2016/01/07 by Karl Liechty, Dong Wang, Liechty, Karl +1
Mathematics · Physics and Astronomy · #30E20 #33E17 #34M55 #60B20 #Classical Analysis and ODEs (math.CA) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.CA #math.MP #math.PR #msc:30E20 #msc:33E17 #msc:34M55 #msc:60B20 #nlin.SI
paper · pdf · doi:10.48550/arxiv.1601.01603
46 pages, 20 figures
arxiv created 2016/08/19 · arxiv updated 2016/08/22
We consider two Lax systems for the homogeneous Painlevé II equation: one of size 2× 2 studied by Flaschka and Newell in the early 1980's, and one of size 4× 4 introduced by Delvaux-Kuijlaars-Zhang and Duits-Geudens in the early 2010's. 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.