A double scaling limit for the d-PII equation with boundary conditions

TL;DR

This paper investigates a double scaling limit for the discrete Painlevé II equation with boundary conditions, showing convergence to a specific tronquée solution of the continuous Painlevé II equation in a critical regime.

Contribution

It demonstrates that in a critical boundary regime, the discrete Painlevé II solution converges to a tronquée solution of the continuous Painlevé II, extending understanding of discrete-to-continuous transitions.

Findings

Discrete solutions converge to a tronquée Painlevé II solution

The convergence occurs in a double scaling limit near the critical boundary

A new approximation method supports the convergence proof

Abstract

We study a double scaling limit for a solution of the discrete Painlev\'e II equation with boundary conditions. The location of the right boundary point is in the critical regime where the discrete Painlev\'e equation turns into the continuous Painlev\'e II equation. Our main results it that, instead of the Hastings-McLeod solution (which would occur when the right boundary point is at infinity), the solution to the discrete equation converges in a double scaling limit to a tronqu\'ee solution of the Painlev\'e II equation that behaves like the Hastings-McLeod solution at minus infinity and has a pole at a prescribed location. Our proof of the double scaling limit is based on finding an approximation that is sufficiently close in order to apply the Kantorovich theorem for Netwons method. To meet the criteria for this theorem, we will establish a lower bound for the solutions to the Painlev\'e II equation that occur (including the Hastings-McLeod solution).