On differential equations invariant under a projective transformation group: integrability and reductions
Marianna Euler, Norbert Euler, Francesco Oliveri
TL;DR
The paper analyzes differential equations invariant under a projective (Mobius) transformation, derives invariants up to order seven, and constructs four invariant evolution- equation classes tied to the Schwarzian S. It identifies a unique 3rd-order symmetry-integrable equation, u_t = λ u_x / sqrt(S), with a recursion operator generating a hierarchy linked to the Schwarzian KdV, and proves higher-order integrable members belong to this hierarchy (notably for 5th- and 7th-order quasilinear cases). It moreover computes 27 one-dimensional optimal subalgebras to obtain explicit symmetry reductions to ODEs and establishes a hodograph-type transformation linking the fully nonlinear 3rd-order equation to Schwarzian KdV. Section 6 recasts invariant ODEs under the same projective symmetry, obtaining solvable 5th-order ODEs in S and reducing 6th- and 7th-order cases to lower-order forms; the fully nonlinear 7th-order integrability remains open. Overall, the work provides a comprehensive symmetry-algebraic framework for projective-invariant evolution equations and their reductions, with explicit connections to Schwarzian dynamics and potential for further extensions via SL(2,R) and SL(3,R) transformations.
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.