On the Perturbed Second Painlevé Equation

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