Takasaki’s rational fourth Painlevé-Calogero system and geometric regularisability of algebro-Painlevé equations. (English) Zbl 1532.34091
K. Okamoto [Proc. Japan Acad., Ser. A 56, 264–268 (1980; Zbl 0476.34010)] provided an equivalent first-order non-autonomous Hamiltonian system for each of the Painlevé equations, and constrcuted an extended phase space on which the system is everywhere regular. In this paper, the authors propose a notion of regularisation for the class of systems with globally finite branching about movable singularities in the sense of the algebro-geometric property. They start with a case study, a non-autonomous rational Hamiltonian system obtained by K. Takasaki [J. Math. Phys. 42, No. 3, 1443–1473 (2001; Zbl 1016.34089)]. This system does not possess the Painlevé property, but is mapped by a rational transformation to the fourth Painlevé equation. They start constructing a family of rational surfaces which provide a manifold on which the system is either regular or regularisable; then they provide a global Hamiltonian structure on such manifold and compare it to that of Painlevé IV on Okamoto space’s due to T. Matano et al. [J. Math. Soc. Japan 51, No. 4, 843–866 (1999; Zbl 0941.34076)]. The notion of global regularisability on bundles of rational surfaces by similar algebraic transformations is then introduced and applied to a suite of examples including one by R. Halburd and T. Kecker [“Local and global finite branching of solutions of ordinary differential equations”, in: Reports and studies in forestry and natural sciences. University of Eastern Finland. 57–58 (2014)].
Reviewer: Simonetta Abenda (Bologna)
MSC:
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
Keywords:
Painlevé equations; space of initial conditions; rational surface; algebro-Painlevé propertyReferences:
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