Riccati-type equations, generalised WZNW equations, and multidimensional Toda systems

TL;DR

This paper explores the connection between Riccati-type differential equations, generalized WZNW equations, and multidimensional Toda systems, establishing their integrability and providing examples of integrable multidimensional Riccati equations.

Contribution

It introduces a framework linking Riccati equations with Toda systems via Lie algebra gradations and studies their integrability in multiple dimensions.

Findings

Established connection between Riccati-type equations and Toda systems.

Extended Riccati equations to multidimensional cases.

Provided examples of integrable multidimensional Riccati equations.

Abstract

We associate to an arbitrary $\mathbb Z$-gradation of the Lie algebra of a Lie group a system of Riccati-type first order differential equations. The particular cases under consideration are the ordinary Riccati and the matrix Riccati equations. The multidimensional extension of these equations is given. The generalisation of the associated Redheffer--Reid differential systems appears in a natural way. The connection between the Toda systems and the Riccati-type equations in lower and higher dimensions is established. Within this context the integrability problem for those equations is studied. As an illustration, some examples of the integrable multidimensional Riccati-type equations related to the maximally nonabelian Toda systems are given.