Two Lax systems for the Painlevé II equation, and two related kernels in random matrix theory
Abstract
We consider two Lax systems for the homogeneous Painlevé II equation: one of size $2\times 2$ studied by Flaschka and Newell in the early 1980's, and one of size $4\times 4$ introduced by DelvauxKuijlaarsZhang and DuitsGeudens in the early 2010's. We prove that solutions to the $4\times 4$ system can be derived from those to the $2\times 2$ 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 FlaschkaNewell Lax system: the tacnode kernel arising in models of nonintersecting paths, and a critical kernel arising in a twomatrix model.
 Publication:

arXiv eprints
 Pub Date:
 January 2016
 arXiv:
 arXiv:1601.01603
 Bibcode:
 2016arXiv160101603L
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 Mathematical Physics;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Probability;
 34M55;
 33E17;
 30E20;
 60B20
 EPrint:
 46 pages, 20 figures