Integrability and solvability of polynomial Li\'{e}nard differential systems

TL;DR

This paper characterizes when polynomial Lie9nard systems are Liouvillian integrable, showing most are not, but identifying specific subfamilies with integrals and exploring non-autonomous integrability aspects.

Contribution

It provides necessary and sufficient conditions for Liouvillian integrability of polynomial Lie9nard systems and identifies new integrable subfamilies.

Findings

Most polynomial Lie9nard systems are not Darboux integrable.

Certain subfamilies with fixed polynomial degrees possess Liouvillian first integrals.

New Liouvillian integrable subfamilies are identified.

Abstract

We provide the necessary and sufficient conditions of Liouvillian integrability for Li\'{e}nard differential systems describing nonlinear oscillators with a polynomial damping and a polynomial restoring force. We prove that Li\'{e}nard differential systems are not Darboux integrable excluding subfamilies with certain restrictions on the degrees of the polynomials arising in the systems. We demonstrate that if the degree of a polynomial responsible for the restoring force is greater than the degree of a polynomial producing the damping, then a generic Li\'{e}nard differential system is not Liouvillian integrable with the exception of linear Li\'{e}nard systems. However, for any fixed degrees of the polynomials describing the damping and the restoring force we present subfamilies possessing Liouvillian first integrals. As a by-product of our results, we find a number of novel Liouvillian integrable subfamilies. In addition, we study the existence of non-autonomous Darboux first integrals and non-autonomous Jacobi last multipliers with a time-dependent exponential factor.