Nonlocal symmetries of Riccati and Abel chains and their similarity reductions

TL;DR

This paper investigates nonlocal symmetries in Riccati and Abel chains, revealing shared symmetry forms and reduction patterns, and provides methods to derive general solutions for these differential equations.

Contribution

It identifies common nonlocal symmetries in Riccati and Abel chains and establishes their role in similarity reductions and solution derivation.

Findings

All Riccati chain equations share the same nonlocal symmetry form.

Similarity reduction of Nth order Riccati chain yields (N-1)th order ODE in the same chain.

Abel chain equations share a different nonlocal symmetry form, with reductions ending in Riccati chain equations.

Abstract

We study nonlocal symmetries and their similarity reductions of Riccati and Abel chains. Our results show that all the equations in Riccati chain share the same form of nonlocal symmetry. The similarity reduced $N^{th}$ order ordinary differential equation (ODE), $N=2, 3,4,...$, in this chain yields $(N-1)^{th}$ order ODE in the same chain. All the equations in the Abel chain also share the same form of nonlocal symmetry (which is different from the one that exist in Riccati chain) but the similarity reduced $N^{th}$ order ODE, $N=2, 3,4,$, in the Abel chain always ends at the $(N-1)^{th}$ order ODE in the Riccati chain. We describe the method of finding general solution of all the equations that appear in these chains from the nonlocal symmetry.