Abstract
The main subject of the paper is the so-called Discrete Painlevé-1 Equation (DP1). Solutions of DP1 are classified under the criterion of their behavior while the argument tends to infinity. The Isomonodromic Deformations Method (IDM) yields asymptotic formulae for regular solutions of DP1. DP1 is an integrable system, which allows to develop the appropriate Whitham theory. Asymptotics of singular solutions of DP1 are calculated by using the Whitham method.
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