Asymptotics of Solutions of the Discrete String Equation

TL;DR

This paper studies the asymptotic behavior of solutions to the Discrete Painlevé-1 Equation, classifying solutions and deriving asymptotic formulas for both regular and singular solutions using isomonodromic deformations and Whitham theory.

Contribution

It introduces a classification of solutions of DP1 based on their behavior at infinity and develops asymptotic formulas for both regular and singular solutions using advanced integrable systems methods.

Findings

Asymptotic formulas for regular solutions of DP1

Classification of solutions based on behavior at infinity

Asymptotic analysis of singular solutions using Whitham theory

Abstract

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