Mathematical Sciences: Asymptotic Solutions of Differential Equations

在这个项目中，主要研究人员将致力于建立作为代数微分方程解的形式幂级数的多重可和性。多重可和概念是Borel可和概念的推广。最近，W.Balser给出了形式幂级数的多重和的一个简单刻画。主要研究人员将利用Balser的这些思想和Gevrey的渐近演算来研究非线性代数微分方程解的结构。研究代数微分方程组的解的性质是有趣的，因为它本身以及它对代数和分析中的一些问题的启发。大约在世纪之交，像庞加莱和佩恩列夫这样的数学家能够为几大类微分方程式提供形式级数解。这项工作反过来导致了渐近微积分的进步，因为它表明发散的级数解往往与收敛的级数解一样好，甚至更好。在这个项目中，主要研究人员将应用一些关于散度级数的最新结果来研究非线性代数微分方程解。

In this project the principal investigator will work on establishing the multisummability of formal power series that are the solutions of algebraic differential equations. The notion of multisummability is a generalization of the concept of Borel summability. Recently W. Balser gave a simple characterization of multisummability of formal power series. The principal investigator will employ these ideas of Balser along with the asymptotic calculus of Gevrey to study the structure of solutions of nonlinear algebraic differential equations. The study of the properties of solutions of systems of algebraic differential equations is of interest for its own sake and for the light it sheds on a number of problems in algebra and analysis. Around the turn of the century mathematicians like Poincare and Painleve were able to provide formal series solutions to several broad classes of differential equations. This work led in turn to advances in asymptotic calculus, since it showed that very often divergent series solutions are as good as or better than convergent series solutions. In this project the principal investigator will apply some of the latest results on divergent series to study the solutions of nonlinear algebraic differential equations.