标题： 标量Riccati方程的解 显示英文标题

标题： Solution of the scalar Riccati equation

摘要： 标量Riccati方程是一个具有多种数学联系的典型非线性常微分方程。 自其最初提出以来，根据标准教科书理论，如果已知一个特解，则可以确定通解；但除了在零星的特殊情况下，不存在一般的方法来显式确定一个特解。 本文的目的是通过显式构造直接求解标量Riccati方程以及二阶一般线性常微分方程。 在Riccati方程的情况下，解在实线上局部可积函数的三元组与黎曼球面自同构群中通过单位的局部绝对连续路径之间建立了一个双射对应关系。 作为结果的应用，我们得到了一维Schrödinger方程的显式解，Miura变换的逆公式，以及Airy函数的新公式。 显示英文摘要

摘要： The scalar Riccati equation is a prototypical nonlinear ODE having diverse mathematical connections. In the centuries since its initial formulation, a standard textbook theory has emerged according to which the general solution may be determined if a particular solution is known; but no general method exists to determine a particular solution explicitly, except in sporadic special cases. The purpose of the present article is to solve the scalar Riccati equation in general form, as well as the general linear ODE of second order, directly by explicit construction. In the case of the Riccati equation, the solution sets up a bijective correspondence between triples of locally integrable functions on the real line, and locally absolutely continuous paths through the identity in the automorphism group of the Riemann sphere. As applications of the results, we obtain an explicit solution to the one-dimensional Schr\"odinger equation, an inversion formula for the Miura transform, and a new formula for Airy functions.