Two Lax systems for the Painlevé II Equation, and Two Related Kernels in Random Matrix Theory

We consider two Lax systems for the homogeneous Painlevé II equation: one of size 2×2 studied by Flaschka and Newell in the early 1980s, and one of size 4×4 introduced by Delvaux, Kuijlaars, and Zhang and Duits and Geudens in the early 2010s. We prove that solutions to the 4×4 system can be derived from those to the 2 × 2 system via an integral transform, and consequently relate the Stokes multipliers for the two systems. As corollaries we are able to express two kernels for determinantal processes as contour integrals involving the Flaschka–Newell Lax system: the tacnode kernel arising in models of nonintersecting paths and a critical kernel arising in a two-matrix model.