The Asymptotic Theory of Solutions of Differential Equations

我们的研究计划“微分方程解的渐近理论”有四个研究目的。针对高阶常微分方程的复WKB方法，将Fedoryuk理论应用于三阶微分方程BNR方程。对于由积分定义的函数的渐近展开式的研究，我们处理了用拉普拉斯积分表示的BNR方程的解。在这些分析中，我们有效地使用了可动鞍点法的概念。毕竟，从1982年首次出现的BNR方程开始，经过20年的时间，我们可以得到BNR方程的几乎完全的渐近分析，即在整个复平面上构造渐近展开式并得到connectIon公式。我们的结果将在不久的将来公布。对于项目的其他两个目的，即收敛WKB方法和偏微分方程的渐近理论，我们没有取得任何实质性的进展。一是发现了由6张复z平面组成的BNR方程特征根的黎曼曲面与1张复w平面之间的映射。由此，我们可以在一张纸上表示整个Stokes曲线或Stokes区域，从而可以以可见的方式构造存在解的渐近展开式的容许域，或存在基本解系统的正则域。在分析过程中，我们发现了二阶微分方程不存在的阴影带的存在。另一个突破是可移动鞍点法的概念。利用可动鞍点法对用拉普拉斯积分表示的BNR方程的解进行求解，得到了拉普拉斯积分在可容许域上对z具有一致有效的渐近展开式。此外，柯西积分定理给出了描述BNR方程若干解之间线性关系的连接公式。少

There are four purposes of studies of our Research Project Asymptotic Theory of Solutions of Differential Equations. Regarding the complex WKB method for higher order ordinary differential equations, We applied Fedoryuk theory to the third order differential equation named BNR equation. And as for studies of the asymptotic expansion of the functions defined by the integrals, we treated solutions of BNR equation expressed by the Laplace integral.In these analyses we effectively use the notion of movable saddle point method. After all, we could obtain almost complete asymptotic analyses for the BNR equation, that is to construct asymptotic expansions on the whole complex plane and to get the connectIon formulas, after twenty years from the BNR equation firstly appeared in 1982.Our results will be published in the near future. Regarding other two purposes of the project, confluent WKB method, and asymptotic theory for partial differential equation, we could not obtain any essential progre … More ss.There are two break through in our analyses. The one is the discovery of the mapping between a Riemman surfase of characteristic roots of BNR equation, which composed of 6 sheets of the complex z plane, and one sheet of the complex w-plane. By this, we can express whole Stokes curves or Stokes domains on one sheet of paper, and then it becomes possible to construct admissible domains where asymptotic expansions of solutions exist, or canonical domains where fundamental systems of solutions exist, in a visible manner. In the course of the analyses, we find the existence of shadow zone which does not exist in the case of second order differential equations.The another break through is the notion of the movable saddle point method. By applying the movable saddle point method to the solution of the BNR equation expressed by the Laplace integral, we find that the Laplace integral has asymptotic expansion uniformly valid for z in the admissible domain. Moreover, the Cauchy s integral theorem gives us connection formulas which describe linear relation between several solutions of the BNR equation. Less