Interplay of symmetries and other integrability quantifiers in finite dimensional integrable nonlinear dynamical systems

TL;DR

This paper explores the relationships between various analytical methods and symmetries in finite-dimensional integrable nonlinear dynamical systems, enabling easier identification of integrable cases without solving complex equations.

Contribution

It establishes a comprehensive connection between multiple symmetry-based and integrability quantifiers for nth-order nonlinear ODEs, facilitating their combined use.

Findings

Unified framework linking symmetries and integrability quantifiers

Examples demonstrating the interconnections between methods

Characteristic quantities deduced without solving determining equations

Abstract

In this work, we establish a connection between the extended Prelle-Singer procedure with other widely used analytical methods to identify integrable systems in the case of $n^{th}$-order nonlinear ordinary differential equations (ODEs). By synthesizing these methods we bring out the interlink between Lie point symmetries, contact symmetries, $\lambda$-symmetries, adjoint-symmetries, null forms, Darboux polynomials, integrating factors, Jacobi last multiplier and generalized $\lambda$-symmetries corresponding to the $n^{th}$-order ODEs. We also prove these interlinks with suitable examples. By exploiting these interconnections, the characteristic quantities associated with different methods can be deduced without solving the associated determining equations.