On the perturbed second Painlevé equation *

Abstract

Abstract We consider a perturbed version of the second Painlevé equation ( P II ), which arises in applications, and show that it possesses solutions analogous to the celebrated Hastings–McLeod and tritronquée solutions of P 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 P II and are surprising because the perturbed equation does not possess additional distinctive properties that characterise P II , particularly the Painlevé property.