Integrable systems and modular forms of level 2

Mark J Ablowitz; Sarbarsh Chakravarty; Heekyoung Hahn

paper

Integrable systems and modular forms of level 2

arXiv:math/0609210

Abstract

A set of nonlinear differential equations associated with the Eisenstein series of the congruent subgroup $Γ_{0} (2)$ of the modular group $S L_{2} (Z)$ is constructed. These nonlinear equations are analogues of the well known Ramanujan equations, as well as the Chazy and Darboux-Halphen equations associated with the modular group. The general solutions of these equations can be realized in terms of the Schwarz trianle function $S (0, 0, 1/2; z)$ .

PACS numbers: 02.30.Ik, 02.30.Hq, 02.10.De, 02.30.Gp