Computable Integrability. Chapter 2: Riccati equation

TL;DR

This chapter explores the Riccati equation as a key example of integrability, demonstrating transformations, solution approaches, and connections to solitonic and finite-gap potentials within the broader context of integrability in quadratures.

Contribution

It introduces new equivalent forms of the Riccati equation, compares solution methods, and links classical and linear equations to solitonic solutions and finite-gap potentials.

Findings

Constructed equivalent forms of Riccati equation.

Compared three solution approaches for Riccati and its forms.

Linked classical Riccati solutions to solitonic and finite-gap potentials.

Abstract

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.