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Integrable systems and modular forms of level 2

Mark J Ablowitz·Sarbarsh Chakravarty·Heekyoung Hahn
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Abstract

A set of nonlinear differential equations associated with the Eisenstein series of the congruent subgroup $Γ_0(2)$ of the modular group $SL_2(\mathbb{Z})$ 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 $S(0,0,1/2; z)$.

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Mark J Ablowitz, Sarbarsh Chakravarty, Heekyoung Hahn. 2006-09-07. Integrable systems and modular forms of level 2. https://arxiv.org/abs/math/0609210

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