Abstract
It is now well known that a deep connection exists between soliton equations and ODEs of Painlevé type. As a consequence there has been a significant reemergence of interest in the study of such ODEs and related issues. In this paper we demonstrate that a novel class of nonlinear ODEs, Darboux-Halphen (DH) type systems, can be obtained as reductions of the self-dual Yang-Mills (SDYM) equations. We show how to find by reduction from SDYM the associated linear pair for DH. This linear system is found to be monodromy evolving, which is different from the linear systems associated with the Painlevé equations, which are isomonodromy. The solution of the DH system can be obtained in terms of Schwarzian equations, which are themselves linearizable. The DH system has solutions that are related to Painlevé equations, but the solutions can have complicated analytic singularities such as natural boundaries and dense branching.
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Ablowitz, M.J., Chakravarty, S., Halburd, R. (1999). On Painlevé and Darboux-Halphen-Type Equations. In: Conte, R. (eds) The Painlevé Property. CRM Series in Mathematical Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1532-5_9
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DOI: https://doi.org/10.1007/978-1-4612-1532-5_9
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