Multivariable Painleve'-II equation: connection formulas for asymptotic solutions

TL;DR

This paper demonstrates the integrability of a generalized Painleve-II system, using a Lax pair and WKB methods to connect asymptotic behaviors, with applications to quantum models and phase transitions.

Contribution

It introduces a new integrable multivariable Painleve-II system and relates its asymptotic solutions using exact quantum solutions and asymptotic analysis.

Findings

Established integrability of the generalized Painleve-II system.

Connected asymptotic behaviors at different infinities via a Lax pair and WKB approach.

Applied the analysis to quantum decay and phase transition problems.

Abstract

It is shown that a generalization of the Painlev\'e-II equation (P-II) to a system of coupled equations with symmetry breaking terms is integrable. A Lax pair for this system is used to relate the asymptotic behavior of the solutions at different infinities via an asymptotically exact WKB approach. The analysis relies on an exact solution of the quantum mechanical Demkov-Osherov model (DOM). An application to the problem of unstable vacuum decay during a second order phase transition provides precise scaling of the number of excitations, including subdominant contributions.