Christos Efthymiopoulos

Soranou Efessiou 4, GR-11527, Athens, Greece
Research Center for Astronomy and Applied Mathematics, Academy of Athens, Athens


Efthymiopoulos C., Bountis A., Manos T.
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 $\tau = t – t_0$ 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.
Citation: Efthymiopoulos C., Bountis A., Manos T.,  Explicit construction of first integrals with quasi-monomial terms from the Painlevé series, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 385-398

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