Large-Parameter Asymptotics of Generalized Hasting-McLeod Functions

The generalized Hastings-McLeod solutions to the inhomogeneous Painlev\'{e}-II equation arise in multi-critical unitary random matrix ensembles, the chiral two-matrix model for rectangular matrices, non-intersecting squared Bessel paths, and non-intersecting Brownian motions on the circle. We establish the leading-order asymptotic behavior of the generalized Hastings-McLeod functions as the inhomogeneous parameter approaches infinity using the Deift-Zhou nonlinear steepest-descent method for Riemann-Hilbert problems. This analysis is done in both the pole-free region and pole region. The asymptotic formulae show excellent agreement with numerically computed solutions in both regions.