The Asymptotic Theory of Solutions of Differential Equations

我们的研究项目研究微分方程解的渐近理论有四个目的。对于高阶常微分方程复WKB方法，我们将Fedoryuk理论应用于三阶常微分方程组BNR方程。对于积分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，我们有效地使用了移动鞍点方法的概念。毕竟，在1982年首次出现BNR方程20年后，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联系公式。我们的结果将在不久的将来发表。至于项目的另外两个目的，合流的wkb方法和偏微分方程的渐近理论，我们没有得到任何必要的进展…更多。我们的分析有两个突破。其一是发现了BNR方程特征根的Riemman曲面与复w平面之间的映射，该特征根由复z平面的6片组成。这样，我们就可以在一张纸上表示完整的Stokes曲线或Stokes区域，然后以可见的方式构造存在解的渐近展开式的可允许域或存在基本解系的正则域。在分析过程中，我们发现了在二阶微分方程解的情况下不存在的影子区域的存在。另一个突破是活动鞍点方法的概念。将活动鞍点方法应用于拉普拉斯积分表示的BNR方程的解，我们发现拉普拉斯积分在可容许区域内对z具有一致有效的渐近展开式。此外，柯西·S积分定理还给出了描述BNR型方程几个解之间的线性关系的联系公式。较少

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