misa`part les articles originaux, de pre´sentation accessible couvrant tous les aspects. Ainsi Divergent Series, Summability and Resurgence comble une lacune. Ces volumes pre´sententunlargepanoramadesrechercheslesplusre´centessurunvastedomaine classique et passionnant, en pleine renaissance, on peut meˆme dire en pleine ex- plosion.Ilssontne´anmoinsaccessiblesa` toutlecteurposse´dantunebonnefamilia- rite´ avec les fonctions analytiques d’une variable complexe. Les divers outils sont soigneusement mis en place, progressivement et avec beaucoup d’exemples. C’est unebellere´ussite. A` Toulouse,le16mai2014, Jean-PierreRamis Preface to the Three Volumes Thisthree-volumeseriesaroseoutoflecturenotesforthecourseswegavetogether ataCIMPA1 schoolinLima,Peru,inJuly2008.Sincethen,thesenoteshavebeen used and developed in graduate courses held at our respective institutions, that is, the universities of Angers, Nantes, Strasbourg (France) and the Scuola Normale SuperiorediPisa(Italy).Theoriginalnoteshavenowgrownintoself-containedin- troductionstoproblemsraisedbyanalyticcontinuationandthedivergenceofpower seriesinonecomplexvariable,especiallywhenrelatedtodifferentialequations. A classical way of solving an analytic differential equation is the power series method,whichsubstitutesapowerseriesfortheunknownfunctionintheequation, then identifies the coefficients. Such a series, if convergent, provides an analytic solution to the equation. This is what happens at an ordinary point, that is, when we have an initial value problem to which the Cauchy-Lipschitz theorem applies. Otherwise, at a singular point, even when the method can be applied the resulting series most often diverges; its connection with “actual” local analytic solutions is notobviousdespiteitsdeeplinktotheequation. The hidden meaning of divergent formal solutions was already pondered in the nineteenthcentury,afterCauchyhadclarifiedthenotionsofconvergenceanddiver- genceofseries.Forordinarylinear differentialequations,ithasbeenknownsince thebeginningofthetwentiethcenturyhowtodetermineafullsetoflinearlyinde- pendentformalsolutions2atasingularpointintermsofafinitenumberofcomplex powers, logarithms, exponentials and power series, either convergent or divergent. Theseformalsolutionscompletelydeterminethelineardifferentialequation;hence, they contain all information about the equation itself, especially about its analytic solutions.Extractingthisinformationfromthedivergentsolutionswastheunderly- 1 Centre International de Mathe´matiques Pures et Applique´es, or ICPAM, is a non-profit inter- nationalorganizationfoundedin1978inNice,France.Itpromotesinternationalcooperationin highereducationandresearchinmathematicsandrelatedsubjectsforthebenefitofdeveloping countries.ItissupportedbyUNESCOandIMU,andmanynationalmathematicalsocietiesover theworld. 2Onesaysaformalfundamentalsolution. ix
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