On the Perturbed Second Painlev\'{e} Equation

TL;DR

This paper studies a perturbed version of the second Painlevé equation, demonstrating that it admits solutions similar to classical special solutions, with properties like holomorphicity and positivity, despite lacking the Painlevé property.

Contribution

It introduces and analyzes solutions of a perturbed PII equation that resemble classical solutions, revealing unexpected properties despite the absence of the Painlevé property.

Findings

Existence of Hastings-McLeod-type solutions for the perturbed equation.

Existence of tritronquée-type solutions with large sector holomorphicity.

Solutions share key properties with classical PII solutions despite perturbation.

Abstract

We consider a perturbed version of the second Painlev\'{e} equation ($\textrm{P}_{\textrm{II}}$), which arises in applications, and show that it possesses solutions analogous to the celebrated Hastings-McLeod and tritronqu\'ee solutions of $\textrm{P}_{\textrm{II}}$. The Hastings-McLeod-type solution of the perturbed equation is holomorphic, real-valued and positive on the whole real-line, while the tritronqu\'ee-type solution is holomorphic in a large sector of the complex plane. These properties also characterise the corresponding solutions of $\textrm{P}_{\textrm{II}}$ and are surprising because the perturbed equation does not possess additional distinctive properties that characterise $\textrm{P}_{\textrm{II}}$, particularly the Painlev\'e property.