Geometrical and Dynamical Aspects of Nonlinear Higher-Order Riccati Systems

TL;DR

This paper explores the geometric and dynamical properties of nonlinear second-order Riccati equations, linking them to projective vector fields, Darboux integrability, and Hamiltonian structures, with applications to Painlevé II.

Contribution

It provides a novel geometric formulation of the second-order Riccati equation and establishes its Lagrangian and Hamiltonian descriptions, connecting Darboux polynomials and symmetries.

Findings

First-integral of the SORE obtained

Lagrangian and Hamiltonian structures identified

Relation to Painlevé II analyzed

Abstract

We study a geometrical formulation of the nonlinear second-order Riccati equation (SORE) in terms of the projective vector field equation on $S^1$, whichn in turn is related to the stability algebra of Virasoro orbit. Using Darboux integrability method we obtain the first-integral of SORE and the results are applied to the study of its Lagrangian and Hamiltonian description. We unveil the relation between the Darboux polynomials and master symmetries associated to second-order Riccati. Using these results we show the existence of a Lagrangian description for the related system, and the Painlev\'e II equation is analysed.