可计算可积性。第二章：黎卡提方程
Computable Integrability. Chapter 2: Riccati equation
摘要：
本章以Riccati方程为例，尝试证明第一章中介绍的一些思想和概念--求积可积性、守恒律等。关于RE的变换群和解的奇异性，我们构造了Riccati方程的一些等价形式。比较了Riccati方程及其等价形式的三种不同解法。RE的经典形式使我们能够很容易地构造形式级数表示的渐近解。二阶线性方程被证明是更方便地描述有限间隙势的精确孤子解，这将是一个更复杂的任务，一个RE本身，而推广的类孤子势有限间隙势需要修改Schwarzian方程。
英文摘要：
In this Chapter, using Riccati equation as our main example, we tried to demonstrate at least some of the ideas and notions introduced in Chapter 1 - integrability in quadratures, conservation laws, etc. Regarding transformation group and singularities of solutions for RE, we constructed some equivalent forms of Riccati equation. We also compared three different approaches to the solutions of Riccati equation and its equivalent forms. The classical form of RE allowed us to construct easily asymptotic solutions represented by formal series. Linear equation of the second order turned out to be more convenient to describe finite-gap potentials for exact solitonic solutions which would be a much more complicated task for a RE itself while generalization of soliton-like potentials to finite-gap potentials demanded modified Schwarzian equation.
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