Christos Efthymiopoulos
Soranou Efessiou 4, GR11527, Athens, Greece
Research Center for Astronomy and Applied Mathematics, Academy of Athens, Athens
Publications:
Efthymiopoulos C., Bountis A., Manos T.
Explicit construction of first integrals with quasimonomial terms from the Painlevé series
2004, vol. 9, no. 3, pp. 385398
Abstract
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 "quasipolynomial" functions, from the information provided solely by the Painlevé–Laurent series solutions of a system of ODEs. Restrictions on the number and form of quasimonomial terms appearing in a quasipolynomial 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 quasipolynomial 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 nonexistence of a quasipolynomial first integral. Examples from specific dynamical systems are given.
