Space of initial conditions for a cubic Hamiltonian system

2017/04/29 by Thomas Kecker, Kecker, Thomas · 1 citation
Mathematics · Physics and Astronomy · #34M35 #34M55 #Classical Analysis and ODEs (math.CA) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #math.CA #msc:34M35 #msc:34M55 #nlin.SI

paper · pdf · doi:10.48550/arxiv.1705.00234

8 pages

arxiv created 2017/12/29 · arxiv updated 2018/01/03

Abstract

In this paper we perform the analysis that leads to the space of initial conditions for the Hamiltonian system q' = p² + zq + α, p' = -q² - zp - β, studied by the author in an earlier article. By compactifying the phase space of the system from ℂ² to \mathbbCP² three base points arise in the standard coordinate charts covering the complex projective space. Each of these is removed by a sequence of three blow-ups, a construction to regularise the system at these points. The resulting space, where the exceptional curves introduced after the first and second blow-up are removed, is the so-called Okamoto's space of initial conditions for this system which, at every point, defines a regular initial value problem in some coordinate chart of the space.

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• Modified Painlevé systems with meromorphic solutions for polynomial Hamiltonians of all degrees

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