On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation

We demonstrate the way to derive the second Painlevé equation P2 and its Bäcklund transformations from the deformations of the Nonlinear Schrödinger equation (NLS), all the while preserving the strict invariance with respect to the Schlesinger transformations. The proposed algorithm allows for a construction of Jordan algebra-based completely integrable multiple-field generalizations of P2 while also producing the corresponding Bäcklund transformations. We suggest calling such models the JP-systems. For example, a Jordan algebra JMat(N,N) with the Jordan product in the form of a semi-anticommutator is shown to generate an integrable matrix generalization of P2, whereas the VN algebra produces a different JP-system that serves as a generalization of the Sokolov’s form of a vectorial NLS.

On the quasi-Ablowitz–Segur and quasi-Hastings–McLeod solutions of the inhomogeneous Painlevé II equation

We consider the quasi-Ablowitz–Segur and quasi-Hastings–McLeod solutions of the inhomogeneous Painlevé II equation [Formula: see text] These solutions are obtained from the classical Ablowitz–Segur (AS) and Hastings–McLeod (HM) solutions via the Bäcklund transformation, and satisfy the same asymptotic behaviors when [Formula: see text]. For [Formula: see text], we show that the quasi-Ablowitz–Segur (qAS) and quasi-Hastings–McLeod (qHM) solutions possess [Formula: see text] simple poles on the real axis, which rigorously justifies the numerical results in Fornberg and Weideman (A computational exploration of the second Painlevé equation, Found. Comput. Math. 14(5) (2014) 985–1016).

Painlevé transcendents

This article discusses the history and modern theory of Painlevé transcendents, with particular emphasis on the Riemann–Hilbert method. In random matrix theory (RMT), the Painlevé equations describe either the eigenvalue distribution functions in the classical ensembles for finite N or the universal eigenvalue distribution functions in the large N limit. This article examines the latter. It first considers the main features of the Riemann–Hilbert method in the theory of Painlevé equations using the second Painlevé equation as a case study before analysing the two most celebrated universal distribution functions of RMT in terms of the Painlevé transcendents using the theory of integrable Fredholm operators as well as the Riemann–Hilbert technique: the sine kernel and the Airy kernel determinants.