The formal and analytic classification of integrable singular linear differential equations has been studied among others by R. Gerard and Y. Sibuya. We provide a simple proof of their main result, namely: For certain irregular systems in two variables there is no Stokes phenomenon, i.e. there is no difference between the formal and the analytic classification.
R. Gerard和Y. Sibuya等人研究了可积奇异线性微分方程的形式和解析分类。我们对他们的主要结果提供了一个简单的证明，即:对于某些具有两个变量的不规则系统，不存在Stokes现象，即形式分类与解析分类之间没有区别。
{"title":"Singular Linear Differential Equations in Two Variables","authors":"B. Braaksma, M. Put","doi":"10.1619/FESI.51.459","DOIUrl":"https://doi.org/10.1619/FESI.51.459","url":null,"abstract":"The formal and analytic classification of integrable singular linear differential equations has been studied among others by R. Gerard and Y. Sibuya. We provide a simple proof of their main result, namely: For certain irregular systems in two variables there is no Stokes phenomenon, i.e. there is no difference between the formal and the analytic classification.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2008-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74102676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we shall deal with hyperbolic operators whose principal symbols can be microlocally transformed to symbols depending only on the fiber variables by homogeneous canonical transformations. We call such operators "hyperbolic operators with nearly constant coefficient principal part." Operators with constant coefficient hyperbolic principal part and hyperbolic operators with involutive characteristics belong to this class of operators. We shall give a necessary and sufficient condition for the Cauchy problem to be C∞ well-posed under some additional assumptions. Namely, we shall generalize "Levi condition" and prove that the generalized Levi condition is necessary and sufficient for the Cauchy problem to be C∞ well-posed.
{"title":"On the Cauchy problem for hyperbolic operators with nearly constant coefficient principal part","authors":"S. Wakabayashi","doi":"10.1619/FESI.51.395","DOIUrl":"https://doi.org/10.1619/FESI.51.395","url":null,"abstract":"In this paper we shall deal with hyperbolic operators whose principal symbols can be microlocally transformed to symbols depending only on the fiber variables by homogeneous canonical transformations. We call such operators \"hyperbolic operators with nearly constant coefficient principal part.\" Operators with constant coefficient hyperbolic principal part and hyperbolic operators with involutive characteristics belong to this class of operators. We shall give a necessary and sufficient condition for the Cauchy problem to be C∞ well-posed under some additional assumptions. Namely, we shall generalize \"Levi condition\" and prove that the generalized Levi condition is necessary and sufficient for the Cauchy problem to be C∞ well-posed.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2008-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78804327","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We state sufficient conditions for a certain nonlinear planar differential equation to have only oscillatory solutions. We also obtain a result of this type for a case where the equation is subjected to an impulsive condition.
{"title":"Oscillations of Planar Impulsive Delay Differential Equations","authors":"M. C. Gadotti, P. Táboas","doi":"10.1619/FESI.48.35","DOIUrl":"https://doi.org/10.1619/FESI.48.35","url":null,"abstract":"We state sufficient conditions for a certain nonlinear planar differential equation to have only oscillatory solutions. We also obtain a result of this type for a case where the equation is subjected to an impulsive condition.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2005-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84954028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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