Order preserving contact transformations and dynamical symmetries of scalar and coupled Riccati and Abel chains

TL;DR

This paper introduces new contact transformations that linearize Riccati and Abel chains of nonlinear ODEs, enabling the derivation of dynamical symmetries and extending to higher-dimensional cases.

Contribution

The paper presents novel contact transformations that are not of Cole-Hopf type, explicitly linearize Abel chains, and facilitate symmetry analysis for scalar and coupled Riccati and Abel equations.

Findings

New contact transformations identified for Riccati and Abel chains

Explicit linearization of Abel chain demonstrated for the first time

Method extended to two-dimensional generalizations

Abstract

We identify contact transformations which linearize the given equations in the Riccati and Abel chains of nonlinear scalar and coupled ordinary differential equations to the same order. The identified contact transformations are not of Cole-Hopf type and are \emph {new} to the literature. The linearization of Abel chain of equations is also demonstrated explicitly for the first time. The contact transformations can be utilized to derive dynamical symmetries of the associated nonlinear ODEs. The wider applicability of identifying this type of contact transformations and the method of deriving dynamical symmetries by using them is illustrated through two dimensional generalizations of the Riccati and Abel chains as well.