2015/05/05 by Akane Nakamura, Nakamura, Akane · 1 citation
Mathematics · Physics and Astronomy · #14H70 #33E17 #34M55 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1505.00885
openalex publication_date 2015/05/05 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Higher dimensional analogs of the Painlevé equations have been proposed from various aspects. In recent studies, 4-dimensional analogs of the Painlevé equations were classified into 40 types. The aim of the present paper is to geometrically characterize these 40 types of equations. For this purpose, we study the autonomous limit of these equations and degeneration of their spectral curves. We obtain two functionally independent conserved quantities H1 and H2 for each system. We construct fibrations whose fiber at a general point hi is the spectral curve of the system with Hi=hi for i=1, 2. The singular fibers at Hi=∞ are one of the degenerate curves of genus 2 classified by Namikawa and Ueno. Liu's algorithm enables us to give degeneration type of spectral curves for our 40 types of integrable systems. This result is analogous to the following observation; spectral curve fibrations of the autonomous 2-dimensional Painlevé equations P\rm I, P\rm II, P\rm IV, P\rm IIID8, P\rm IIID7, P\rm IIID6, PV and P\rm VI are elliptic surfaces with the singular fiber at H=∞ of Dynkin type E8(1), E7(1), E6(1), D8(1), D7(1), D6(1), D5(1) and D4(1), respectively.