arXiv subjects

Mark J Ablowitz

Publications and source records attributed to Mark J Ablowitz.

2 recordsLinked to original sources

arXiv2009

Parameterizations of the Chazy equation

The Chazy equation $y''' = 2yy'' - 3y'^2$ is derived from the automorphic properties of Schwarz triangle functions $S(α, β, γ; z)$. It is shown that solutions $y$ which are analytic in the fundamental domain of these triangle functions, only correspond to certain values of $α, β, γ$. The solutions are then systematically constructed. These analytic solutions provide all known and one new parametrization of the Eisenstein series $P, Q, R$ introduced by Ramanujan in his modular theories of signature 2, 3, 4 and 6.

nlin.SI↗

arXiv2006

Integrable systems and modular forms of level 2

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 $S(0,0,1/2; z)$.

math.NT↗