M. G. Tseitlin, “Explicit solutions of $O(3)$ and $O(2,1)$ chiral models and the associated equations of the two-dimensional Toda chain and the Ernst equation when the solutions are parametrized by arbitrary functions”, TMF, 1983, Volume 57, Number 2,Pages 238

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Explicit solutions of $O(3)$ and $O(2,1)$ chiral models and the associated equations of the two-dimensional Toda chain and the Ernst equation when the solutions are parametrized by arbitrary functions

M. G. Tseitlin

Abstract: By means of elliptic solutions of the $O(3)$ and $O(2,1)$ $\sigma$ models parametrized by arbitrary holomorphie functions (generalization of a singular harmonic mapping) and the previously considered [1] correspondence between chiral models and systems with exponential interaction, elliptic solutions are obtained for one of the two-dimensional Toda chains corresponding to the Kac–Moody algebra parametrized by a holomorphie or an antiholomorphic function. Solutions of the sinh-Gordon equation are given. For the Ernst equation, a solution is generated by the meron sector of the $O(2,1)$ $\sigma$ model which is parametrized by two real functions (cylindrical waves) or a holomorphic function (stationary axisymmetric solutions). A solution of Liouville's equation on a torus is given.

Received: 04.04.1983