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On non-isospectral flows, Painlevé equations, and symmetries of differential and difference equations

D. Levi^a, O. Ragnisco^b, M. A. Rodriguez<sup>b

a</sup> INFN — National Institute of Nuclear Physics
^b Universidad Complutense, Departamento de Fisica Teorica II

Abstract: We identify the Painlevé Lax pairs with those corresponding to stationary solutions of non-isospectral flows, both for partial differential equations and differential-difference equations. We discuss symmetry reductions of integrable differential-difference equations and show that, in contrast with the continuous case, where Painlevé equations naturally arise, in the discrete case the so-called “discrete Painlevé equations” cannot be obtained in this way. Actually, symmetry reductions of integrable differential-difference equations naturally provide “delay Painlevé equations”.

Received: 18.06.1992

Language: English

 English version: Theoretical and Mathematical Physics, 1992, 93:3, 1409–1414

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