Explicit Construction of First Integrals with Quasi-monomial Terms from the Painlev\'{e} Series

TL;DR

This paper presents an explicit algorithm to construct first integrals of dynamical systems using Painlevé series, enabling identification of integrable systems and their quasi-polynomial invariants.

Contribution

The paper introduces a novel method to derive quasi-polynomial first integrals directly from Painlevé Laurent series solutions, expanding tools for analyzing integrability.

Findings

The method successfully constructs first integrals for specific dynamical systems.

It can also demonstrate the non-existence of such integrals in certain cases.

Examples illustrate the effectiveness of the approach.

Abstract

The Painlev\'{e} and weak Painlev\'{e} conjectures have been used widely to identify new integrable nonlinear dynamical systems. For a system which passes the Painlev\'{e} test, the calculation of the integrals relies on a variety of methods which are independent from Painlev\'{e} analysis. The present paper proposes an explicit algorithm to build first integrals of a dynamical system, expressed as `quasi-polynomial' functions, from the information provided solely by the Painlev\'{e} - Laurent series solutions of a system of ODEs. Restrictions on the number and form of quasi-monomial terms appearing in a quasi-polynomial integral are obtained by an application of a theorem by Yoshida (1983). The integrals are obtained by a proper balancing of the coefficients in a quasi-polynomial function selected as initial ansatz for the integral, so that all dependence on powers of the time $\tau=t-t_0$ is eliminated. Both right and left Painlev\'{e} series are useful in the method. Alternatively, the method can be used to show the non-existence of a quasi-polynomial first integral. Examples from specific dynamical systems are given.