Interconnections between various analytic approaches applicable to third-order nonlinear differential equations

TL;DR

This paper explores the connections between different analytical methods used to identify integrable third-order nonlinear differential equations, unifying various techniques through established interrelations and demonstrating their practical application with examples.

Contribution

It establishes a novel interconnection between the extended Prelle-Singer procedure and {\

Findings

Unveiled links between multiple analytical methods for third-order nonlinear ODEs.

Demonstrated how to derive various quantities from a single starting point.

Validated the approach with three illustrative examples.

Abstract

We unearth the interconnection between various analytical methods which are widely used in the current literature to identify integrable nonlinear dynamical systems described by third-order nonlinear ordinary differentiable equations (ODEs). We establish an important interconnection between extended Prelle-Singer procedure and {\lambda}-symmetries approach applicable to third-order ODEs to bring out the various linkages associated with these different techniques. By establishing this interconnection we demonstrate that given any one of the quantities as a starting point in the family consisting of Jacobi last multipliers, Darboux polynomials, Lie point symmetries, adjoint-symmetries, {\lambda}-symmetries, integrating factors and null forms one can derive the rest of the quantities in this family in a straightforward and unambiguous manner. We also illustrate our findings with three specific examples.