TY - JOUR

T1 - Explicit construction of first integrals with quasi-monomial terms from the Painlevé series

AU - Efthymiopoulos, C.

AU - Bountis, T.

AU - Manos, T.

N1 - Copyright:
Copyright 2009 Elsevier B.V., All rights reserved.

PY - 2004

Y1 - 2004

N2 - 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.

AB - 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.

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U2 - 10.1070/RD2004v009n03ABEH000286

DO - 10.1070/RD2004v009n03ABEH000286

M3 - Article

AN - SCOPUS:63549130690

VL - 9

SP - 385

EP - 398

JO - Regular and Chaotic Dynamics

JF - Regular and Chaotic Dynamics

SN - 1560-3547

IS - 3

ER -