Explicit construction of first integrals with quasi-monomial terms from the Painlevé series

C. Efthymiopoulos, T. Bountis, T. Manos

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The Painlevé and weak Painlevé conjectures have been used widely to identify new integrable nonlinear dynamical systems. For a system which passes the Painlevé test, the calculation of the integrals relies on a variety of methods which are independent from Painlevé 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 Painlevé - 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 τ = t-t 0 is eliminated. Both right and left Painlevé 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.

Original languageEnglish
Pages (from-to)385-398
Number of pages14
JournalRegular and Chaotic Dynamics
Issue number3
Publication statusPublished - 2004

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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