{"title":"应用指数渐近学的复苏问题","authors":"Samuel Crew, Philippe H. Trinh","doi":"10.1111/sapm.12660","DOIUrl":null,"url":null,"abstract":"<p>In many physical problems, it is important to capture exponentially small effects that lie beyond-all-orders of an algebraic asymptotic expansion; when collected, the full asymptotic expansion is known as a trans-series. Applied exponential asymptotics has been enormously successful in developing practical tools for studying the leading exponentials of a trans-series expansion, typically for singularly perturbed nonlinear differential or integral equations. Separately to applied exponential asymptotics, there exists a related line of research known as Écalle's theory of resurgence, which, via Borel resummation, describes the connection between trans-series and a certain class of holomorphic functions known as resurgent functions. Most applications and examples of Écalle's resurgence theory focus mainly on nonparametric asymptotic expansions (i.e., differential equations without a parameter). The relationships between these latter areas with applied exponential asymptotics have not been thoroughly examined—largely due to differences in language and emphasis. In this work, we establish these connections as an alternative framework to the factorial-over-power ansatz procedure in applied exponential asymptotics and clarify a number of aspects of applied exponential asymptotic methodology, including Van Dyke's rule and the universality of factorial-over-power ansatzes. We provide a number of useful tools for probing more pathological problems in exponential asymptotics and establish a framework for future applications to nonlinear and multidimensional problems in the physical sciences.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"152 3","pages":"974-1025"},"PeriodicalIF":2.6000,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.12660","citationCount":"0","resultStr":"{\"title\":\"Resurgent aspects of applied exponential asymptotics\",\"authors\":\"Samuel Crew, Philippe H. Trinh\",\"doi\":\"10.1111/sapm.12660\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In many physical problems, it is important to capture exponentially small effects that lie beyond-all-orders of an algebraic asymptotic expansion; when collected, the full asymptotic expansion is known as a trans-series. Applied exponential asymptotics has been enormously successful in developing practical tools for studying the leading exponentials of a trans-series expansion, typically for singularly perturbed nonlinear differential or integral equations. Separately to applied exponential asymptotics, there exists a related line of research known as Écalle's theory of resurgence, which, via Borel resummation, describes the connection between trans-series and a certain class of holomorphic functions known as resurgent functions. Most applications and examples of Écalle's resurgence theory focus mainly on nonparametric asymptotic expansions (i.e., differential equations without a parameter). The relationships between these latter areas with applied exponential asymptotics have not been thoroughly examined—largely due to differences in language and emphasis. In this work, we establish these connections as an alternative framework to the factorial-over-power ansatz procedure in applied exponential asymptotics and clarify a number of aspects of applied exponential asymptotic methodology, including Van Dyke's rule and the universality of factorial-over-power ansatzes. We provide a number of useful tools for probing more pathological problems in exponential asymptotics and establish a framework for future applications to nonlinear and multidimensional problems in the physical sciences.</p>\",\"PeriodicalId\":51174,\"journal\":{\"name\":\"Studies in Applied Mathematics\",\"volume\":\"152 3\",\"pages\":\"974-1025\"},\"PeriodicalIF\":2.6000,\"publicationDate\":\"2023-12-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.12660\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Studies in Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1111/sapm.12660\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Studies in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1111/sapm.12660","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Resurgent aspects of applied exponential asymptotics
In many physical problems, it is important to capture exponentially small effects that lie beyond-all-orders of an algebraic asymptotic expansion; when collected, the full asymptotic expansion is known as a trans-series. Applied exponential asymptotics has been enormously successful in developing practical tools for studying the leading exponentials of a trans-series expansion, typically for singularly perturbed nonlinear differential or integral equations. Separately to applied exponential asymptotics, there exists a related line of research known as Écalle's theory of resurgence, which, via Borel resummation, describes the connection between trans-series and a certain class of holomorphic functions known as resurgent functions. Most applications and examples of Écalle's resurgence theory focus mainly on nonparametric asymptotic expansions (i.e., differential equations without a parameter). The relationships between these latter areas with applied exponential asymptotics have not been thoroughly examined—largely due to differences in language and emphasis. In this work, we establish these connections as an alternative framework to the factorial-over-power ansatz procedure in applied exponential asymptotics and clarify a number of aspects of applied exponential asymptotic methodology, including Van Dyke's rule and the universality of factorial-over-power ansatzes. We provide a number of useful tools for probing more pathological problems in exponential asymptotics and establish a framework for future applications to nonlinear and multidimensional problems in the physical sciences.
期刊介绍:
Studies in Applied Mathematics explores the interplay between mathematics and the applied disciplines. It publishes papers that advance the understanding of physical processes, or develop new mathematical techniques applicable to physical and real-world problems. Its main themes include (but are not limited to) nonlinear phenomena, mathematical modeling, integrable systems, asymptotic analysis, inverse problems, numerical analysis, dynamical systems, scientific computing and applications to areas such as fluid mechanics, mathematical biology, and optics.