Computable Integrability. Chapter 2: Riccati equation

Details Computable Integrability. Chapter 2: Riccati equation Computable Integrability. Chapter 2: Riccati equation

2005-09-12

arxiv.org/abs/math-ph/0509024v1

Physics

Mathematical Physics

This text is preliminary version of Chapter 2 of the ook "Computable Integrability", it consists of 31 pages

Scientific paper

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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