Note on algebro-geometric solutions to triangular Schlesinger systems

2016/04/06 by Dragovic, Vladimir, Shramchenko, Vasilisa · 1 citation
#14H70 #34M55 #34M56 #Algebraic Geometry (math.AG) #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1604.01820

Abstract

We construct algebro-geometric upper triangular solutions of rank two Schlesinger systems. Using these solutions we derive two families of solutions to the sixth Painlevé equation with parameters (1/8, -1/8, 1/8, 3/8) expressed in simple forms using periods of differentials on elliptic curves. Similarly for every integer n different from 0 and -1 we obtain one family of solutions to the sixth Painlevé equation with parameters ((9n²+12n+4)/(8), -(n²)/(8), (n²)/(8), (4-n²)/(8)).

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