On differential equations invariant under a projective transformation group: integrability and reductions

TL;DR

This paper investigates projective invariants for differential equations, constructs a class of nonlinear evolution equations, identifies symmetry-integrable cases, and explores their reductions and hierarchy, revealing connections to the Schwarzian KdV.

Contribution

It introduces a method to find invariants under projective transformations and classifies symmetry-integrable equations within this framework, including the fully nonlinear third-order equation.

Findings

Identified invariants up to order seven for projective transformations.

Found the unique third-order symmetry-integrable evolution equation and its recursion operator.

Established hierarchy and reductions of higher-order integrable equations.

Abstract

We consider a projective transformation and establish the invariants for this transformation group up to order seven. We use the obtained invariants to construct a class of nonlinear evolution equations and identify some symmetry-integrable equations in this class. Notably, the only symmetry-integrable evolution equation of order three in this class is a fully-nonlinear equation for which we find the recursion operator and its connection to the Schwarzian KdV. We furthermore establish that higher-order symmetry-integrable equations in this class belong to the hierarchy of the fully-nonlinear 3rd-order equation and prove this for the 5th-order case as well as for the quasi-linear 7th-order case. We list all symmetry reductions of this 3rd-order fully-nonlinear symmetry-integrable evolution equation to ordinary differential equations by exploiting the 1-dimensional optimal Lie symmetry subalgebras of the transformation group. We also identify the ordinary differential equations that are invariant under this projective transformation and reduce the order of these equations.