Equivalence classes and Linearization of the Riccati and Abel chain

TL;DR

This paper investigates the linearization of equations within the Riccati and Abel chains up to fourth order, identifying which are linearizable and characterizing their equivalence transformations.

Contribution

It provides a complete solution for linearization by point transformations for these chains up to fourth order and characterizes their equivalence classes and transformation groups.

Findings

Third and fourth order equations are not linearizable by point transformations.

The Lie pseudo-group of equivalence transformations is explicitly determined.

Some equivalence classes and subgroup structures are identified.

Abstract

The problem of linearization by point transformations is solved for equations in the generalized Riccati and Abel chain of order not exceeding the fourth. It is shown in particular that nonlinear third order and fourth order equations from the chain are not linearizable by any point transformations. The Lie pseudo-group of equivalence transformations for equations of arbitrary orders from the chain are then found, together with expressions for the transformed parameter functions. An important subgroup of the group of equivalence transformations found is considered and some associated equivalence classes are exhibited.